This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
0.
1.4.2
We establish sufficient conditions for all solutions of (1.4.1) - (1.4.2) to have the asymptotic behavior
limN(t) = N*,
1.4.3
t-+oo
In the special
cas~,
when either
at
or a2 is zero, one can reduce (1.4.1) to the form
dyd(t) = -8[1 +y(t)]y(t -1); t .
1.4.4
if, both T1 and T2 are nonzero, then a change of time scale to convert (1.4.1) to the form (1.4.4) is not possible and hence (1.4.1) has to be analysed in its own right. A continuous analogue of (1.4.1) is of the form 1.4.5
Our study of (1.4.1) is based on a Lyapunov functional together with an analysis of the nature of solutions of (1.4.1) - (1.4.2). It is easy to see that if we let N(t) == N*[l + y(t)] 1.4.6 in (1.4.1), then y is governed by
dy(t)
*
-d- = -N [1 t
+ y(t)][aty(t - Td + a2y(t - T2)]
°
1.4.7
with 1 + y(t) > for t ;:::: 0. The initial condition of y will be inherited from (1.4.2) through (1.4.6). We recall that a solution of (1.4.7) is said to be oscillatory, if y has an infinite sequence of zeros {t n } ~ 00 as n ~ 00 and y is called nonoscillatory, if there exists a T = T(
°
> for t > T.
§1. 4. Glo bal stability
56
Lelnma 1.4.1. If y is a nonoscillatory solution of (1.4.7), then
lim yet) = O.
1.4.8
t ....... co
Proof. Suppose y is nonoscillatory, and say, is an eventually positive solution of (1.4.7). Since 1 +y(t) > 0 for t 2: 0, d~~t) < 0 for t 2: T = T(
"*
limt . . . . co yet) = P., 2: O. Since y is bounded on [-7,00), it follows that and ~ are bounded for t 2: 27. By lemma 1.2.3, it will follow that d~~t) -+ 0 as t -+ 00 and hence we have from (1.4.7)
Thus f
= O.
If y is an eventually negative solution the proof is similar.
[J
Lelnma 1.4.2. Let b, at, a2, 71,72 be positive real numbers and let y be an arbitrary solution of (1.4.7). Then there exists Tl = Tl (
for
t > Tl
1+ yet) 2: exp [-b(7l + 72)
(e b(Tl+ T2)
1.4.9
-1)]
for
t
> T1 •
1.4.10
Proof. It is sufficient (in view of Lemma 1.4.1) to consider only oscillatory solutions of (1.4.7). Let {tn} -+ 00 as n -+ 00 be a sequence of zeros of an oscillatory solution y. Let y(t~) denote a local maximum of y; we have from (1.4.7) that
1.4.11 Since 1 + y(t~) > 0, (1.4.11) implies at least one of y(t~ - 71) or y(t~ - 72) is nonpositive. As a consequence, there exists cr > 0 such that t~
- (71
+ 72) < cr < t~,
Integrating (1.4.7) over [cr, t~], 1 + y(t~) * log l+y(cr) =-N
Jut~
< N*(a1 S b( 71
(aly(s-7d+ a2Y(S-72)) ds +a2)(t~
+ 72)
-cr) on using
yet) >-1
1.4.12
57
§1.4. Global stability and therefore
1.4.13 Since the right side of (1.4.13) is independent of that
t~,
we can claim from (1.4.13),
where t1 is the first zero of the oscillatory solution y. The derivation of (1.4.10) is similar; for instance, if y(s~) denotes a local minimum of y, we then have
which shows that at least one of y( S;l there exists 1] such that.
-
T1), y( s~ - T2) is nonnegative, and hence
Integrating (1.4.7), 1 + y(s*) log 1 (n) +Y1]
= -N* ls~ [a1Y(s -
T1)
+ a2Y(s -
T2)]ds
11
2 -N*(a1
+ a2) (e b(Tl+T2)
-
1) (T1 + T2)
= -b(Tl + T2) (e b(Tl+T2) -1) implying
1+ y( s~) 2 exp [-b( T1 + T2) (e b(
Tl +T2)
-
1) ]
from which we can conclude (1.4.10) as we did for (1.4.9), and this completes the proof. [] The next result provides a sufficient condition for all solutions of (1.4.1) (1.4.2) to satisfy (1.4.3). Theorem 1.4.3. Assume the following: (i) al,a2,TI,T2, b E ( 0,00 ) ;
(ii) 1.4.14
§1.{ Global stability
58
Then all solutions of (1.4.1) - (1.4.2) satisfy (1.4.3).
Proof. It is sufficient to prove that all solutions of (1.4.7) satisfy
lim yet) = o.
1.4.15
t-oo
We define a functional V = V(y)(t) as follows:
V(y)(t)
=
[yet) - aJN"
l~r' [1 + yes + TJlJy(s)ds - a'N'l~rY + yes + T,)JY(S)dSj'
+ (N")' [a; l r Y + yes + 2TdJ ([[1 + y(u + TdJY'(U)dU) ds + aJa'l~r' {1 + yes + TJ + T,)}
(J.'[l
+ y(u + TdJy'(u)du) ds
+ aJa'l~r, {1 + yes + TJ + T,)}
(J.'[l
+ y(u + T,)JY'(U)dU) ds
+ a~ lr,P + yes + 2T,)} ( [ (1 + y(u + T,)}y'(u)du) dS].
1.4.16
Calculating the rate of change of V along the solutions of (1.4.7),
~V = 2 [yet) t
a}N*
it
t-rl
[ - ai N ' {1
{I
+ yes + T})}y(s)ds - a2N*
it
{I
+ yes + T2)}Y(S)dS]
t-T2
+ yet + TJl} - a,N' {I + yet + T,)}] yet)
+ (al N ·)2[1 + yet + TtlJy'(t) l~r, [1 + y( s + 2TdJ ds - (aJN')'[l
+ yet + TtlJ l~r, [1 + yes + TtlJ y'(s) ds
+ al a2(N')'[1 + yet + T,)Jy'(t) l~r, [1 + yes + TJ + T,)Jds - aJ a,(N')'[l
+ yet + T,)J l~r. {1 + y( s + TJ)}Y'( s) ds
+ aJ a,(N')' [1 + yet + TJ)Jy'(t) l~rY + y( s + 1'J + 1'2)Jds - aJ a,(N')'[l
+ yet + 1'1 )ll~r, {I + y( s + 1',)}y2(S )ds
X
59
§1.4. Global stability
+ (a2N*)2[l + yet + T,)]y'(t) 1~r, [1 + yes + 2T,l]ds - (a,N*)'[l
+ yet + T'l]l~T' [1 + yes + T,l]y2(s)ds.
1.4.17
The right side of (1.4.17) can be estimated using the inequality 2ab S; a2 + b2 so that
dV
dt
+ yes + rd]y2(t)Fl(y)(t) + a2N*[1 + yet + r2)]y2(t)F2(y)(t)
S; alN*[l
where
F1(y)(t) = [ - 2 +
1.4.18
alN'l~T' {1 + y( s + TIl} ds
+ a2N'1~T' {1 + yes + T2)}ds 1.4.19
+ alN·l~,., {1 + yes + 2TIl}ds + a2N'1~T' {1 + yes + Tl + T,)} ds ] and
F,(y)(t) = [ - 2 + a1N*
l~T' {1 + yes + TIl} ds
+ a2N'1~T' {1 + yes + T,)}ds + alN·l~,., {1 + yes + Tl + T,)}ds
1.4.20
+ a2N* 1~r, {1 + yes + 2T,)} ds ]. Using the result of Lemma 1.4.2, one can estimate the right sides of (1.4.19) and (1.4.20) for t ~ Tl where Tl is defined in Lemma 1.4.2;
[-2 + 2N*{alTl + a2T2}eb(Tl+T2)] ::; [-2 + 2b( T1 + T2)e b +T2)] for t ~ Tl
Fl(y)(t)::;
(r 1
=
-j..l
(say)
where
j..l
1.4.21
> O.
Similarly, 1.4.22 for t
~
T1 •
§1.4. Global Btability
60
It follows from (1.4.18) - (1.4.22),
V(y)(t)
+ l1alN*
t {I + yes + Td}y2(s)ds
iT
1
+ l1a2N*
1.4.23
t {I + yes +T2)}y2(s)ds :::; V(y)(T iT
1 ).
1
If we now define
J such that 1.4.24
then (1.4.23) implies limt--+(X) J(t) exists. Also d~~t) = [1 + yet + T2)]y2(t) is uniformly bounded for t 2. T1 by Lemma 1.4.2. Hence, by Lemma 1.2.3,
=0
1.4.25
+ yet + T2)]y2(t) -+ 0 as t -+ 00.
1.4.26
lim dJ(t) t-(X)
dt
implying
dJ(t)
---;J,t =
[1
Since 1 + yet) is bounded away from zero uniformly on [0,(0), the result (1.4.8) follows and this completes the proof. [J We remark that the method proposed above can be used for deriving sufficient conditions for the global attractivityof the positive steady state of the hyperlogistic delay differential equation
where T, aI, a2, T1, T2 E (0,00) and () is an odd positive integer. More model equations of hyper-growth are provided in the exercises. We conjecture that the conclusion of Theorem 1.4.3 holds if the right side of (1.4.14) is replaced by ~. It is found that the result of Theorem 1.4.3 is conditional on the size of the
delay. It is possible to show that if there is a delay independent negative feedback term which dominates other terms, then global asymptotic stability can be "delayindependent". For instance, we have an example in the next result.
§1.4. Global stability
61
Theorem 1.4.4. Assume the following:
(i)
r, a E (0,00) i 71,72 E [0,00) ; bI , b2 E ( -00,00);
(ii) 1.4.27 Then all positive solutions of
du(t) dt u(t) = tp(t)
- - = u(t)[r - au(t)
tp
~
°,t
+ b1 u(t -
71)
E [-7,0] , tp(O)
+ b2 u(t -
12)]
> 0 , 7 = max( 71,72)
1.4.28
E C([-7,0],1R+)
satisfy lim u(t) =
t-+oo
u=
r/[a - (b i
+ b2 )].
1.4.29
Proof. Define a Lyapunov functional V(u)(t) such that
V(u)(t) = [u(t) - U - iilog(u/ii)] + d]
+ d21~T' [u(s) -
l~T' [u(s) -
ii]2ds 1.4.30
ii]2ds
where d 1 , d2 are to be selected below suitably. Calculating the rate of change of V along the solutions of (1.4.28) one can derive dV < _UT AU
dt -
in which
uT
= {u(t) - U , u(t - 7I) - U , u(t - 72) - u}
a - (d 1 A
1.4.31
,
=
[
+ d2 )
-¥ -¥
-
¥ -¥ ] d1
0
0 d2
1.4.32 .
Since the off-diagonal elements of A are nonpositive, the matrix A in (1.4.32) is positive definite, if and only if all the principal minors of A are positive (Plemmons [1977]). This suggests that we have to choose the constants d 1 , d2 such that
a> d 1 + d 2 a > d1 + d2 + (bi/4d1 ) a> d1 + d2 + (bi/4d1 ) + (bV4d 2 ).
§1.4. Glo bal stability
62
These requirements can be satisfied by choosing d 1 = Ib l l/2 and d z = Ib2 1/2, since by hypothesis a > I b1 1 + ,b2 1. For such a choice of d1 and d z , we will have 1.4.33
where A is the smallest positive eigenvalue of A. We obtain from (1.4.33),
V(u)(t) + A
J.'
2
Ju(s) - "J ds S; V(u)(O).
1.4.34
It follows from the definition of V and (1.4.34) that, u is uniformly bounded on [0,00) implying the uniform boundedness of ~~ on [0,00). By Lemma 1.2.2, we can conclude from (1.4.34) that lim [u(t) - u]2 =
t--oo
° []
and this completes the proof.
We remark that a result more general than that 6f Theorem 1.4.4 has been obtained by Lenhart and Travis [1986J. We shall now consider a logistic integrodifferential equation of the form
d~~t) = x(t) [<> x(o) =
Xo
J.'
E (0,00),
f(s)x(t - s)ds] }
1.4.35
a E (0,00)
and obtain sufficient conditions for all solutions of (1.4.35) to converge to an equilibrium. The next result is a special case of one due to Yamada [1982]. Theorem 1.4.5. Assume the following: (i) f is a strongly positive kernel; (ii)
1
00
1
00
f(t)dt = (3;
(0:1{3)
tf(t)dt < 1.
1.4.36
Then every solution of (1.4.35) satisfies lim x(t) = x*
t-+oo
= (al fJ).
1.4.37
63
§1.4. Global stability
Proof. vVe rewrite (1.4.35) in the form
d~~t) =
-x(t)
[I.'
1
f( s) (X(t - s) - x") ds - x" ["" f( s)ds
1.4.38
and consider a Lyaptmov ftmction V = V (x )( t) defined by
V(x)(t)
=
x(t) - x* - x* log{x(t)jx*}.
1.4.39
Calculating the rate of change of V along the solutions of (1.4.38),
~~ =
-[x(t) -
x"l
I.'
f(s )[x(t - s) -
+ x*[x(t) - x*]
s -[x(t) - x"l
I.'
100
x"l ds
f(s)ds 1.4.40
f(s)[x(t - s) - x"lds
+ x*x(t) /.00 f(s)ds. An integration of both sides of (1.4.40) leads to
x(t) - x" - x" log
(x;~») s -
I.'
([rX(u - s) - x"lf(s )dS) du + x" I.' x( u) (J.= f( s )ds) du + V (OJ. [x( u - s) -
x"l
1.4.41
If we let
net)
sup xes),
=
0~s9
then we have from (1.4.41),
n(t)S x" log n(t) + x"
(J.= sf( s)dS) n(t) + YeO) + x*
Since by hypothesis nl > 0, n2 > 0, nl <
(x * Jooo sf (s )ds < 1), n2
1.4.42
- x* logx*.
it follows that n( t) will lie between
determined by
[1- x" J.= sf(s)ds1nj =
x"lognj
+ V(O) + x" -
x" log x"
j = 1,2.
1.4.43
64
§1.4. Global stability
°
The boundeclness of x(t) for t :::: will now follow from nl ~ x(t) ~ nz implying the boundedness of ~~ on [0,00). As a consequence, we obtain the uniform continuity of x on [0,00). By the boundedness of x on [0,00) and the strong positivity hypothesis of I, (1.4.41) leads to
x(t) - x* - x' log
(x~:)) ::: -I'
1.'
[x( u) - x*J'du 1.4.44
+ x*nzl°O sI( s )ds + V(O), and therefore J1
I
t
[X(u) - x*Jzdu
~ x* + x* log(n2/x*) + n21°O sI(s)ds + 1,1(0)
from which we can conclude (as in the proof of Theorem 1.4.4) that
[xC u) - x*]2
--7
°
as u
--7
00.
[]
This completes the proof.
We have not so far discussed the asymptotic behavior of positive solutions of the delay logistic equation with variable delays such as
dN(t) dt
=
r
N(){ _ N(t - r(t»} t 1 ]{
1.4.45
where r ,K E (0,00) and r(.) denotes a bounded continuous nonnegative function defined for t E [0,00). We shall suppose that
supr(t) = ro 2:: t~O
°
1.4.46
and (1.4.45) is supplemented with the initial condition
N(s)
= ¢(s) ,s
E
[-ro ,0],
¢(o) > 0,
¢ E C([-ro,O],R+).
1.4.47
We want to derive a sufficient condition under which every nonconstant positive solution of (1.4.45) - (1.4.47) will satisfy
lim N(t)
t-+oo
= K.
We recall from Lemma 1.4.2 that there exists a constant Tl solution of (1.4.45) - (1.4.46) satisfies the estimates
1.4.48
>
°
such that every
1.4.49 We use the following result whose proof can be found in Halanay [1966].
65
§1.4. Global stability
Lemma 1.4.6. If J : [to, co)
dJd(t) 5 -aJ(t) t and
1--1-
[0,(0) is continuous such that
+ f3[
J(S)]
sup t-To S;sS;t
t 2:: to
for
> f3 > 0, then there exist positive numbers I > 0, k >
if a
J(t) < ke-"(t
1.4.50
.
for
°
such that
t > to.
1.4.51
Theorem 1.4.7. Suppose the delay T in (1.4.45) is bounded and continuous satisfying (1.4.46). If furthermore T( t) is small enough to satisfy 1.4.52
then every positive solution of (1.4.45) satisfies (1.4.48).
Proof. We let
N(t) = I<[1
+ y(t)]
1.4.53
in (1.4.45) and derive that y is governed by
dy(t) -;It
= -rA(y)(t)y(t = -rA(y)(t)y(t)
T(t)
+ rA(y)(t) [yet) - yet - T(t»]
= -rA(y)(t)y(t) + rA(y)(t)T(t) dy~?» = -rA(y)(t)y(t) + rT(t)A(y)(t)rA(y)(~(t»y(~(t) -
T(~(t))
1.4.54
in which
A(y)(t)
= 1 + yet)
and
t - T(t) :::;
~(t)
< t.
For convenience let us rewrite (1.4.54) in the form
d~~t) = -o:(t)y(t) + f3(t)y(~(t) - T(~(t») and note t -
270
< ~(t) -
J1
T(~(t»)
1.4.55
< t. We let [to, (0) = J 1 U J 2 where
= {t 2:: tol yet) 2:: OJ,
Now for t E J 1 ,y(t) = I yet) I and therefore we have from (1.4.55) that
!I
y( t) I :s;
-"0 I y(t) I + f3t-2~~f'$1
y( s) ]
1.4.56
§1.4. Global stability
66 where ao and
130
are defined by ao
13°
= t?:to inf
aCt)
= t?:to inf A(y)(t) = inf r[l + yet)] t?:to = re-rro(errO_1)
= sup f3(t) = r2ro sup [1 t?:to
1.4.57
+ y(t)]2
t?:to
= r2roe2rro
1.4.58
By hypothesis ao > 13° and hence by Lemma 1.4.6 above, there exist numbers Cl > 0 , /1 > 0 such that 1.4.59 Now for t E Jz we have -yet) = can derive that
I yet) I and
repeating the above arguments, one 1.4.60
for some
C2
> 0,
/2
> O. Combining (1.4.59) and (1.4.60), we obtain t > to
1.4.61
where C = max( Cl , C2) and / = mine /1 , /2). The assertion (1.4.48) will follow [] from (1.4.61) and (1.4.53); this completes the proof. We conclude this section with the remark that the result of Theorem 1.2.14 can be used to derive a stronger result than that of Theorem 1.4.7; the interested reader can try to establish such a result by deriving sharper solution bounds.
1.5. Oscillation and nonoscillation We recall from Proposition 1.3.1 that all nontrivial solutions of dx(t) -;u+ ax(t -
r)
= 0,
a,r E (0,00)
are oscillatory, if and only if the associated characteristic equation
67
§1.5. Oscillation and nonoscillation
has no real roots. We shall exploit such a knowledge of the linear equation for studying the oscillatory characteristics of a class of nonlinear equations. In particular, we are now concerned with the derivation of conditions for all positive solutions of
1
dN( t) = N(t) [ -;It an - ~ bjN(t - Tj) ,
1.5.1
}=l
a,bj,TjE(O,oo), j= 1,2, ... ,n to be oscillatory about the positive equilibrium N* that if we let
N(t)
= a/ E J=l bj .
It is easy to see
= N*ex(t)
1.5.2
in (1. 5.1), then x is governed by
dx(t) = -N* t bj d t .}=1
[eX(t-Tj)
-1]
=-
tpjj(x(t - Tj)) .
(say).
1.5.3
}=1
Oscillation or nonoscillation of N about N* is now equivalent to that of x about zero. The next result will lead to the derivation of necessary and sufficient conditions for all positive solutions of (1.5.1) to be oscillatory about N*. The following Theorems 1.5.1 and 1.5.2 are due to Kulenovic et al. [1987aJ.
Theorem 1.5.1. Consider
dx(t)
---;It
n
+ ~ Pj(t)x(t -
Tj)
= 0;
t 2 to
1.5.4
}=1
where
Pj E C([to, 00), R+) , t~ Pj(t)
Tj E [0,00), j If the characteristic equation
= Pj,
= 1,2, ... ,n.
j = 1,2, ... , n
1.5.5
n
A + LPje- ATj = 0
1.5.6
j=l
associated with the limiting equation of (1.5.4) has no real roots, then all the nontrivial solutions of (1.5.4) are oscillatory. Proof. Define F : R 1-+ R as follows: n
F(A) = A + LPje- ATj • j=l
68
§1.5. Oscillation and nonoscillation
Suppose F(A) = a has no real roots. We note that F(A) -4 00 as A -4 00 and since F( A)# a by hypothesis for A E IR, we conclude F( A) > 0 for all A E IR. In particular, F(a) = I:,j=l pj > 0 and therefore Pj > 0 for some) E {I, 2, 3, ... , n}. Also Pjo > 0 for some )0 E {I, 2, ... ,n} and the corresponding Tjo is positive since otherwise A = - I:,j=l Pj will be a real root of (1.5.6); as a consequence, we have F( -00) = 00 and so m = min'xEilR F( A) exists and is positi";'e. (Can there be a sequence An i=- 0, F( An) i=- 0, such that An -4 A* and F( An) -4 O?) Thus, n
A + 2:Pje-'xTj ~ m
for
AE R
j=l
or equivalently n
2:Pje'xTj ~ A + m)
A E R.
1.5.7
j==l
Suppose now for the sake of contradiction that (1.5.4) has a nonoscillatory solution x which we shall assume, is eventually positive. We have immediately from (1.5.4) that dx(t) 1.5.8 -;u + pjo(t)x(t - Tjo) ::; 0 where by choice Pio > 0 , Tjo > O. Define a set A so that
A = {A ~ 0;
dx(t)
-;u + Ax(t) ::; O}.
1.5.9
Clearly 0 E A and A is a subinterval of IR+. The proof is completed by showing that A has the following contradictory properties (for a similar technique we refer to Fukagai and Kusano [1983a,b]). Q 1. A is bounded above; Qz. A E A => (A + m/2) E A) where m is the positive constant satisfying (1.5.7). We have from (1.5.8) that, for all sufficiently large t) 1.5.10 Applying the Lemma from Ladas, Sficas and Stavroulakis [1983b] to (1.5.10), we can derive
x(t - Tjo) < [16/(pj oTjo)]X(t), which together with the decreasing nature of x(t) leads to
X(t-Tj) < kx(t) , j
= 1,2, ... ,n
69
§1.5. Oscillation and nonoscillation for some k > O. But then eventually,
showing that
(k '2:/;=1 Pj + 1) is an upper bound of A. = eAtx(t) where ,\ E A and find
Now to establish Q2, we let 'ljJ(t)
d'ljJ(t) \ ( )] < 0 - = e At [dX(t) - - + AX t dt dt implying, 'ljJ is decreasing. Choose e > 0 such that Pj(t) 2: Pj - e > 0 for each = 1,2, ... , nand t sufficiently large where e L:7=1 e>"Tj < m/2 which is possible since ,\ E A and A is bounded. We have from (1.5.7),
pj > 0, j
d (t)
n
~t + (,\ + m/2)x(t) = - ?=Pj(t)x(t - Tj) + (,\ + m/2)x(t) )=1
= e-" [-
t,
pj( t)e'Tj 1/;( t - T;)
S e-"1/;(t) [-
t,(P; -
e)e'Tj
s e-At'ljJ(t) [- tPje>"Tj + e )=1
S e-At'ljJ(t) [-,\ =
°
====? ,\
m
+ 2"
m
+ (,\ + m/2)1/;( t)]
+ (,\ + m/2)]
t)=1
eATj
+,\ + m/2]
+ ; +,\ + ;]
E A.
It follows that Q1 and Q2 hold; as noted before, this completes the proof.
[]
§ 1. 5. Oscillation and nonoscillation
10
Theorem 1.5.2. Assume the following:
Pi E (0, co) , Ii E [0, co)
(i) (ii) (iii)
j
j = 1,2, ... ,n.
A+ EJ=l pje->"Tj = 0 has no real roots. fEC(IR,R) , uf(u»O for
ui-O.
1. ~ Then every solu tion of
. ) (lV
l'Imu_o
f( u) -
1.5.11 oscillates about zero.
Proof. Let us suppose for the sake of contradiction that, (1.5.11) has a nonoscillatory solution y which we shall assume to be eventually positive (if y is eventually negative the proof is similar). Since uf( u) > 0, we note that f(y( t - Ii» > 0 for j = 1,2,3, ... , n and so d~~t) < 0 eventually. Thus, limt_oo yet) = e 2: 0 exists. We shall first show that e = OJ otherwise e > 0 and fee) > 0 implying lim dy(t) = -
t->oo
dt
(~p.) fee). 6 J
1.5.12
j=l
Since A + E ::;;1 pje->"Tj = 0 has no real roots, (1.5.12) will imply that yet) -7 -00 as t -7 yet) -7 0 as t -7 00.
1
2.:1=1 Pi > 0 which together with 00
and this is impossible. Thus
We rewrite (1.5.11) in the form 1.5.13 where
Pi(t) = pjf(y(t - Ii» > 0 and yet - Ij) -
lim PJ·(t) = Pj. t-+oo
By Theorem 1.5.1, it will then follow that every solution of,(1.5.13) oscillates ~bout zero contradicting y( t) > 0 and hence the result follows. [J As a corollary to Theorem 1.5.2, one can derive that if n
A + N*
L j=l
bje ->"Tj
= 0
1.5.14
§1.5. Oscillation and nonoscillation
71
has no real roots, then all positive solutions of (1.5.1) oscillate about the positive equilibrium; while a number of sufficient conditions for the nonexistence of the real rootsof (1.5.14) can be derived (see exerCises 10-15), it is an open problem to derive sufficient conditions in terms of the parameters for all the roots of (1.5.14) to be nonreal when n 2 2. One of the sufficient conditions for all the roots of (1.5.14) to be nonreal is given in the following: Proposition 1.5.3. Let bj , Tj E (0,00) be such that n
eN*
L bjTj > 1.
1.5.15
j=l
Then (1.5.14) has no real roots. Proof. Suppose the result is not true. Then, (1.5.14) has a real root and su~h a root has to be negative; we let .A = -J-L, J-L > 0 in (1.5.14) and note that n
f-t
= N*
2:=
or
bjel'Tj
j=l n
1 = N*
L
el'Tj
Tjb j - - . j=l f-tTj
~ N* (t bjTi)e }=1
and this contradicts (1.5.15) and hence the result follows.
[J
We remark that it is a consequence of Theorem 1.5.2 that if (1.5.15) holds, then all positive solutions of (1.5.1) oscillate about the positive equilibrium N*. Let us consider a nonautonomous delay logistic equation of the form
duet) dt
= r(t)u(t) [1 _u(t - T(t»] K
1.5.16
where r, T are positive continuous functions defined on [0,00) and K is a positive constant. We assume that together with (1.5.16), we have
u(s) = rp(s) 20 , rp(O) > 0 , rp E C([-T*,O],R+)
T*
= sup T(t). t:;~o
1.5.17
72
§1.5. Oscillation and nonoscillation
If we let
Vet)
u(t)
= -}~ \.
1,
t :2 r*
1.5.18
in (1.5.16), then V is governed by
dV(t) dt
- - = -/(t)[l
+ V(t)]V(t -
ret))
1.5.19
whose initial conditions will be inherited from (1.5.17) via (1.5.18). The oscillation of u about K is equivalent to that of V about zero. We note from (1.5.16) - (1.5.17) that u(t) > 0 for t :2 0, which implies that 1 + Vet) > 0 for t :2 O. Theorem 1.5.4. Assume the following: I, r are continuolls positive functions defined on [0,00); (i) (ii) t - ret) - t 00 as t - t 00; (iii) for some to :2 0 ,
fOO I( s )ds = 00.
1.5.20
lto
Then every solution of (1.5.19) is either oscillatory or converges to zero monotonically as t -+ 00. Proof. Suppose that V is not oscillatory and Vet) > 0 for t :2 T. It follows from (1.5.19) that d~~t) < 0 for t > T* where T* > T such that T* - r(T*) > T and hence 1.5.21 lim Vet) = a 2 0 exists. t---oo
If a > 0 then we have from (1.5.20) and (1.5.21),
dV(t) < _ -a (1
~
+ a )r (t)
t _> T*
f or
1.5.22
leading to
V(oo) - V(t*) :::; -a(l + a)
foo
It·
r(s)ds
1.5.23
where t* = max{to, T*}. But (1.5.23) contradicts (1.5.19). Now suppose Vet) for t 2 T. Then d~;t) > 0 for t > T* and hence lim Vet)
t--oo
= (J 2 O.
<0
1.5.24
If (J < 0, we have dV(t)
~ ~
-[1
+ y(O)]V(t -
r(t))r(t)
2 -[1 + yeO)] (Jr(t) eventually which on integration will again lead to a contradiction as before, and this completes the proof. [J
§1.5. Oscillation and nonoscillaiion Theorem 1.5.5. In addition to the assumptions of Theorem 1.5.4.,
73
if
r(s)ds> lie,
liminfjt
1.5.25
t-r(t)
t-OCJ
then every nontrivial solution of (1.5.19) is oscillatory.
Proof. First we define 8( t) as follows:
8(t)
= sE[O,t] max {s -
T(S)}
1.5.26
and observe (see the proof of Theorem 1.3.4) that (1.5.25) is equivalent to liminf t-+OCJ
t
r(s)ds> lie.
1.5.27
leCt)
Suppose now the assertion of Theorem 1.5.5 is not true. Then, there exists a nonoscillatory solution, say y of (1.5.19) such that
y(t»O, y(t-T(t»>O for t>T*.
1.5.28
A consequence of (1.5.28) is that the linear differential inequality
d~~t) + r(t)y(t -
T(t»
sO
1.5.29
has an eventually positive solution when (1.5.27) holds. But this is not possible due to a result of Koplatadze and Chanturiya [1982].
t
~
Let us suppose that (1.5.19) has an eventually negative solution yet) T. Since 1 + yet) > 0, we have
dy(t)
dt
+ y(t)]y(t - T(t» -r(t)[l + y(t)]y(8(t»
= -r(t)[l ~
and hence
i
t
e(t)
< 0 for
dyes)
~ds S -
J.t
yes)
r(s)[l
e(t)
(8( »
+ y(s)]~ds yes)
implying log [Y(8(t»]
yet)
~
t
lc(t)
r(s)[l
+ y(s)]y(8(s» ds. yes)
1.5.30
74
§1.5. Oscillation and nonoscillaiion
Let w be defined by
wet) = y(8(t» yet) , and note that wet) ~ 1 since d~~t)
log[w(t)]~w(O
t
> T*
> 0 for t ~ T*.
t
1.5.31
From (1.5.30) and (1.5.31)
r(s)[l+y(s)]ds for
J6(t)
~E(8(t),t).
We shall show that w is bounded; by Theorem 1.5.4, yet) -+ 0 as t -+ nonoscillatory. Hence for large enough T*,
1 + y( t) ~ -1 , 2 For any t*
~
it
r( s )ds
~ c
> -1
t
for
~
00
since y is
T*.
e
o(t)
T*, there exists atE [8(t*), t*] such that
i
t
~
r(s)ds
6(t*)
c
t.
-, 2
i
J.t
r(s)[l
C
r(s)ds 2
t
2'
We have from (1.5.19),
yet) - y(8(t*» 2 -
6( t*)
~
~
1 -[-y(8(t)] 2
+ y(s)]y(8(s)ds
it
r(s)ds
6(t*")
C
4[-y(8(t»]
and hence
y(8(t*» ~ ~y(8(t». Similarly, again from (1.5.19),
y(t*) - yet)
~
c
'4 [-y(8(t*»]
implying
yet) Since yet)
< 0,
~ ~y(8(t*» ~ (~)2 y(8(t».
we have from (1.5.32),
wet)
= y( 8(t» ~ (~) 2 yet)
c
1.5.32
§1.5. Oscillation and nonoscillation
75
which implies the boundedness of w. We define
e=
e < 00.
liminf wet), t-+oo
Taking liminft->oo of both sides of (1.5.30),
lo~e ~ liminf .(.
t
r(s)ds
t->oo J8(t)
and this leads to liminf
t
r(s)ds
t-oo J8(t)
~ lie
which contradicts (1.5.27), and hence (1.5.25). This completes the proof.
[]
The following result due to Vescicik [1984] generalizes the result of Theorem 1.5.5.
Theorem 1.5.6. In tile equation
d~~t) + p(t)f(y(t - ret»~ =
0
assume the following:
pEC(R+,IR+)i lim (t - ret»~ = t-+oo
t - ret)
rEC(IR+,IR+)
)
00
is non decreasing in
t E [0,00)
f E C(IR, IR) f is non decreasing on yf(y) > 0 for y f 0 liminf t-oo
it
pes) ds = P > 0
t-r(t)
liminf feu) = F > 0 u-o U 1 PF> e liminf t-oo
it
t-[r(t)f2]
ret - [r(t)/2])
~
pes) ds >
ret).
o}
§1.5. Oscillation and nonoscillation
16 Then all solutions of
d~~t) + p(t)f(y(t -
T(t))
=
°
are oscillatory.
Proof. Details of proof are similar to those of Theorem 1.5.5 and therefore are [] left to the reader to complete.
A sufficient condition for the existence of a nonoscillatory solution of (1.5.16) is formulated in the following result due to Zhang and Gopalsamy [1988]. Theorem 1.5.7. Let r, T be positive continuous functions on [0,(0) such that limsup
t
r(s)ds < lie.
1.5.33
Jt-r(t)
t-oo
Then (1.5.19) has a nonoscillatory solution on [0,(0). Proof. Our proof is based on an application of the well known Schauder-Tychonoff fixed point theorem. Let C[to, 00) denote a locally convex linear space of all continuous real valued functions on [to, (0) endowed with the topology of uniform convergence on compact subsets of [to, (0). Define a set S as follows:
s=
I
yEC[to,OO)
Y
is nondecreasing on
-(1 - e) ::; yet) y(t)=-(l-e)
yet) ::; -(1- e)expfp(t)] on for
y(t)e ::; yet - T(t)) t
where pet) = [-e Jt l r(s)ds)j t1 is sufficiently large such that JLr(t) r(s)ds :S lie for t 2 t1 and e is a fixed positive number such that 1 - c > O. We note that S is a nonempty closed convex subset of C[to, (0). We define a map F: S -7 C[to, (0) as follows:
F(y)(t)
={
-(1 - e) -(1 _ c)exp [_
ft r(.'l)[l+y(s))y(s-r(s))) Jtl
y(s)
dS]
1.5.34
We first verify that F S C Sj it is easy to see that
F(y)(t) 2 -(1 - e) for t 2 to
1.5.35
77
§1.5. Oscillation and nonoscillation
and
tr(s)[l
lt1
+ y(s)]y(s -
l ~ it tl ~e e
res)) ds
yes)
r( s)
[1 - (1 - c)exp (-e l
r(s)ds.
From (1.5.34) and (1.5.35),
-(1 - e)
1
r( U)du) ds
~ F(y)(t) ~ -(1 -
e)exp [-e
l
1 for
r( s )ds
t::: t1.
It is also found that
F(y)(t) F(y)(t-r(t))
= exp [-1t
r(s)[l
~
t
for
(lie)
+ y(s)]y(s -
res)) dS]
yes)
t-T(t)
~ tl
.
It follows from the above, FS C S.
The continuity of F : S ~ S C G[to, 00) will be verified now: let Yn E S , y E S and let Yn -+ Y as n -+ 00. Let t2 be a fixed number such that tl < t2 < 00. We have from the uniform convergence of Yn -+ Y on [tl' t 2 ), that for any Cl > 0 there exists no( cd satisfying
[1
sup SE[tl,t2j 1
+ Yn(S)]Yn(S -
res))
Yn(S)
-
[1 + y(s)]y(s - res)) I < yeS)
for n
C1
> nO(cI)'
From the definition of F, for t E [t1, t 2],
c)1 exp
IF(Yn)(t) - F(y)(t)1 = (1 - exp
1 t
1t
l'tl
( ) Y s
r(s)1 [1
+ Yn(S)]Yn(S -
~ (1 -
c)c1
t
res)) ds
I + y(s)]y(s -
res)) IdS
y( s)
r(s)ds for n > no(cI)
ltl c )c1 1t2 r( s) ds. tl
s
reS)) _ [1
Yn (s)
tl
~ (1 -
+ Yn(S~l~n(s -
Yn s r(s)[l + y(s)]y(s - r(s))d
tl
~ (1- c)
r(s)[1
1.5.36
§1.5. Oscillation and nonoscillation
78
Since C1 is arbitrary, the continuity of F on S follows from (1.5.36) and Yn -- y. It is easy to see that F(y)( t) is unifonnly (in y) bounded for t §: [t1, 00 ) showing the equiboundedness of the family FS. Now by the Arzela-Ascoli theorem, the precompactness of F S follows. All the requirements of the Schauder-Tychonoff fixed point theorem are sat~sfied and hence there exists a y~ E S such that F(y*)(t) = y*(t). One can see from the definition of F that this y* is a nonoscillatory solution of (1.5.19), if we identify to of G[to, (0) with -7*. The proof is complete. []
l-it
I
We remark that it is open to discuss the oscillation and nonoscillation of equations of the type in (1.4.35) and the delay logistic equation dx(t) _ ()[ _ X(At)] dt - rx t 1 ]{ ,
O
1.6. Piecewise constant arguments and impulses In this section we are concerned with a study of the oscillatory and asymptotic properties of solutions of the equation
d~it) = rN(t) (1-
t. -ill), ajN([t
t 2:
0
where [ . ] denotes the greatest integer function, r E (0,00) and [0, (0) with 2: =0 aj > 0.
1.6.1
aD, al,
... , am E
1
The possible complex behavior of the solutions of (1.6.1) can be demonstrated by looking at a simple special case of (1.6.1) namely, 1.6.2 On any interval of the form [n, n obtain
y( t) for n :::; t obtain
<
n
+ 1 and
n
+ 1), n =
0,1,2, ... one can integrate (1.6.2) and
= y( n )exp {r [1 - yi) 1(t = 0,1,2, ....
yIn + 1) = y(n)exp
Taking limits as t __ n
{r [1- Y~)]},
1.6.3
n) }
+ 1, in
n = 0,1,2,....
(1.6.3) we
1.6.4
§1.6. Piecewise constant arguments and impulses
19
The first order difference equation (1.6.4) has been considered in its own right, with no reference to (1.6.2) as a discrete population model of single species with nonoverlapping generations. It is known from the works of May (1975) and May and Oster [1976) that for certain parameter values of r, the asymptotic behavior of the solutions of (1.6.4) is complex and "chaotic". Since (1.6.2) inherits all the behavior of (1.6.4) in a natural way through (1.6.3), the equation (1.6.2) (and also (1.6.1)] provides a simple example of a continuous one-dimensional dynamical system capable of displaying complicated and "chaotic" behavior. We establish below, necessary and sufficient conditions for the oscillation of all positive solutions of (1.6.1) about the positive steady state N* =
1 m
L.:j=o aj
.
We also obtain sufficient conditions for all positive solutions of (1.6.1) to be attracted to the positive steady state N*; these results are due to Gopalsamy, Kulenovic and Ladas [1990b]. It appears that literature about models of differential equations with regular and piecewise constant arguments is scarce and we hope to provoke some interest in the study of model systems, such as (1.6.1), containing both regular and piecewise constant arguments and in the study of "chaotic behavior" in continuous systems. For recent literature on differential equations with piecewise constant arguments and their applications we refer to Cooke and Wiener [1984], Aftabizadeh and Wiener [1985], Gyori and Ladas [1989] and the references cited therein.
We shall first obtain necessary and sufficient conditions for the oscillation of all positive solutions of (1.6.1) about the positive steady state N*. As usual~ we say that a solution N(t) of (1.6.1) oscillates about N*, if the function N(t) - N* has arbitrarily large zeros. By a solution of (1.6.1), we mean a function N which is defined on the set
{-m,-m
+ 1, ... ,-1,0} U(O,oo)
and which possesses the following properties: N is continuous on (0,00). (ii) The derivative d~;t) exists at each point t E [0,(0) with the possible exception of the points t E {a, 1,2, ... } where one-sided derivatives exist.
(i)
§1. 6. Piecewise constant arg1.tments and impulses
80
(iii) Equation (1.6.1) is satisfied on each interval [n,n
+ 1) with n = 0,1,2, ....
We assume that (1.6.1) is supplemented with initial conditions of the form
N(O)
= No> 0 and N(-j) = N_ j
~
0, j
= 1,2, ... ,m.
1.6.5
The following lemma implies that, (1.6.1) with (1.6.5) has a unique positive solution. Lemma 1.6.1. Let No> 0 and N_j ~ 0 forj = 1,2, ... ,m be given. Then (1.6.1) and (1.6.5) have a unique positive solution N(t) given by
and n
= 0,1,2, ...
where the sequence {N n } satisfies the difference equation
Proof. For every n = 0,1,2, ... and for n S; t < n dN(t) --;It
[m
+ 1, (1.6.1)
becomes
],
= ;N(t) 1- ~ ajNn _ j
1.6.8
)=1
where we use the notation
Nn=N(n)
for
nE{-m, ... ,-l,O,l, ... }.
By int~grating (1.6.8) from n to t we obtain (1.6.6) and by continuity, as t ~ n+1, (1.6.6) implies (1.6.7). Conversely, let {Nn } be the solution of the difference equation (1.6.7) defined on {-m, ... , -1, O} U (0, 00) by (1.6.5) and (1.6.6). Then one can show by direct substitution into (1.6.1) that N satisfies (1.6.1) and (1.6.5). It is also clear that No > implies that N(t) > 0 for t > O. The proof is complete.
°
We note that No > 0 implies N(t) > 0 for t > 0 and for any N_j E IR, j = 1,2, ... , m. However, we assume in (1.6.5) that N _j ~ 0 only for "biological reasons" .
§1.6. Piecewise constant arguments and impulses
81
Let N(t) be the positive solution of (1.6.1) and (1.6.5); set
N(t)
= N*exp[x(t)],
t
~
O.
Then xCi) satisfies the equation
dXd(i)
t
+
t ,
rN*ajf(x((t - j]))
= 0,
t
~
m
1.6.9
J=O
where 1.6.10 together with the initial conditions
xU)
= log [~~)
1 for
= 0,1,2, ... , m.
j
Clearly, N(t) oscillates about N*, if and only if x(t) oscillates about zero. Observe that the function f defined by (1.6.10) satisfies the following properties:
uf(u»O
fEC(IR,IR],
lim
for
feu) = 1
1.6.11 1.6.12
u
u->O
uto,
and feu)
:s; u
for
u:S;
o.
1.6.13
We recall the following result which is extracted from Gyori and Ladas [1989]. Lemma 1.6.2. Consider tbe equation
d:~t) +
t
qjf(x([t - jD)
= 0,
1.6.14
j=O
wbere qo, ... , qm ~ 0, 2:7=0 qj > 0, m+qo =f 1 and tbe function f satisfies (1.6.12) and (1.6.13). Tben every solution of (1.6.14) oscillates about zero, if and only if tbe equation m
). -1
+L
qj).-kj
=0
1.6.15
j=O
bas no roots in (0,1). Applying Lemma 1.6.2 to (1.6.9), we obtain the following necessary and sufficient condition for the oscillation of all positive solutions of (1.6.1) about its positive equilibrium N* .
82
§1.6. Piecewise constant arguments and impulses
= 1,2, ... , m be given and assume that i E (O,oo),ao,al, ... ,am E [0,(0) with 2::i=o aj > 0 and r + m =/I. Then the unique positive solution of (1.6.1) and (1.6.5) oscillates about its positive equilibrium N* if and only if the equation
Theorem 1.6.3. Let No > 0 and N _j 2: 0 for j
1.6.16
has no roots in (0,1). Applying Theorem 1.6.1 to two simple special cases of (1.6.1), one obtains the following corollaries. Corollary 1.6.4. Let Yo > 0 be given and assume that rand I{ are positive constants. Then the unique solution of (1.6.2) with y(O) = Yo oscillates about K if and only if i > 1. Corollary 1.6.5. Let No > 0 and N _j 2:: 0 for j = 1,2, ... , £ be given and assume that i and K are positive constants. Then the unique solution of
with N(O)
= No
and
N(-j) = N_ j
for
j = 1,2, ... ,m
oscillates about K if and only if
Let us proceed to obtain sufficient conditions for all positive solutions of (1.6.1) to converge to the positive steady state N* as t -+ 00. Our result is precisely formulated as follows: Theorem 1.6.6. Assume the following: (i) r E (0,00), ao, al, ... , am E [0, (0),
2::]:'1 aj > 0,
r
+m
=/1;
(ii) er (m+l) < 2.
1.6.17
§1.6. Piecewi.'3e constant arg'uments and impulses
83
Then all solutions of (1.6.1) corresponding to initial conditions of the type (1.6.5) satisfy lim N(t) = N*
1.6.18
t-oo
where
Proof. As we have seen earlier, the change of variables
= N*exp[x(t)],
t 20
rN*aj[ex([t-iD - 1]
= 0,
N(t) reduces (1.6.1) to
d~~t) +
f
t 2 m.
1.6.19
j=O
It suffices to show that (1.6.17) implies
lim x(t) = O.
1.6.20
t-oo
First, we assume that x(t) is eventually nonnegative. From (1.6.19) we see that
dx(t)
-dt- -< 0 for n _< t < n + 1
1.6.21
where n is sufficiently large, say n 2 no. It follows that x(t) is nonincreasing for n 2 no and so f == lim x(t) t-oo
exists and f
2 O. Assume, for the sake of contradiction, that f > O. Then m
a = ~rN*aj(el -1)
= reel -1) > 0
j=O
and (1.6.19) yields
dx(t)
-dt- + a < 0, n _< t < n + 1
for
We derive
x(t) - x(n) :::; -aCt - n)
n >_ no.
84 and as t ---" n
§1.6. Piecewise constant arguments and impu18es
+1 x(n
+ 1) - x(n)
S; -a,
n 2: no.
1.6.22
As n ---" 00, (1.6.22) implies that 0 = e-e S; -a < 0 which is impossible and so (1.6.20) holds for nonnegative solutions. In a similar way it follows that (1.6.20) is true for nonpositive solutions. Finally, assume that x(i) is neither eventually nonnegative nor eventually nonpositive. Hence, there exists a sequence of points {en} such that m
< 6 < e2 < ... < en < en+l < ... , lim en
n-oo
= 00,
x( en) = 0 for n = 1,2, ... , and in each interval (en, en+d the function x(i) assumes both positive and negative values. Let in and Sn be points in (en,en+l) such that for n = 1,2, ...
x(i n ) = max[x(i)] for
en < i < en+l
x( Sn) = min[x( t)]
en < t <
and for
~n+l'
Then for n = 1,2, ... 1.6.23
while 1.6.24
where D-x is the left derivative of x. Furthermore, if in
ti. N, 1.6.25
and if tn E N, m
OS; D-x(t n ) = - 'LrN*aj [eX(tn-i-l) -1].
1.6.26
j=O
Similarly, if Sn rf:. N,
0= dX~;n) = D-x(sn) = - t j=O
rN*ai [eX(sn-i- l )
-1]
1.6.27
85
§1.6. Piecewi3e con3tant argument3 and impulse3
and if
8n
E N, then
o ~ D-X(8 n )
m
= -
L
rN*aj [ex(Sn-j-l)
-1] .
1.6.28
j=O
Next, we claim that for each n = 1,2, ...
1.6.29 and
1.6.30 If for instance (1.6.29) were false, then (1.6.25) and the hypothesis that 2: =0 aj > o together would lead to a contradiction; (1.6.30) will also be true due to a similar
1
reason. By integrating (1.6.19) from Tn to tn and using the fact that tn - Tn :::; m we note,
o=x(t n ) -
x(Tn)
+
t
r N'aj [ " [e,([,-m - lJds > x(t n ) Tn
j=O
~
X(t n )
-
f
+1
rN' aj(t n - Tn)
j=O
rem + 1).
That is,
x(t n ) <
rem + 1),
n
= 1,2, ...
and so
x(t)
Sn
t~6.
and using the fact that
Sn -
Sn
m
~ X(8 n )
+L
rN*aj[er(m+l) -
l](m + 1)
j=O
+ rem + l)[e r (m+l) - 1] since by hypothesis < x(sn) + rem + 1)
= x(sn)
e r (m+l)
< 2.
~ m
+1
§1.6. Piecewise constant arguments and impulses
86 That is,
X(Sn) > -rem + 1), n = 1,2, ... and so
x(i»-r(m+1),
i2:6.
So far, we have established that
-M < xCi) < M,
i 2: 6
1.6.31
where
11/1 = rem
+ 1).
By using (1.6.31) and an argument similar to that given above we find
-M(-e- M
+ 1) < xCi) < M(e M
-1),
i 2: ~1'
One can show, by induction, that
1.6.32 where
Lo = Ro = M and for n
= 0,1,2, ... 1.6.33
along with
1.6.34 Set
L = lim Ln n->oo
and
R = lim Rn. n-oo
In view of (1.6.32), the proof of lim xCi)
t--.oo
=0
will be complete if we show that
L = R = O.
1.6.35
§1.6. Piecewi8e con8iani argumeni8 and imp'ulu8
87
To this end, from (1.6.33) and (1.6.34) we have -L = lvI(e- L
-
1),
-M
~
-L
R = M(e R ~
0 ::; R
~
-
1)
and 1.6.36
M.
Hence, - L and R are zeros of the function
in the interval - M
~
A~
jVf.
'P( -00)
We have
= 'P( (0) = 00,
'P(O)
= 0;
also
'P is decreasing in ( -00, -log M) and 'P is increasing in (-log lvI, (0). Note also that in view of the hypothesis (1.6.17), M E (0,1) and 'P(M)=M(e M -1)-M<M(2-1)-M=0. Therefore, 'P(A) has exactly one zero in (-00, MJ, namely A = O. Thus, -L and R which are zeros of 'P(A) in [-M,M] are both zero. This proves (1.6.35) and completes the proof of the theorem. [] We suggest that the reader can try to improve the above result; for example, it can be proved that if + l)e r (m+l) < 1 instead of (1.6.17), then also the conclusion of the above Theorem 1.6.6 holds; a condition such as this is comparable with rre rT < 1 for the logistic equation (1.2) with a single constant delay r.
rem
We remark that although the delay-logistic equation (1.6.1) with piecewise constant arguments has the potential for complex behavior, it follows from the above result, that if the delays are small as in (1.6.17), then complex behavior or even persistent (undamped or periodic) oscillations are not possible due to the global attractivity of the positive steady state of (1.6.1). Differential equation models such as (1.6.1) are of interest since they contain intrinsically other discretetype models mentioned in the beginning of this section. Though the subject matter of this monograph does not include "difference equations" we shall briefly consider the logistic difference equation (1.6.4) due to its relation with (1.6.2). We refer to Fisher and Goh [1984] for a discussion of difference equation models in population dynamics. It is expected that a detailed
88
§1. 6. Piecewise constant argument.3 and impulses
exposition of the oscillatory and asymptotic behavior of difference equations including the possible chaotiG behavior will appear in a subsequent work. We remark that the theory of nonlinear difference equations and the related method of Lyapunov functions have been developed by LaSalle [1977a, b]. Following the ideas of LaSalle, we' proceed to discuss (1.6.4). First we simplify (1.6.4) by rescaling and obtain
u(n + 1)
= u(n)er[l-u(n)},
n = 0,1, ....
1.6.37
The linear asymptotic stability of the equilibrium u( n) = 1 of (1.6.37) can be studied by means of the associated variational system which in this case is
v(n
+ 1) = (1 -
r)v(n).
1.6.38
All solutions of (1.6.38) are of the form v(n) = v(O)(1 - rt
n
= 0,1, ...
and as a consequence, a necessary and sufficient condition for the linear asymptotic stability of u( n) = 1 is that 0 < 11 - rl < 1 or equivalently that 0 < r < 2. We can, however, derive the following somewhat stronger result by the method of Lyapunov functions (see LaSalle [1977b] and Fisher and Goh [1984]) .
Theorem 1.6.7. If 0 < r < 2, tben all positive solutions of (1.6.37) satisfy lim u(n)
n-+oo
= 1.
1.6.39
Proof. We consider the Lyapunov function V = V(u(n)) calculate V' where
V' = V(u(n
+ 1)) -
[u(n) - 1]2 and
V(u(n)) = u(n) (er[l-U(n)) -l)H(U(n))
and
H(u(n))
= u(n){ er[l-u(n)] +
I} -2.
1.6.40
1.6.41
It is easy to see from (1.6.40) - (1.6.41) that if u( n) ~ 2 then V' < 0 and if u(n) = 1, V' = O. Thus, if we can show that V'(u(n)) ::; 0 for all u E (0,2), then
89
§1.6. Piecewise constant arg'uments and impulses
(1.6.39) will follow from LaSalle's invariance principle since V' = 0 ¢=} u( n) = 1. Let us first consider the case u(n) E (0,1); since e r [l-u(n)]_l > 0, we shall estimate
H(u(n)) for this case first. 'tie let R
= 1 ~ u log
[~ -
1] ,
1.6.42
u E (0,1)
and note that
1) ,
1+ R= ylog ( --i 1--y
1 y=-->O
1-u
= 2Y{~y + ~ +~ = ... } 3y 5y >2>
1.6.43
I.
From (1.6.42) and (1.6.43) , _I_log 1- u
(~-1) > u
1
=}
uer{l-u)
+u -
2
<
0
and therefore V I < 0 for u E (0, 1). A similar estimation of R for u E (1, 2) will show that ue r (1-u) + u - 2 > 0 which with
er(l-u) -
1 < 0 will again imply that V' < 0 for u E (1,2). Thus
V'(u(n)) S; 0
for
all u E (0,00)
and hence by LaSalle's invariance principle lim u(n) E M,
n ...... oo
where M is the largest invariant set contained in
E = {xIV'(u) The result of (1.6.39) follows since V' (x) completes the proof.
= a}.
= 0 has the only solution x ::::: 1 and this []
We remark that the stability characteristics of the following types of equations have not been studied in detail (see Gopalsamy et al. [1989a]):
dx(t)
-;It + ax(t) + bx(t - r) + ex([t - nJ) = 0
§1.6. Piecewise constant argumeni3 and impulJes
90
d~~t) = N(t) {a -
f31N(t)
+ fhN(t -
r)
+ f33N([t -
n])} .
We shall consider these types of equations in Chapter 3 briefly. The simultaneous appearance of both the regular delay r and a piecewise constant argument (t - n] in the above makes it impossible to associate any kind of "characteristic equation" which has b~en instrumental in the development of a major portion of the theories of oscillations, perturbations and stability. We proceed to present a brief discussion of "equations with impulses". Stability of certain ordinary differential equations with impulses has been considered by Barbashin [1970], Pandit and Deo [1982], Gurgula (1982]' Borisenko (1983], Perestyuk and Chernikova [1984] and Bainov and Siemeonov [1989]. We shall examine the following aspect of delay differential equations with impulses; "if the trivial solution of a delay differential system is asymptotically stable, in the absence of impulsive perturbations, under what conditions such perturbations can maintain asymptotic stability." Another related question of nonoscillation of delay differential equations with impulses is not considered here; certain problems in this regard are posed in the exercises. We consider the following delay differential equation with impulses
dx(t)
-;It + ax(t - r)
= bjx(tj- )8(t - tj),
t
# tj
1.6.44
where a,bj,r (j = 1,2, ... ) are real numbers such that r
2 0,
0 < t1 < t2 < ... < t j
-jo
00
as J
-jo
00.
lt is known that when all bj (j = 1,2,3, ... ) are zero, the trivial solution of (1.6.43) is exponentially asymptotically stable whenever < ar < 7r /2. In fact the characteristic equation associated with (1.6.44) when bj = O,j = 1,2,3 ... is of the form 1.6.45
°
and that for 0 < ar <
7r /2,
all the roots of (1.6.45) have negative real parts. Let
sup{?Re(,X)1 A + ae- Ar
= O} =
-ao
1.6.46
where ao is a positive number. The solutions of (1.6.44) are piecewise continuous functions which are left continuous at {td, i = 1,2,3, ... and satisfy (1.6.44). The following result due to Gopalsamy and Zhang [1989] provides a set of sufficient conditions for the asymptotic stability of the trivial solution of (1.6.44).
91
§1.6. Piecewise constant arguments and impulses
Theorem 1.6.8. Assume the following: 0 < a1' < 1f/2. tj+l - tj ~ T >0, j = 1,2, ... : and l' < T .. 1 + Ibjl S: M for j = 1,2,3, ... (iv) (t) log(M) < Q for some Q < Qo· Then the trivial solution of (1.6.44) is exponentlally asymptotically stable.
(i) (ii) (iii)
Proof. It is known from Corduneanu and Luca [1975] that the solution of (1.6.44) corresponding to an initial condition of the form
t < 0;
x(t) = ¢;(t),
x(O+)
=
xo
1.6.47
where ¢; E C([-1', 0], IR) is given by
x(l) = U(I)xo
+ y(l, 1» +
l
U(I - 5)h( 5) ds
1.6.48
in which U is defined by
dUet) d t + aU(t -
1') = 0 t > 0
U(t)
=0
t E [-1',0);
for
U(O+)
=1
1.6.49
and
Y(I,1» = -a JOT U(I -
T -
5)1>(5) d5;
t >0
t > O.
1.6.50 1.6.51
In the following analysis of (1.6.48), we can without loss of generality assume that ¢;(t) E C([-1', 0), R), yet, ¢;) -+ 0 as t -+ 00 by the exponential asymptotic stability of the trivial solution of (1.6.44) in the absence of impulses (due to the condition o < aT < 1f/2). We have from (1.6.48) and (1.6.51)
x(t)
= U(t)x(O+)
x(t)
= U(t -
on
tl)b1x(tl-)
[0, t 1 ) on
1.6.52
(it, t z ).
1.6.53
It is not difficult to see from (1.6.44) and (ii) that
j = 1,2, ...
1.6.54
§1.6. Piecewii3e con8iani arg'umenis and impu[i3es
92
From (1.6.52) - (1.6.54),
1.6.55 Similarly, one derives that
1.6.56 and hence by induction n
x(t) = U(t)
IT(1 + bj)x(O+)
on
t E (tn,tn+1)'
1.6.57
j=l
As a consequence, n
j=l
::; J{ e- ot Mn(t)
Ix(O+)1
1
log(M) ::; K e- ot [ e-r-t Ix(O+)1
::; Klx(O+ )Ie-{o-~}t
1.6.58
where n(i) denotes the number of jumps in the interval (0, i). Now if on [-7,0), then one can easily see from (1.6.48) and (1.6.49) that y(t,
=f: ~
0 0 []
We have already derived sufficient conditions for the oscillation of all solutions of delay differential equations of the type
dx(t)
d1 + p(t)x(t - 7) =
1.6.59
O.
I
Let us now consider an impulsive perturbation of (1.6.59) as follows;
d:(t)
+ p(i)x(t -
7)
t X(ti+) _ X(ti-)
o < t1 < t2 ... ti where
T
is a positive number.
~ 00
= 0, =
t
=1=
ti
biX(ti-) as
~ ~ 00
1.6.60
§1.6. Piecewise constant arguments and impulses Theorem 1.6.9. Assume the following: (i) ti+l - ti 2: T; i = 1,2, ... and 7 < T. (ii) 0 ~ bi ~ M, i = 1,2, .. . (iii) p is continuous on [0,(0) and p( t) ~ 0 for t
(iv) lim inf t-+oo
t
Jt-T
93
1.6.61 ~
O.
p( s ) ds > 1 + M .
1.6.62
e
Then every solution of (1.6.60) is oscillatory. Proof. Suppose the result is not true; then there exists an eventually positive solution say y( t) > 0 for t > t*. Define
yet - 7) yet)
=
wet) Considering the interval [t -
7,
t ~ t*
for
t] and ti E (t -
yet - 7) 2: yeti)
T,
+7
1.6.63
t),
1
1
= 1 + bi y(ti+) 2: 1 + bi yet)
1.6.64
implying
wet) = yet - 7) > _1_ > _1_ yet) -1+bi- 1 + M '
1.6.65
We shall first show that w{t) is bounded above. Let tk be a jump point in [t - 7). Integrating (1.6.60) on [t - I' t],
27, t
7 y(t)-y(t-2')+
it
p(S)Y(S-7) =0
tI 2
from which we have
yet -
~) ~ 2
it I it
p(s)y(s - 7) ds
t-j tk
2: ~
+
T
-
o
t-t
p( s )y( S - T) ds
M
t-~
On integrating (1.6.61) over [t -
it
p( s )y( s - 7) ds .
tk+ T+ O
yet - 7) 1+
+
7,
pes) ds.
t-
~],
37) yet - T) 2: yet - 2
I
t -
t-T
t
p(s)ds.
1.6.66
94
§1. 6. Piecewise constant argumentj and impuls es
Thus, 3 ) y( t - 2:.) 2:: y( t - ~ 2
2
and hence
[1
t
y( t -
1[jt
11 +
1 p( S ) ds - M
T
t-2"
¥) < 1') - [Ji.:}
yet -
p( s ) ds
-.;.
t-r
1+M
p( s ) ds
1[JL~ p( s ) ds1:5, N.
1.6.67
1.6.68
We have from (1.6.61) for large enough t,
i
yl(S)
t
t-T
But
j
t
t-r
+
-( ) ds Y s
yl(S) --ds= yes)
jt t-r
p( s)
y(s-r) () Y s
o.
y'(s) y'(s) --ds+ --ds t-T yes) t,,+O yes) = 10 y(tk - 0) yet) g yet - r) y(tk + 0) ,,-0
yet) 1 gy(t-r)l+b k
= 10
From (1.6.69) and (1.6.70), log [
1.6.69
it
1 t
ds =
y( t - r) yet) (1
1
+ bk ) =
it
t-r
pes)
670 1..
y( S - r) yes) ds.
1.6.71
If f
= liminf wet), t-oo
1.6.72
then f. is finite and positive; also (1.6.71) leads to
log[(l
+ M)w(t)] ?: el~T p( s) ds
which implies that
(1
+ M) > _ e
flt
log[(l f.+ M)f.] 2: l'1m In . t-+oo
and this contradicts (1.6.62). Thus the result follows.
p
()d S s
1. 673 .
t-r
[]
95
§1.6. Piecewise constant arguments and impulses
We remark that there exists almost no literature on delay differential equations with impulses although nondelay equations with impulses have been considered recently (see the monograph by Ladde et al. [1987J). We have formulated a number of exercises on delay differential equations with impulses as well as their applications. The results of Theorems 1.6.8 and 1.6.9 are due to Gopalsamy and Zhang [1989]; it is an interesting, nontrivial and worthwhile exercise to remove the assumption T < T (the delay is smaller than the length of the inter-impulse time intervals) from the hypotheses of the above results. The reader is now required to generate and develop nonoscillation results for delay equations with impulses. We refer to Gopalsamy and Zhang [1989] for a discussion of the asymptotic behavior of the following delay logistic equation with impulses,
dx(t) --;It
= rx(t) [ 1x - ( t J{
T)] + ~ ~ bj [x(tj-) -Ii.'"] b(t -
tj).
1. 7. Feedback control We have seen that all positive solutions of
dn(t) dt
= rn ()t [1 _ (a1n(t) + Ka2n(t - T»)]
1.7.1
satisfy lim net) = n*,
if
al
> a2 2:
°
and
t-+oo T
E [0,00) where n* is the positive equilibrium of (1.7.1)
satisfying (
a1
+ a2 )
I{
n
*
= 1.
1.7.2
We suppose that it is desired to reduce the equilibrium level of (1.7.1) and maintain the population size at a reduced level by means of a feedback regulator (or feedback control) .. We can model such a regulated (or controlled) system by
where a, b, c E (0,00) and u denotes an "indirect" feedback control. It is not difficult to see that solutions of (1.7.3) corresponding to initial conditions of the
§1.1. Feedback control
96 form
N(s)
=
u(O) = uo > 0
E C([-T,O]'R+)}
1.7.4
.
satisfy
N(t) > 0, u(t) > 0
t> O.
for
1.7.5
We note that (1.7.3) has an equilibrium (N*, u*) where N* > 0, u* > 0 and u*
(a
1 = (~)N* ' K + a2 + bC)N* = 1. a a
It follows from a,b,c E (0,00) and (1.7.6) that N*
The purpose of this section is to show that if solutions of (1.7.3) satisfy lim [N(t) , u(t)]
t-oo
=
1.7.6
< n*. al
>
a2
2: 0 then all positive
1.7.7
[N*, u*].
We shall first show that positive solutions of (1.7.3) are bounded for all t 2: O. Suppose lim sup N(t) =
OOj
t-co
let {t m
}
be a sequence such that as
m
-+
00
and
dNI dt
2: OJ
1.7.8
tm
then
o:s
d:L
< rN(tm)[l- a,Nitm)] <0
1.7.9
for m is large enough; but (1.7.9) is impossible and thus we have limsup N(t) < 00. t-co
It is possible that one can find explicit bounds for N(t) also; we shall not do this here. The second of (1.7.3) can be written as
:t (u(t)e
a
,)
= bN(t)e a '
97
§1.7. Feedback control
and therefore
u(i)e"' = u(O) + b ~ u(O)
1.'
N(s)e"' ds
+ bN(e at -
l)/a
where
N = sup N(s). 82:0
We derive from the above that
u(t) ~ u(O)e- at
b + -N(l a
e- a
t
1.7.10
).
For convenience in the proof of our next result we introduce new variables x, y, ()" as follows:
x(t) = u(t) - u* N*
) N(t)
yet) = --;:- [log ( IF ) CT r (}"(t) = -x(t) - -N yet). a
-
cr
--;x(t)j
1.7.11
*
One can verify by direct calculation, that
dx(t)
--;it" d(}"(t)
-;It dy(t) dt
= -ax(t) + b¢((}"(t))
= -CTX(t) -
1.7.12
aIr a2T K ¢((}"(t)) - T¢((}"(t - 7))
1.7.13
= (a~ + bC)N*¢((}"(t)) + (a;)N*¢((}"(t _ T)) Ii
a
1.7.14
I~
where
1.7.15
Theorem 1.7.1. Suppose T, K, aI, a2, a, b, C E (0,00) and positive solutions of (1.7.3) satisfy
al
> a2
~
O. Then all
1. 7.16
§1.7. Feedback control
98
Proof. It is sufficient to show that
lim (x(t),
t ...... co
yet») =
1..7.17
(0,0)
and this is what we shall do. We consider a Lyapunov functional V ·defined by
= V (x, 0")(')
1.7.18
where B,j3 and 6 are positive numbers to be selected below suitably. Calculating the rate of change of V along the solutions of (1.7.12) - (1.7.15), dV
dt
+ x(t)¢>(O"(t» [2Bb - j3er]
= -2Bax 2 (t)
+ 12(17(/)) [0 - ,8~r]
+ ¢>2(0"(t -
r)[-6]
+ ¢>(O"(t)¢>(O"(t ~ -2Bax 2
j3a2r] r» [ - K
+ x(t)¢>( O"(t» [2Bb - j3er]
+ 12(17(1)) [0 - ,8~r] + 12(17(/ -
r))[-o]
r 1 ¢>2( (» j3 a2 j3 a2r 1 ¢>2( O"t-r. ( » O"t +--+J( 2 ]{ 2
1. 7.19
Choose 1. 7.20
and let T7 be defined by 2
T7
f3r
= ]{ ( a 1
-
a2).
1.7.21
One can simplify (1.7.19) so that
dV at
~
- ( Bax 2(t) + Bax 2 (t)-2x(t)
+ 1]2
f3er r;:;Bb- = vBa1] 2
1.7.23
99
§1.7. Feedback control
where f3 is any positive number; we note that it is sufficient to choose B to be the positive root of the quadratic equation 1.7.24 For such a choice of B, we have from (1.7.22), 1. 7.25 leading to
V(x,
+
l
(Bax'(s)
+ [VEax(s) -
ry.p(
s: V(x,
1.7.26
It follows from (1.7.26) that x 2 E L 1 (0,00). To derive the uniform continuity of x on [0,00), it is sufficient to establish the uniform boundedness of x on [0,00); this will be accomplished if we can show that O"(t) is bounded above for all t 2: O. Suppose lim SUPt_oo O"(t) = 00; then there exists a sequence {t m } ~ 00 as m ~ 00 such that
dO' dt (trn) 2: 0;
O"(tm - r) :::; O'(tm),
O'(t m ) ~
00
as
m ~
00.
1.7.27
We have from (1.7.13) and (1.7.27) that 1. 7.28 A consequence of (1.7.26) is that V is nonincreasing and therefore
Bx 2 (t) :::; V(x,O")(t) :::; V(x,O')(O) from which we can conclude the uniform boundedness of x( t) for all t 2: O. Thus x(t m) is uniformly bounded; since O"(t m) ~ 00 as m ~ 00, <jJ(O'(tm» ~ 00 as m ~ 00; hence, we have from (1. 7.28), if m
is large enough
and this contradicts (1. 7.27). We can conclude that lim sup O"(t) t-oo
< 00.
§1. 7. Feedback control
100
From the boundedness of x and a on [0,(0) we can assert that x is bounded and therefore x is uniformly continuous on [0,00). (Note that if a is negative and unbounded below, our argument holds since a enters (1.7.12) through ¢(a)). The uniform continuity of x on [0,00) and x 2 E L 1 (0, (0) together lead to lim x(t) = 0.
1.7.29
t-ex:>
From the uniform boundedness of x and a on [0,(0), follows the uniform boundedness of .J]3ax-Tj¢(a) and its derivative on (0, (0). Thus -v'J!3a,x-Tj¢(a) is uniformly continuous on [0,(0). Since (1.7.26) implies
it will follow by Lemma 1.2.2 that lim [~x(t) - Tj¢(a(t))] = 0.
1.7.30
t--ex:>
But x(t) -+ 0 as t -+ 00; thus ¢(a(t)) -+ 0 as t -+ 00; and therefore aCt) -+ 0 as -+ 00 implying yet) -+ 0 as t -+ 00. This completes the proof. []
t
Recently Gopalsamy and Weng [1992] have considered the convergence characteristics of the system
dn(t) dt = 1'n(t) [n(t-T) 1J{
1
cu(t)
-
1.7.31
duet)
dt = -au(t) + bn(t -
T)
in which r,K,c,a,b,T are positive numbers. The system (1.7.31) has a positive equilibrium (n*, u*) where *
n
aK
= a + Kbe'
bK
u* = a + Kbc'
1.7.32
One can show (for details we refer to Gopalsamy and Weng [1992]) that if }
rT) e <-1 \. (be -+a n* 2' T
rT
1.7.33
then all positive solutions of (1.7.31) satisfy lim
t-ex:>
{n (t ), u(t )} = {n * , u* } .
1.7.34
§1.7. Feedback control
101
The author believes that the conclusion (1.7.34) can hold, if the condition (1.7.33) is relaxed to the form be rT i -+e <1 }
_r (
a-
rT) n*
which reduces to the condition rre rT < 1 when e = 0 in (1.7.31). The problem of delay induced oscillation _to periodicity in (1.7.31) and its generalization to integrodifferential equations are left as exercises to the reader. The following system is a model of feedback regulation of logistic growth by impulsive regulators and we suggest the interested reader should try to investigate the asymptotic behavior of the system: (if the unregulated system is not asymptotically stable, examine whether it is possible to stabilize the system by means of appropriate feedbacks)
[X(t K-
dx(t) --;It
= rx(t) 1 -
d~~t)
= -ay(t)
r)
1
- ey(t)5(t - tn)
+ bx(t) as
n
-+ 00.
For a number of model systems with feedback controls related to population dynamics, the reader is referred to Exercise 54.
102 EXERCISES I
I be continuous on (0,00) with a bounded derivative on [0,00). Let C(O) == C( x) > for x =f O. Let G be continuous on ( -00,00). Prove that
1. Let
0,
°
roo G[f(t)] dt
Jo 2. Let a j, Tj
< 00
I(t)
==::;. lim t-oo
= 0.
(j = 0, 1, 2... n) be real constants such that n
ao < 0; laol >
Laj;
Tj2::0,
j=1,2, .... ,n.
j=l
Prove that the trivial solution of the linear difference differential equation
is asymptotically stable. 3. Let a, b be real constants and let K : [0,00) uous such that
1---+ (
-00, 00) be piecewise contin-
/.00 IK(s)lds = 1; /.00 sIK(s)lds < 00. If a < 0, lal >
Ibl
then prove or disprove that any solution of roo
dx
dt = ax(t) + b Jo
K( s )x(t - s )ds
corresponding to bounded piecewise continuous
xes) = (s),
on (-00, OJ with
s E (-00,0]
has the property limt_oo x(t) = 0. 4. Let bj, Tj
(j
= 0,1,2, ... ,n) be real constants such that n
bo <0,
IboTol < 1;
Tj >0
;j =0,1,2,· .. ,n
lbol > I::lbjl. j=1
Exerci.~e.'J
I
103
Prove or disprove that the trivial solution of
dx(t) -dt
= box(t -
+ L bjx(t - Tj) n
TO)
j=l
is asymptotically stable. 5. Let a, b, T be real constants b f. 0, T >
°
such that
-lie < bTe-aT" < e. Prove that the equation A = a + be-)..1' has a root (say) J-l in the interval (a - 11r, (0). If x denotes a solution of
dx(t)
-;It = ax(t) + bx(t - r); x(s)
= (s),
SE[-T,O);
t>
°
EC[-r,O]
then prove that lim [x(t)e-J.lt] =
t-oo
1-
1 b _
re J.lT"
[(O)+be-J.l1'jO e-J.lS(s)dS] -1'
the limit being approached exponentially. If rj and r are real constants such that Tj ::; r, j = 1,2, ... ,n then generalize the above result assuming
°: ;
n
r
L bjexp( -a + 1lr)rj < 1 j=l
for a system of the form (see Driver et al. [1973])
6. Prove the asymptotic stability of the trivial solution of
dx(t) -a:t = -ax(t if the positive constants a and r satisfy ar
if aT < 7r/2
or
aT
< 3/2?
r),
< 1; can you prove the same result
Exercises I
104
7. If aj, Tj (j = 1,2"", n) are positive constants such that then prove that the trivial solution of
".2:;=1 ajTj <
1,
is asymptotically stable; can you prove the same result, if 3
n
'"""' a -T- < 7r/2 or ~ 112
?
J=l
8. Let J{ : [0, (0) such that
1-+
[0,(0) be piecewise continuous and a be a positive constant
1
1
00
00
K (s ) ds
= 1;
a
s K ( s ) ds < 1.
Prove that the trivial solution of
=
dxd(t) t
_aft
K(t - s) xes) ds
-00
is asymptotically stable; can you prove the same result, if
rOO
aJo
3
sJ{(s)ds«7r/2)or'2
9. If a, T are positive constants satisfying (aeT)
dx(t) ---;{t
+ ax(t -
~
T)
?
1, show that the linear system
=0
is nonoscillatory about zero; prove that if aeT > 1 then the a.bove system is oscillatory about zero. Can you prove similar results if x(i - T) is replaced by xCi) = sup {x( s )Is E [t - T, t]}? 10. If aj, Tj
(j = 1,2, ... ,n) are positive constants such that n
Te 6 "'" a-1< - 1,, j=l
show that the linear system
dx(t)
n
- dt + 6 ~) a-x(i j=l
T-) )
=0
105
Exercises I
is nonoscillatory about zero; if e L::j=l aj Ij > 1, then prove that the linear system is oscillatory about zero. Prove also that when e
n }l/n (n~/j ) > 1 {!1 aj
the above system is oscillatory about zero (Ladas and Stavroulakil' [1982]). 11. Let a be a positive constant and K : [0,00) such that
rOO K(s) ds =
.}O
f-+
[0,00) be piecewise continuous
1,00 sI«s)ds = a < 00;
1;
Then the linear integrodifferential system
d~(t) t
+ajt
K(t-s)x(s)ds=O
-00
oo
has at least one solution without a zero on R; if however ea Jo s K(s) ds > 1 then the above system cannot have solutions of the same sign on (-00,00). Prove or disprove these assertions.
12. Let aj, Ij ,j = 1,2, ... , n be positive constants. Prove that a necessary and sufficient condition for the system
to be oscillatory is that n
-..\ + L
aj e Arj
>0
for all
..\
> O.
j=l
Also deduce that a sufficient condition for the above system to be oscillatory is that n
e
2:= ai Ii > 1. j=l
(j = 1,2 ... n) be positive numbers. Prove that each one of the following (A), (B), (C) is a necessary and sufficient condition for the oscillation of
13 Let aj,'i
Exercises I
106
= >.. + 'L.j=l aje->"Tj = 0
(A)
F(>..)
(B)
).. + ~j=l
aje->"Tj
>0
has no real roots.
for all ).. E R.
n
1-
I::
ajTje->"Tj
= O.
j=1
14. If aj, Tj (j = 1,2 ... , n) are positive constants, then prove or disprove that each one of the following (i) - (v) is a sufficient condition for all solutions of
dx(t) -;It
+
?: ajx(t n
Tj)
= 0
J=1
to be oscillatory: (i)
ajTj > l/e for some j E {I, 2, ... , n};
(ii)
( 2::;'=1 aj)T > lIe where T = min{Tl' T2,·.· Tn};
(iii)
2:j=1 ajTj > 1/ e;
15. Prove or disprove that a necessary and sufficient condition for all solutions of
dx(t)
n
-dt- + '" a-x(t - T-) = ~ J J
0
j=1
to be oscillatory is the following: there exist numbers Ni
i = 1,2, ... ,n,
> 0, n
II i=1
( aiTie )
N-
Ni/Ti
~?=l Ni
=1
such that
1
> .
a
16. If v'et) ::; get, vet - T), vet»~, where v : IR ~ IR+ and get, x, y) : 111 3 ~ III are continuous and 9 increases with respect to the last two variables and z(t) is a solution of Zl(t) = g(t,z(t - T),Z(t»)
z(s) =
s E [-T, 0],
Exercises I for t 2:: 0, where
107
= z(s), for s E [-7,0], then prove that z(t) 2:: vet) for
17. Can you derive a result similar to the one in the previous problem with the inequalities reversed? 18. Discuss the asymptotic (as t -+ (0) oscillatory and convergence· characteristics of the positive solutions of the following generalized food limited model systems: dN(t) dt
(i)
= r N(t) (
K -N(t-r) ) 8 l+rcN(t-r)
(r,I<,c,7 E (0,00):
f)
= 1,3,5, ....
dN(t) - rN(t) [K-N(t)-E~_ aiN(t-rJ ) ] __ 1-1
( ii)
dt
(r,
-
l+r
I<, aj,"7j E (0,00),
f)
8
I:;i=1 aj N(t-rj)
= 1,3,5, .... ) 8
(iii)
dN(t) (It
= r(t)N(t)
K(t)-N(t-mw) [ K(t)+c(t)r(t)N(t-mw) ]
.
Assume that m is a positive integer, f) = 1,3,5, ... and K, r, c are continuous positive periodic functions of period w (Gopalsamy et al. [1988]). 19. Investigate the asymptotic behavior of positive solutions of the following respiratory model systems (for details see Gopalsamy et al. [1989b]):
(1)
dx(t)
dt
(n,a,fj"
(2)
dx(t) dt
= {_ rx (t) + a/3x(t)x"(t-r)}8 "Y+xn(t-d E (0,00),7 E (O,oo),f)
= 1,3,5, ... )
= [_ rx(t) + x(t) { ~~ /3i x(t-rj} }] L.JJ=laj+x(t-rj)
(J
(r,fjj,aj,7j E (O,oo),m E N,f) = 1,3,5, ... )
(3)
d~~t)
+ rx(t) =
x(t) J~oo K(t - s)"Y~:~s{s) ds;
K(t) = te-t.
20. Examine the asymptotic behavior of the positive solutions of the following models of haematopoiesis: dP(t)
+ r P( t ) -_
dP(t)
a/3p (t-r) + r P( t ) = /3+[p(t-r)]n.
(1)
([t
(2)
([t
a/3 /3+(P(t-r)]n' m
Exercises I
108
(3)
dP(t) dt
+ T'P(t") --
ex It-00
(4)
dP(t) dt
+ r pet) --
ex
It
-00
(Assume ex, /3, n, E (0,00),
(5)
d~~t)
(6)
dP(t) _ dt -
K(t-s} !3+P(s}
d
s.
K(t-s)pn(s) f3+pn(s)
K(t)
~ !3+pn(t)
s
= te- t ,
= -(/3 + b)P(t) + 2/3P(t -8P(t) _
d
t 2: 0).
T)e--YT.
+ !3+pn(t-T) 201P(t-T) --yT e .
21. Prove or disprove, that a necessary and sufficient condition for all solutions of the equation
d:~t) + a
l'
K(t - s)x(s) ds = 0
to have at least one zero on (-00,00) is that for all where a is a positive constant and K : [0,00) on [0,00) such that
100
~
/.00
sK(s) ds < 00;
[0,00) is piecewise continuous
K(s) ds = 1.
Also deduce that a sufficient condition for all solutions of the above system to have at least one zero on (-00,00) is that
1
00
ea
K(s)s ds > 1.
22. Obtain sufficient conditions for all positive solutions of the integrodifferential equation
dP(t)
--;It + rP(t) = ex
/.00 K(s) (3 + pet/3 _ s) ds r
0
to converge to a positive steady state as t 23.
--7
00.
that if yeO) > -1 and the zeros of yare bounded, then y(x) 00 where y is a solution of
~rove
x
--7
dy(x)
--a:;- + exy(x -
1)[1
+ y(x)] =
0
--7
0 as
109
Exercises 1
corresponding to continuous initial conditions on [-1,0] where Q' is a positive constant. If yeO) > -1 for -1 < x < 0, then prove that the zeros of y are unbounded. If y > 0 for -1 < x < 0, then show that the distance between the successive zeros of y (if any) is greater than unity; derive also a similar result, if yeO) > -1 and y < for -1 < x < O.
°
24. Consider the two delay logistic equation
dy(t)
dt"
= -[1
+ y(t)][ay(t - 1) + by(t - r»)
together with bounded integrable initial conditions on [-r,O] (assume r > 1) where a, b, r are positive constants and prove the following (see Braddock and Van den Driessche [1983)): (i) a unique solution y is defined for all t > 0; (ii) if a + b f:. 0, then the only possible constant limits of and -1;
y
as t
---t 00
are
°
(iii) yeO) 2: -1 =? yet) 2: -1 and yeO) ~. -1 =? yet) ~ -1 for t > 0; (iv) if yeO)
> -1 and if the zeros of yare bounded, then yet)
---t
°
as t
---t
00;
(v) if yeO) > 1 and if the zeros of yare unbounded, then -1 < yet) <
(vi) if yeO)
e(a+b)r -
1;
> 1 and a + b > 1, then y is oscillatory on [0,(0);
(vii) if (a + b)r ~ 1, then prove that yet) (viii) is it true that if (a + b)r
---t
°
as t
< 7r/2, then yet)
---t
---t
°
00;
as t
---t
oo?
25. Discuss the various possible types of behavior such as convergence, instability, oscillation and nonoscillation of solutions of the following:
(i) (ii)
nE[l,oo);
d~~t)
= r
[1 -
(N(;;r)
rJ
N(t)
(r, K, r, n are positive constants).
110
Exercises I d~~t)
(iii)
(a, (iv)
= -aN(t) + e-N(t-r);
are positive constants ).
T
d~;t) = [a + bN(t - T2)]N(t - T1) ( a, b, T1, T2 are positive constants).
(v)
d~;t) = [a - bN(t - T)]N(t - T) - J1.N(t) (a, b, T, J1. are positive constants).
(vi)
d~~t) = N(t)[a -c- bN(t) J(t - s)N(s)ds] where a, b are positive constants and J : [0,(0) 1--+ [0,00) is piecewise oo oo continuous such that Jo J(s)ds = 1; Jo sJ(s)ds < 00.
J;
-1JooH(s)N(t-s) dS] . 00
dN(t) _ - d- -
( vii)
r N ( t)
t
[ K
1+rc
0
H(s)N(t-s) ds
26. IT r, T, K are positive constants and satisfy disprove that all solutions of
°<
rT
dx(t) _ ()[ _ X(t-T)]. dt - rx t 1 K ' X(S)=¢(S);:::O,SE[-T,O];¢(O»O;
¢
< 7r /2, then· prove or
t>o
is
continuous
[-T,O]
on
satisfy the following (global attractivity of the positive steady state)
lim x(t)
t-+oo
= K.
27. Discuss the global attractivity of the positive steady st,ate of the generalized delay logistic equation d (t)
:t
+ a[x(t) -
x*]
= x(t)[r -
L bjx(t 00
Tj)]8
j=1
assuming that () is an odd positive integer and the constants a, x*, r, bj, Tj are nonnegative such that
a ;::: 0,
x* > 0,
r > 0,
bjTj > 0,
j
= 1,2,3, ... and n
00
0< "" ~ b· } < 00', j=1
infTj J
=
T*
S;
T*
=
SUpTj j
< 00,
r
= x*Lbj j=1
.
111
Exercises]
28. Derive sufficient conditions for the global attractivity of the trivial solution of the following;
(i)
d~\t) = where
(ii)
il,
((1- a)x(t - r.) + ax(t - T,)) [1+ x(t)] °<
r2, a are positive constants such that
dd~t) = -0: [r=-; where
0:,
0'.
< 1.
K( 8)x(t + B) dB] (1 + x(t)]
r are positive constants and
K E C([-l, -,], (0,00».
J::O
(iii) d~~t) = -(X ]{(B + i)X(t + B)dB where (X, i are positive constants and K is nonnegative, continuous and bounded on (-00, OJ. (iv)
d~;t)
= - [ J.:-~ K(B+ r)x(t +8)d8][1 + x(l)
where
0:, i,
1
and ]{ are as in (iii) above.
29. Assume that g : [0,00) ~ [0,00), g(O) = 0, g is increasing on [0,00) and continuous. Discuss the asymptotic stability of the positive steady state of the scalar system
where r, bj, Tj are positive constants such that 2:j:l b j < 00 and 0< infj Tj sup j Tj < 00. Can you generalize your discussion to equations of the form
d~;t)
= g( X(t)l(r -
f'
::;
h( x(t - s)) dK(S)]
for suitably defined functions K and h. Discuss also the oscillatory and nonoscillatory nature of solutions of (*) and (**) about their nonzero steady states. 30. Assume the following:
(i) (ii) (iii) (iv)
°
aCt) 2:: 0, t 2:: Jooo cos(wt)a(t)dt > g E C( -00,00) jEL1(0,00).
°
(-00 < w < 00)
Exercises I
112
If x is a locally absolutely continuous bounded solution of
dx(t) dt
t
+
aCt - r)g(x(r»dT
Jo
then prove that g(x(t»
-7
°as t
= J(t);
t > 0, Staffans [1975]
-7 00.
31. Assume the following:
(-l)ja(j)(t) ~ O;j = 0,1,2, aCt) 1= constant, xg(x) > for x t- 0; Jo±oo g(x) dx = 00. J E L1 (0,00) and either Id~~t) I ::; M for all t or IJ(t)1 ::; M for all t.
(i)
(ii) (iii)
°
a
E L1 (0,00) and
Then prove or disprove that every positive solution of
d~~t) + [1+ x(t)] tends to zero as t
-7 00
l'
art - r)g(x( r» dr = I(t)
(for details see MacCamy and Wong [1972]).
32. Assume a, b, c, d, aj, Tj; j = 1,2, ... , n are positive constants and discuss the asymptotic stability (local and global) of the positive steady states of the following:
dx(t)
[
bX(t)]
(i)
-;It = x(t) a - [c + x(t - T)] .
d~~t) = dx(t)
d:t
d~~t) =
x(t)
[a - bx(t) { t,[aj/x(t [
cx(t)
rj )]} -']. ]
= x(t) a - bx(t) - [d + x(t _ T)] .
x(t)
[a - bx(t) {
l=
[k(t - s )/x( s)] ds } -']
( ii)
(iii)
(iv)
(k being a suitable delay kernel). 33. Discuss the oscillatory and nonoscillatory nature of the systems in exercise 32 above. 34. Investigate the asymptotic behavior (convergence to a positive steady state) as t -7 00, oscillations and nonoscillations of the following systems (a, b, c, T, T1, T2 are positive constants):
113
Exercises I
rn - cx(t)x(t -
(i)
d~~t) = ax(t - r)exp{-bx(t -
(ii)
d~~t) =ax(t- rl)exp{-'"-x(t- r 2)}-X 2 (t).
(iii)
r).
d~~t) = a J~oo k(t - s )e-bx(s) ds - cx(t). (k being a suitable nonnegative delay kernel).
35. Examine the stability and asymptotic behavior of the system
dN(t)
----;It
+]1 +N(t1 n
= -,N(t)
)
(
1
rj)
.
In the exercises 36 - 39 below, assume a is a positive constant and r is a nonnegative constant.
36. Let u be a continuous real valued function on [-r, (0) such that.
duet) --;[.t 2: au( t - r) on If u(t) 2:
°
on [-r, 0] then prove that u(t) 2:
[0,(0).
°
on [0,(0).
37. Let u, v be continuous real valued functions on [-r, (0). Suppose u, v also satisfy
duet)
-dt- > au(t - r) dv(t)
on
[0,(0)
--;It = av(t - r). If vet) ~ u(t) on [-r,O], prove that vet) ~ u(t) on [0,(0). 38. Let u, v be continuous real valued functions on [-r, (0). Suppose that
duet) dt dv(t) --=av(t-r) dt
- - < au(t - r) on
[0,00).
If vet) 2: u(t) on [-r,O], then prove that vet) 2:: u(t) on [0,(0). Develop integrodifferential analogues of the results in 36, 37, 38 when the delay terms are replaced by terms like
1,00 J«s)u(t -
s)ds.
Exercises I
114 39. If v is a continuous solution of
dv( t) > av(t _ T) t >0 dt , v(t»O on [-T,O], then prove that v( t) of
>
°
on [0,00). Furthermore if u is a particular solution
du(t) dt
- - = au(t - T)
given by u(t)
= eallt
where J-l is a real root of
then prove that lim u( t ) . exists and t-cc
v(t)
.
u(t)
hm -() t
t---cc V
< 00.
In the exercises 40-43 below, assume ao, al , ... , an are continuous on [to, 00 ) and ai(t) 2:: 0, i == 1,2, ... , n. Let Tl, T2, ... , Tn be positive constants. Define
40. If
liminf t---cc
j
t+r. [
t
L ai(s)1ds > -1e n
i=l
then prove that the following scalar system is oscillatory:
d (t) n _x_ dt == """' L-t aJ'x(t
+ TO). J ,
t > to.
j=l
41. If
li~~p
j
t+T. [
t
n
1ds < ~,1
~ ai(s)
then show that the system
dx(t)
n
- dt = ""'" L-t a Jox(t + TO). J , j=l
t > to
Exercises I
115
has at least one nonoscillatory solution. 42. If liminf t-+oo
I
t
+
T
* [
t
J..+ T' ao(u)du1ds>
n ~ a ·(s)e. L..t )
J
j=1
1 _, e
then show that the system d (t)
~d t
+L . n
= ao(t)x(t)
aj(t)x(t + 7j);
t > to
J=1
is oscillatory. 43. If lim sup t-+oo
l
l
+
T
* [
L aj(s)e.J.6+T' n
J
1ds < -,1
ao(u)du
e
j=1
t
then show that the following system has at least one nonoscillatory solutioni d (t)
~t = ao(t)x(t) +
?= aj(t)x(t + 7j)i )=1 n
t > to.
44. Derive sufficient conditions for the oscillation and nonoscillation of
(i)
7
E (O,oo),a E (0,00).
(ii)
d~~t) = -[ax(t) + bx(t - 7)j3i
a E [O,OO)i
(iii)
d~\t) =x(t)[a-bX(t-T)r
a,b,r E (0,00)
(iv)
d~\t)
= x(i) [a -
2::1=1 bjx(t -
r
Tj)
b E R,7 2:: O.
TI, T2,"
. , Tn E [0,00);
«() is an odd positive integer; a > 0, 2:7=1 bj > 0). (v)
d~~t) = x(t)[ a - bx(t) ]U j x(t)
= sUPSE[t-T,t] x( s) , () =
1,3,5, ..
45. Can you prove that each solution x(t) in the following is such that limt_oo x( t) = constant ? (i) d~~t) = -[x(t)P/3
+ [x(t _ 7)]1/3;
r is a positive constant.
Exercises I
116
(ii) d~~t) = -f(x(t)) + f(x(t - r»); f is defined on ( -00,00), continuous such that r is a positive constant.
f
is an odd function and
(iii) d~~t) = -ax(t) + aJ~oo K(t - s)x(s)ds; a is a positive constant and K : [0,00) 1--7 [0,00), K is piecewise cont'inuoo oo ous on [0,00) such that Jo K(s)ds'= l;Jo sK(s)ds < 00. (iv) d~(/) = - f(x(i» + J~oo K(t - s )f(x( s» ds; f and K are as in (iii) and (iv) above respectively. 46. Obtain sufficient conditions for the asymptotic stability of the trivial solution of the impulsive system
[00
dx(t)
-;]t = ax(t) + bx(i - r) + c Jo
k(s)x(t - s) ds
x(tj+) - x(tj-) = bjx(tj) Assume a, b, bj , c E (-00,00),
100
;
j
k: [0,(0)
r E (0,00)
1
= 1,2, .... 1--7
[0,00);
00
k(s) ds < 00,
sk(s)ds < 00.
Derive also conditions for all solutions of the above impulsive system to have at least one zero on IR. 47. Discuss the oscillatory and asymptotic behavior of the delay-logistic equation subjected to impulsive perturbations;
where
°<
il
< t2 < ... tj
-+
as
00
J -+ 00.
48. Discuss the asymptotic stability of the positive equilibrium of the following food limited model subjected to impulsive perturbations;
dN ( t)
---;It
[ K - N (i - r) = r N (i) 1 + erN (t _ r)
Assume r,K,r,c,b j E (0,00);0
1+ b [N (t
< t l ,t2 , ••• tj
j
-+
j - ) -
K] <5 (t - t j )
00 asj -+ 00.
.
Exercises I
117
49. If a nonnegative piecewise continuous function u satisfies the condition
u(l) ::;
C
+ L p, u(I, ti
0)
+ [v( s) um(s) ds,
>to
m>O,
to
where c ~ 0, f3i ~ 0, v( s) ~ 0, and ti are discontinuity points of the second kind of the function u( t), then prove that
50. Develop sufficient conditions for the asymptotic stability of the trivial solution of
dy(t)
d1 + a(t)y(t -
r)
+ b(t)y([t -
m])
=
°
where [m] denotes the greatest integer contained in mER. Discuss by formulating your own hypotheses, the stability of positive steady state of the nonlinear system
dN(t)
--;It
= rN(t)[a - bN(t - r) - cN([t -
N(tj+) - N(tj-)
m])];
t =/= tj
= pjN(tj-);j = 1,2, ..
Try first without impulses (i.e. with Pj =
°
,j = 1,2, ... ).
51. Discuss the asymptotic behavior of the solutions of each of the following: formulate your own hypotheses.
(1)
(2)
dy(t)
d1
= -ay([t - m]),
d~~t) = -ay(t) ,
y(t)=
a E IR;
sup sE[t-r,
(3)
(4)
dy(t) -dt
= -a
it
y(s)ds,
t1
yes);
rER;
r E (0,00);
t-r
dyd(t) = -ay(t) + by(t - r) t
+ cy([t -
m]) + f3y(t)
+8
t
Jt-T
yes) ds;
Exercises I
118
(5)
d~~t) = a(t)y(t - pet)~ - b(t)y(t - ret));
(6)
dt
dy(t)
= a(t)y(t - pet)) - b(t)y(t - yet));
dN(t) = N() [ _ N(t)]. dt r t 1 K '
(7)
T ,
r , K E (0 , 00 ),
N(t)
=
N ( s ).
sup SE[t-T, tJ
(8) dN ( t) [0'. N (t - T) --=rN(t) 1-
+ j3 N(t) + l' N ( [t -
&
(9)
dP(t)
-;[t
+ r P( t ) =
Q jJ
m])
+ 8 fLT
K
./3 + 0'[p-(t)]n'
-( )
P t
dP(t) P( ) _ - d +r t t a
(
= SE[t-T sup , t]
0'./3
+ fLT
P( s) ds
.
)nl
(12)
dP(t) P(t)a(J( f,t_T P(S)dS) m -d-+rP(t)= n t (J + ( f,'-T P( S) dS)
(13)
dN(t) = rN(t)[l- N(t -1J(N(t)))] dt K'
(14)
dN(t) dt
= rN(t)[1 _
. I
(10)
(11)
N ( s ) dS]
N(t -1J(N(t)))] K
)
P(s;
Exercises I N(t)=
sup
sE[t-r,t]
r,r,KE (0,00);
[0,00)1-t [0, co).
1]:
dN(t) + r N(t) = pe--yN(t-r(t) ); -;u-
(15)
(16)
N(s);
119
dN(t) -;u+ rN(t) = pe--yN(t)
N(t)
dN(t) -;u+ rN(t) = pN(t -
(17)
=
sup
sE[t-r,t]
N(s);
r)e--yN(t-r);
dN(t) -- + rN(t) = pN(t)e-'YN(t); dt .
(18)
(19)
dN(t) -;u+ rN(t) = pN([t -
m1)e--yN(t-r);
(20)
dN(t) -;u+ rN(t) = pN(t -
r)e-'YN([t-mJ);
dN(t) +rN(t) =p(jt
(21)
&
N(s)ds)e-'YN(t-r);
t-T
dN(t) + r N(t) = p N(t -;u-
(22)
dx(t)
aT
(23)
= -a(t)x'Y(t)
r! N(s) d.~ ; r ) e -'Y J!-T
+ b(t)x([t -
m])'Y
where I is the ratio of odd positive integers.
(24)
dx(t)
-;It
= -a(t)x'Y(t)
+ b(t)x'Y(t),
x(t) =
sup
sE[t-r,t]
xes),
r E IR.
52. Prove that if x(t) is an arbitrary solution of
dx(t)
-;It
= ax(t)
+ aox([tJ) + alx([t -
11),
(a)
Exercises I
120 then
x(n + 1) where
= box(n) + b1x(n -
((3)
1),
bo = ea + aoa-l(e a - 1) bI = a-::1al(e a -1).
Prove or disprove the following; "the trivial solution of ( Q') is asymptotically stable, if and only if that of ((3) is asymptotically stable". Prove also that the trivial solution of ((3) is asymptotically stable, if and only if the roots of satisfy I A I < 1 . Can you develop such a stability criteria for an equation with a regular delay T such as that in
dx(t)
---;It
= ax(t) + aox([tJ) + alx([t -
1]) + a2x(t - T)?
Discuss the stability characteristics of the following equations:
d:~t)
(1)
N
= aox(t)
+L
aix([t - iJ).
i=O
(2)
(3)
d (t) T = aox(t - T) + :L aix([t - i)) + t N
c
i=l
sup
xes).
sE(t-r,t]
Examine the asymptotic stability of the nontrivial steady state of the nonlinear equation
(4)
d~?)
= N(t)
(7' - aN(t -
r) - aoN([t]) - alN([t
-1]))
Discuss the local and global attractivity properties of the positive steady state of the logistic equation with an unbounded delay of the type
(5)
dN(t) = rN(t) dt
[1 _N(At)] I{
121
Exercises I where 0 < .A. < 1 ; can you discuss the cases).
= 1 and), > 1 also?
53. Examine the local and global asymptotic stability properties of the positive equilibrium of the impulsive logistic equation
[n
]
dN(t) = N(t) b - ~ ~j log[N(t - Tj)] , -;It where
Tj Tj+l - Tj
- t 00
~
T,
as
J
- t 00
j = 1,2,··· ,
where b, aj, Tj ,j = 1,2, .. are positive constants; Cj is a sequence of real numbers and the sequence t j is increasing. Examine also the existence of nonoscillatory solutions. 54. Derive sufficient conditions for the global asymptotic stability of the following feedback control models of population systems: (assume suitable intitial conditions and let all the parameters be positive constants);
d~?) du(t) ---;It
= rN(t)
[1 _N2~{-; T) - CU(t)]) (1)
= -au(t) + bN(t -
T).
d~it) = rN(t)[l- NiP - cu(t)] du(t) --;It
=
N(t) =
~
-au(t) + bN(t); sup
(2)
N(s).
SE[t-T,t)
[K -
dN(t) = rN(t) N(t - T) - CU(t)]) dt 1 + N(t - T) du(t) ---;It = -au(t) + bN 2 (t).
(3)
dN(t) = rN(t - T) [N(At) ] -;It 1- ~ - cu(t - T) du(t) --;It = -au(t) + bN(J-lt) O
(4)
Exercises I
122
d~; t) = r
N [1 - Nt (t)
logr
duet)
---;u- = -au(t) + blog[N(t -
d~y) = rN(t) [1 - ~ duet)
---;u- =
1
1)
r)] - cu( t)
(5)
r)J.
00
H(s)N(t - s) ds - CU(t)] )
roo H(s)N
-au(t) + b Jo
2
(6)
(t - s) ds.
55. Assume the following: to < tl < t2 < ... and limk-+oo tk = 00. (i) the sequence {tk} satisfies (ii) m is a piecewise continuous function from [0,(0) into [0,(0). (iii) p is a continuous nonnegative function defined on [0,(0). (iv) h is a nonnegative piecewise continuous function on [0,(0); h is left continuous at t = tk; (v) f3k (k = 1,2, ... ,) are nonnegative numbers. If
°: ;
met)
~ h(t) +
It
p(s)m(s)ds
L
+
to
f3km(tk);
to
then prove (see Pirabakaran [1989]) that
m( t) :,; h(t) +
+
J;:« {"ll}l+,8; (l )exp
t IT
p(s)
dS) },8k h(t k)
(J. p(U)dU)P(S)h(S)dS; t
lto s
(l+ f3 k )exp
s
Deduce that if h(t) is a constant c > 0, then
If h is differentiable, then derive that
m(t) :,; h(t,)
,.1.LY + +[
,8k) exp
II to 8
(l
p( s)
dS)
(1 + ,8.)h'(s)exp
(J.t P(U)dU) ds; 8
t~to.
129
Exercises I
Can you develop similar results for inequalities of the form
met)
~ h(t) +
l
pes )m(t - s) ds
to
+
L
flkm(tk),
to
For the delay logistic equation
derive sufficient conditions for the existence of a positive equilibrium. Discuss the convergence and oscillatory characteristics of all positive solutions.
CHAPTER 2
DELAY INDUCED BIFURCATION TO PERIODICITY 2.1. Introduction This chapter is intended to provide a reasonably self~contained demonstration of the various computations associated with Hopf-bifurcation in delay differential equations in which delay becomes a bifurcation parameter. We begin with a brief motivation. In most biological populations, the accumulation of metabolic products may seriously inconvenience a population and one of the consequences can be a fall in the birth rate and an increase in the mortality rate. If we assume (see Volterra [1931]) that the total toxic action on birth and death rates is expressed by an integral term in the logistic equation, one can then, consider the following integrodifferential equation
d~?) = rN(t) -
bN2 (t) - CN(t)([" K(s)N(t - s) ds
r
2.1.1
where K denotes the residual intensity of pollution and n E (0,00). A model related to (2.1.1) in theme has been i1Umerically studied by Borsellino and Torre [1974]. In order to derive the qualitative findings of Borsellino and Torre by means of an analytically manageable model, Cushing [1977] has proposed a model of the form
d~;t) = N(t){ a -
fJN(t)
-,(J.=
K(s)N(t - S)dS),}
2.1.2
where a, /3, I are positive constants and 1
K(s) == - sexp(-s/I), I
I
> O.
2.1.3
It has been found by Cushing [1977J that (2.1.2) - (2.1.3) has a nontrivial steady state and there exist positive numbers say 11,12 such that for 11 < I < 12 the steady state is unstable and for I > 12 the steady state regains its lost stability. The system (2.1.2) - (2.1.3) has been further analysed by Cohen et al. [1979] and they have established that when the steady state loses its stability, stable oscillatory solutions exist.
Another system more general than (2.1.2) - (2.1.3) has been considered by Landman [1980] in the form
d~?) =
AN(t){ 1 - aN(t)
-/=
K(t - S)F(N(s»dS},
t>O
2.1.4
§2.1. Introduction
125
where K is a nonnegative delay kernel, a is a positive constant and F is a nonnegative function of N; A is a parameter with respect to which stability of (2.1.4) is considered. It is shown by Landman [1980] that there exists a positive A* such that for A = A*, a steady state of (2.1.4) becomes unstable and oscillatory solutions bifurcate for A near A*. Since the parameter A appears in all the terms of the growth rate in (2.1.4), it is difficult to attach any biological or ecological interpretation of the parameter A in (2.1.4). For a general discussion of bifurcation of periodic solutions of integrodifferential equations we refer to Cushing (1977] and Simpson [1980] as well as Landman (1980]. In this chapter we consider a generalization of the familiar logistic equation incorporating the effects of pollution resulting in additional mortality acting with a time delay. In particular, we consider the following time delayed logistic equation:
d~;t) = N(t){ <> -
PN(t) - 'YN 2 (t -
r)}.
2.1.5
We keep fr, (3, 'Y fixed and investigate the behavior of solutions of (2.1.5) for a range of positive values of the delay parameter r. In fact, we establish that there exists a critical value of r at which a Hopf-type bifurcation to a periodic solution arises and a constant steady state becomes unstable. In the previous chapter we have seen that small time delays do not destabilize a stable system. Our analysis of (2.1.5) will show that significant time delays in the negative feedback or the death rate will destabilize an otherwise stable system. The system (2.1.5) has a unique positive steady state N* defined by 2.1.6 In order to study the linear stability of N* in (2.1.5), we let
N(t)
= N* + x(t)
in (2.1.5) and derive the variational system
dx(t)
dt + ax(t) + bx(t where
a
= (3N*,
r) = f(x(t), x(t - r))
2.1.7
§2.1. Introduction
126
a20
= -(3,
all
= -2,N*,
a02
= -,N*,
al2
= -,.
It is known from the theory of delay-differential equations that if the trivial solution of the linear approximation
dx(t)
--;It + ax(t) + bx(t - T) = 0
2.1.9
is asymptotically stable, then the zero solution of (2.1.7) and hence the steady state N* of (2.1.5) is (locally) asymptotically stable; this result is similar to the one in ordinary differential equations and can be found in Bellman and Cooke [1963], Hale [1977], Krasovskii [1963] and Halanay [1966]. It is possible to introduce the change of variables t in place of (2.1.9) we have
dy(s)
--;r;- + aTY(s) + bTY(S -
= ST,
x( ST)
= y( s) so that
1) = 0
which can, to some extent simplify our subsequent analysis of (2.1. 7). We will not, however, do this simplification; we note that our method of analysis of (2.1. 7) can be adapted to multispecies systems with two or more delays. Before we proceed further with our analysis of (2.1.5), we recall (for the benefit of the reader) the "Hopf-bifurcation theorem" for a system of autonomous ordinary differential equations. Hopf-bifurcation is one of the ways by which a nonconstant small amplitude periodic solution can arise in autonomous ordinary differential equations. In its simplest form, Hopf-bifurcation can be loosely stated as follows: as a real parameter say fL in an autonomous system of ordinary differential equations passes through some critical value say fL*, two eigenvalues of the linear variational equation about an equilibrium point cross the imaginary axis while all others have negative real parts. It is then shown that a smooth curve S in the parameter space, passing through fL* can be chosen in such a way that the nonlinear differential system has a nonconstant periodic solution near the equilibrium point and for every point fL E S. One of the several precise formulations of the Hopf-bifurcation theorem for autonomous differential equations is as follows (Hopf [1942J) :
"Consider a real system of autonomous ordinary differential equations
dx' dt' = !i(Xl, ... Xn,fL),
i = 1,2, ... n
§2.1. Introduction
127
written compactly in vector notation, dx dt
-
= F(x, p)
2.1.10
where F is assumed to be analytic in x and p for i in a domain G containing the zero vector 8, G c Fin and for Ipi < a, a being a positive c'onstant. (i) Assume that F( 8, p) = 8 for IJLI < a. (ii) For p = 0, the characteristic equation associated with the linear variational system corresponding to (2.1.10) at 8 has a pair of pure imaginary roots while all others have negative real parts. (iii) Let a(p) and o-(p) be a pair of continuous extensions of the pure imaginary roots (0- denoting the conjugate of (J) and let (J(O) = -0'(0)
= OJ
~e
[0"(0)]
=f. O.
2.1.11
Then there exists a real family of periodic solutions x = x(t, E), P = J.l(€) which has the properties p(O) = 0; x(t,O) = Bbut x(t, €) Bfor all sufficiently small e o. The period T( €) also satisfies T(O) = 27r 110'(0)1. Such a set of {x(t, E), p(E), T( e)} is unique and the periodic solution exists only for JL > 0 or only for p < 0 or only for p = 0".
t=
t=
There are several proofs of this theorem based either on the implicit function theorem or on the centre manifold theorem (see Marsden and McCracken [1976] or Hassard et al. [1981]).
Definition. Let pet) be a periodic solution of (2.1.10). Let r denote the closed path x = pet) in IRn. The periodic solution pet) is said to be orbitally stable if for each € > 0 there exists a 8 > 0 such that every solution x(t) of (2.1.10) whose distance from r is less than 8 for t = 0, is defined and remains at a distance less than € from r for all t 2': O. The periodic solution pet) is said to be asymptotically stable (T' is said to be a limit cycle) if, in addition, the distance of x(t) from r tends to zero as t
-4
00.
It is known (Coddington and Levinson [1955]) that the characteristic exponents of the nonconstant periodic solution x(t, €) of (2.1.10) are the eigenvalues of the eigenvalue problem dii +AU \ - = L-U 2.1.12 dt
§2.1. Introduction
128
where u(t) has the same period T = T(e) as the solution x(t, e), L being the linear operator obtained by linearization of (2.1.10) at the periodic solution. The characteristic exponents are determined only mod (21l"ijT) and depend continuously on e, one of which is, of course, zero since A = 0 and u = x( t, €) is a solution of the eigenvalue. problem (2.1.12). By the hypotheses of the above bifurcation theorem, exactly two exponents approach the imaginary axis. But, one of them will be identically zero while the other say /3 = /3( c) must satisfy /3(0) = O. It will follow from the result of the bifurcation theorem that if 2.1.13
/3 = /3 ( €)
= /31 e + /32 c2 + ...
2.1.14
then J-Ll = 0, /31 = O. Furthermore, Hopf [1942] (see Marsden and McCraken [1976]) has established the following formula for exchange of stabilities:
/32
= -2J-L 2 ?Re [0"'(0)].
2.1.15
If we now assume that Jl2 =f 0 and ?Re 0"'(0) > 0, then J-L2 and /32 are of opposite signs. Hence, if nonconstant periodic solutions bifurcate to the right (i.e. for J-L2 > 0) they are then, locally asymptotically stable (since /32 < 0); if the bifurcation is to the left (i.e. J-L2 < 0), then /32 > 0 and this will imply that the bifurcating periodic solution is unstable. These facts are illustrated by the so called 'bifurcation diagram' at the end of section 2.4.
2.2. Loss of linear stability The trivial solution of the linear system
dx(t)
---;It
+ ax(t) + bx(t -
7) = 0
2.2.1
will be asymptotically stable if and only if all the roots of its characteristic equation
z
+ a + be -
ZT
=0
2.2.2
have negative real parts; if there exists a root of (2.2.2) with a zero or positive real part then the trivial solution of (2.2.1) is not asymptotically stable. We first note that z = 0 cannot be a root of (2.2.2) since for z = 0, (2.2.2) leads to a
+b=
N*(/3 + 2,N*) > OJ
129
§2.2. Loss of linear stability
also if z = p + iq (p, q are real numbers) is a root of (2.2.2), then z = p - iq is also a root of (2.2.2). Let z = p + iq be a root of (2.2.2); separating the real and imaginary parts we derive p
+a = q
-be-
= be-
pr
pr
COSqT}
2.2.3
sin qT
and therefore
(p + a)2 For fixed q
~
0 and
T
~
+ q2 = b2e- 2pr .
0 if we plot the functions
It (p)
=
(p + a? 2
h(p) = b e-
2pr
It
2.2.4 and
h
where
+ q2
,
it is then found that p < 0 in (2.2.3) whenever 2.2.5 The inequality (2.2.5) will definitely hold if b < a; we shall assume in the following that b > a or equivalently (2,N*) > f3 .
It is found that in the parameter space of aries is given by a + b = O.
a
and b, one of the stability bound2.2.6
Since zero is not a root of (2.2.2), we look for other stability boundaries in the parameter space of a, b. Such a boundary is obtained when (2.2.2) has a pair of purely imaginary roots say ±iO"o, 0"0 > O. Thus, if we let p = 0 and q = 0"0 in (2.2.3), we get the equations of another stability boundary in the parameter space of a, b in the form a = -bcos O"OT } 2.2.7 0"0 = b sin O"OT. These two equations determine the two unknowns 0"0 and T for (2.2.2) to have two purely imaginary roots. For positive 0"0 it will follow from (2.2.7) that COS(O"OT) < 0 and sin(O"oT) > 0 and hence we derive that
0"0 T
7r 2n7r + "2
satisfies
< O"OT < 2n7r + 7r;
n
= 0,1,2, ...
2.2.8
§2.2. Loss of linear stability
130
-e/b - - - - --
Figure 1
As it is seen from Figure 1, if we solve (2.2.7) for the unknowns obtain 0"0 = (b 2 - a 2 p1 and T = TO +1271"n, n = 0,1,2, ... } TO
= [ arc cos ( -ajb)]/(b2 - a2 )2
0"0,
To,
we
2.2.9
where 'arc cos' denotes the 0 to 71" branch of the inverse cosine function. We also derive from (2.2.2) by implicit differentiation, <;n ;ne
(dZ) I dT
r=ro
--
0"52
o)
(l+ aT
2
2
2.2.10
+TOO"O
It follows that when T is near TO and T < TO, all the roots of (2.2.2) have negative real parts while if T is near TO and T > TO, two roots of (2.2.2) gain positive real parts as T passes through TO. Similarly, whenever T passes through TO + 2n7l" (n = 1,2,3, ... ) from left to right, two roots of (2.2.2) gain positive real parts. Thus, the least positive value of T at which the trivial solution of (2.2.1) loses its asymptotic stability is TO given by (2.2.9). 2.3. Delay induced bifurcation to periodicity We have seen that when T = TO, the linear variational system (2.2.1) has periodic solutions with period 271"/0"0. If instead of (2.2.1) we consider the full nonlinear equation (2.1. 7) as a perturbation of (2.2.1) and if T is considered to be a perturbation of TO, one of the questions will be whether such a perturbed equation has a periodic solution with a period which is a perturbation of that of the linear approximation (2.2.1). Classical Hopf-bifurcation theory (Hopf [1942]) deals with such a problem. Bifurcation of periodic solutions in population dynamic models
§2.3. Bifurcation to periodicity
131
have been investigated in a number of articles and in a monograph by Cushing [1977]; in most of the works by Cushing (see Cushing [1977], [1979]), the period of the bifurcating periodic solution is the same as that of the linear approximation; as in the case of Hopf-bifurcation (Hopf [1942]) we expect a perturbation of the period if the instability causing delay parameter undergoes a perturbation. The classical H'opf-bifurcation theorem has been extended to differential equations with delays, integrodifferential equations and partial differential equations by a number of authors (Hale [1977], Chow and Mallet-Paret [1977]' Stech [1979]). All these works show that the bifurcating periodic solutions will be stable if a certain number is negative and unstable if that a number is positive. The verification of the sign of such a number proves in many cases to be difficult. For this reason, we provide a discussion of the stability of the bifurcating periodic solution along the same lines as that of Hopf's original work on ordinary autonomous differential equations (Hopf [1942]). We consider the nonlinear system
dx(t)
d l + ax(t) + bx(t where a
= (iN*,
= f(x(t),x(t -
2.3.1
T»
b = 2,(N*? and
f(x(t), yet)~ a20
T)
= a20x2(t) + allx(t)y(t) + a02y2(t) + a12x(t)y2(t)
= -(i,
all
== -2,N*,
a02
== -,N*,
a12
= -,.
2.3.2
We have already seen that for T in the range [0, 'To) all the roots of the characteristic equation (2.2.2) have negative real parts and the trivial solution of (2.3.1) is locally asymptotically stable; hence, the only periodic solution of (2.3.1) for T E [0, TO) is the constant solution x(t) == 0 in a sufficiently small neighbourhood of the trivial solution of (2.3.1). However, for T = TO the characteristic equation (2.2.2) has two purely imaginary roots ±io-o where 0-0 == (b 2 - a2 )'! and as a consequence, the linear homogeneous system
dx(t)
-;It + ax(t) + bx(t -
TO)
= 0
2.3.3
will have a pair of linearly independent periodic solutions
'P2(t) = coso-ot
2.3.4
§2.3. Bifurcation to periodicity
132
with period 27r/ao. One may expect that when l' is near 1'0 the nonlinear system, (2.3.1) will have periodic solutions with periods near 27r/ao and we proceed to establish the existence of such periodic solutions for l' near 1'0. We first introduce a new independent variable s defined by
2.3.5 where e is a parameter to be defined below (in the application of an implicit function theorem) and a( e) is defined by
2.3.6 in which T( e) denotes the unknown period of the periodic solution to be determined for (2.3.1). We change the dependent variable in (2.3.1) to y by the substitution
yes)
= XeS/aCE)) = x(t)
2.3.7
so that (2.3.1) becomes
dyes) a d;-
+ ayes) + byes -
ar)
= f(y(s), yes -
aT))
2.3.8
and we will look for (27r / ao )-periodic solutions of (2.3.8) in the new variable s. Proposition 2.3.1. For each T in a one sided neighbourhood of TO there exists a one parameter family of nontrivial periodic solutions z of period 27r / ao for (2.3.8) and this family can be obtained in the fonn
z(s, e) = e'PI(s) + €2wo(s)
+ €3 WI (S, E) }
+ €2T2(€) a(E) = 1 + €2a2(E) r(E) = TO
2.3.9
where a( €) is defined by (2.3.6), Wo and WI are differentiable in S; Wl(·,€),r2(E),a2(€) are continuous in € for I€I < EO for some EO> 0; furthermore, i = 1,2 where
r /(10 woe s )'Pi( s )ds, i = 1,2.
2.3.10
21f
(wo, 'Pi)
= Jo
2.3.11
§2.9. Bifurcation to periodicity
199
Proof. Let us first suppose
y(s, €) = €"p(s)
+ €2wo(s) + €3 W1 (s, €) }
+ Tl € + €2 T2 ( €) O'(€) = 1 + O'I€ + €20'2(€) T( €) =
2.3.12
TO
where Wo and WI satisfy (2.3.10) and "p belongs to the span of epl(S) and ep2(S); we shall determine the various terms in (2.3.12). Supplying (2.3.12) in (2.3.8) and collecting together coefficients of the respective powers of € we get (after some simplification) d?j;(s) 2.3.13 ~ + a"p(s) + b"p(s - TO) = O. We choose "p( s) in (2.3.13) to be the solution "p( s) == epl (s) which we denote by cp( s) in the following. For such a choice of "p( s), woe s) is given by
where 2.3.15 The linear system
dyes) -;z;-
ayes) - byes + TO) = 0
2.3.16
is adjoint to the linear homogeneous system in (2.3.14). By the Fredholm alternative theory (Halanay [1966]), the solvability conditions for (2.3.14) are given by
Jor27fjtIO yes) { -
dep( s) 0'1 ~
+ b(Tl + O'I TO)ep'(s -
TO)
+ F 1(s)
}
ds
=0
2.3.17
for all periodic solutions y( s) of (2.3.16). It can be shown that (2.3.16) and (2.3.17) lead to (for j = 1,2) 2.3.18 It is found that (2.3.18) simplifies to the two equations,
-aT! -
O'I( a
+ aTo) =
0
2.3.19
§2.3. Bifurcation to periodicity
134 which imply
71
=0=
For such a choice of
0"1'
dwo(s)
~
and 0'1, (2.3.14) simplifies to
71
+ awo(s) + bwo(s -
TO)
= F1(s)
2.3.20
for which, we can choose a solution Wo satisfying
(WO(S),CPi(S») == 0,
i = 1,2.
Having selected 7/J, T1 ,0'1 and wo, we have to determine
dWl(s,e) ds
+ aWl(S,€) + bW1(S -
TO, e)
+ b{ TO 0'2 (e) + T2( e) }cpl (s F2(s, e) =all { wo(s)cp(s - TO)
WI (S,
= -0'2(t) TO)
2.3.21 e) such that
d
~
+ F 2(s, e)
+ cp(s)wo(s -
2.3.22 where
TO) }
+ 2a02CP(S - TO)WO(S - TO)
+ a12CP( S )cp2( S -
TO)
+ 2azocp(s)wo(s) + O(e).
2.3.23
We first examine the particular case for e = 0 in (2.3.22) and then, invoke an implicit function theorem for the case e =1= 0 in (2.3.22). Accordingly, we consider
dWl(S,O) ds + aw l(s,O)+ bw l(S-TO,O)
= -0"2(0)
dcp( s)
~
+ b{T00'2(0) + T2(0)}cp (s I
TO)
+ F2(s, 0).2.3.24
0"2(0) and T2(0) in (2.3.24) are not yet known. By the solvability conditions of (2.3.24) we obtain
J.
21r/UO
= 1,2
2.3.25
-(l/7rO"O)(cpl(S), F 2(s, 0») -(1/7r)(cp2(s), F 2(s, 0»)
2.3.26
[<:0 'Pl (s) + 'P'( s), F,( s, 0))1 T2(0) = -(1/") [( 1 ::TO 'Pl (s) + TO'P2( s), F2( s, 0)].
2.3.27
o
cPj(s){right side of (2.3.24)}ds
= 0,
j
which will simplify to
+ T00"2(0) = 0'2(0)(1 + aTo) =
T2(0) -aT2(0) -
0",(0) = (I/")
195
§2.9. Bifurcation to periodicity
For the choice of 0"2(0),72(0) in (2.3.27), we have a solution of (2.3.24) which we make unique by requiring that (<.pj(S),WI(S,O») = 0, j = 1,2. We now return to the solution of (2.3.22) for € =f o. Let B denote the real Banach space of 2n"/0"0-periodic continuous functions with the norm 1I<.p1i = maxl<.p(s)I, 0::; S ::; 27r/0"0; consider a direct sum decomposition of B in th~ form 2.3.28 where 51 is the span of <.pI and c.p2 while 52 is the orthogonal complement of 51 in B defined by 2.3.29 S2 = {g E BI(g, <.pi) == 0, i = 1, 2}. Let P and Q denote the projection operators defined by 2.3.30 Then, Sl is the null space in B of the differential operator L where
L<.p
dc.p(s)
= -----;I;- + ac.p( s) + b<.p( S -
2.3.31
ro).
IT we write WI(S, e) E B in the form 2.3.32 then we have from (2.3.22)
Lw~(s, e) = QF2(s, e) E 52 2.3.33 Since QF2 is in the range of L there exists an operator K defined on the range of L such that K L = I (identity) and hence we have from (2.3.22) - (2.3.33), the following equivalent of (2.3.22);
w~(s, e) -
KQ{ -
<72 ( e)
~~ + b[To <72 ( e) + T2( e)J
TO)
+ F2(S' e)} =
0 2.3.34
{
~~ +b[ro0"2(€)+0"2(e)]
2.3.35
{
~~ +b[r00"2(€)+r2(e)]<.p'(s-ro)+F2(s,€») =0.
2.3.36
§2.3. Bifurcation to periodicity
136
We define a mapping H( W~, 0"2, 1"2, E) of 52 x R x R x R into 52 x IR x IR so that H(w~, 0"2, T2, E) = (HI, H 2, H 3) where HI, H 2, H3 denote respectively the left sides of (2.3.34), (2.3.35), (2.3.36). It can be found that H has a Frechet derivative at (w~ (s, 0), T2(0), T2(0), 0) given by f!..&. Buz
Elb.
D=
2.3.37
BU2
!l.!b. BU2
The right side of (2.3.37) simplifies to
10 [0
KQ
-KQ
2.3.38
The linear operator defined by (2.3.38) is an isomorphism of 52 X IR x III onto itself. Therefore, by the implicit function theorem (see Sattinger [1973]) there exists a one parameter family of maps w~(s,€), 0"2(€),T2(€) defined for € in (-€o,€o) for some EO > 0 such that H(w~(S'€)'0"2(C),T2(€),C) = 0, lEI < co. The solution w~(s, c) thus determined is unique in the subspace 52 c B and since, we set out looking for a solution of (2.3.22) in 52, we have obtained such a solution to be w~ (s, €) which we shall denote in the following by WI (s, €). This completes the proof of the existence of a solution yes, E) bifurcating from the steady state and the detennination of Y(S,€),T(€) and O"(€) in the form (2.3.9). (]
2.4. Stability of the bifurcating periodic solution In the previous section we have constructed a (211' / 0"0) periodic solution z( s, €) of
dyes) 0'( €)d;""
+ aye s) + bye s -
O'T) = f(y(s), y( s - O'T»)
2.4.1
given by
z(s,€) = €'{)(s) + E2wo(S) + E3Wl(S, E)
2.4.2
for I€I < EO· We have also derived expressions for O'(E) and T(€) as follows:
o"(E) T( E)
= 1 + €20'2(€) } = TO + €2 T2 ( E).
2.4.3
§2.4. Stability of the periodic solution
137
We begin our stability investigation with some preliminary definitions from the work of Stokes (1964J. Let C denote the linear space of continuous functions
C
= {h\h : [-£r(e}r(e),O]
1-4
2.4.4
R}
endowed with a norm 11.1\ defined by I\hll = sup{lh(6)\,
~
-a(e)I(<:)
(}
~
hE C.
O},
2.4.5
By the autonomous nature of (2.4.1), whenever z(s, e) is a periodic solution of (2.4.1) with period 21r/£ro, {z(s + e,f), 0 ~ e < 21r/ao} is also a solution. Thus, if we define V C C by
2.4.6 then V is compact in C since V is defined by a continuous map from [O,2n"/aol into C. In the phase space C, if we identify the solutions which differ only by a translation in the variable s, then V will be a closed trajectory. We note that a closed trajectory V C C is said to be asymptotically stable if there exists a neighbourhood N of V such that 7j; E N implies dist.{YtIJ(s), V} ~
°
as
where YtIJ(s) is a solution of (2.4.1) with YtIJ(O) dist.(1/7, V)
= min.{\\1/7 -
s ~
00
= 7j; and ZI\, Z E V}.
2.4.7
A closed trajectory V is said to be asymptotically stable with aysmptotic phase if it is asymptotically stable and given 7j; E N there exists a constant n = n( 1/7) such that
IIYtIJ(S) - z(s
+ n, e)1I ~
°
as
s~
00.
2.4.8
To investigate the asymptotic stability of z( s, e), we substitute
y(s,e) = z(s,e)+v(s,e)
2.4.9
in (2.4.1) where v is a perturbation term whose behavior will decide the stability of z( s, e). Such a change of variable leads to the variational system
dv(s,e) a () € --a:;-
+ av(s,) e + b( v s-
aI, e)
= f:r;(z(s,e),z(s - ar,e))v(s,e)
+ Jy(z( s, e), z( s - aT, e)) v( s - aI, e) .
2.4.10
§2.4. Stability of the periodic solution
138
Now the linear stability of z(s, €) is determined by the stability of the zero solution v(s, €) == of (2.4.10). Therefore, we have to study the behavior of the solutions v(s,€) of (2.4.10) as S -+ 00 given small initial conditions for v on [-0"1,0]. In (2.4.10) the coefficients are periodic in s of period 27r/{70. Thus, to study (2.4.10) we will have to use the Floquet technique, well known for periodic_ ordinary differential equations; this technique has been used by several authors in discussing stability of bifurcating time periodic solutions in partial differential equations (see Crandall and Rabinowitz [1972], Sattinger [1973]). The Floquet technique has been extended to functional differential equations by Stokes [1964].
°
We shall look for solutions of (2.4.10) in the form
v( S, €) = q( s, €) exp[1]( €)s]
2.4.11
where q is periodic in s with period 27r / (70; the numbers 1]( €) which can be complex are known as Floquet exponents and the numbers exp[(27r/{7o)1](e)] are known as Floquet multipliers. The sign of the real part of 1]( e) will obviously determine whether or not v( s, e) -+ as s -+ 00 and hence the stability of the trivial solution of (2.4.10). Clearly a solution v(s, €) of the type in (2.4.11) will be a periodic solution if there is a Floquet multiplier equal to unity. In fact, such a periodic solution of (2.4.11) exists as a consequence of the autonomous nature of (2.4.1). Since z(s, e) is a solution of (2.4.1), we have
°
(7(e)
dz( s €) ds' +az(s,e)+bz(s-{7T,€)=J(z(s,e),z(s-{7T,e»
2.4.12
which also leads to
2.4.13
showing that {dZ~:,E)} is a periodic solution of (2.4.10) with period 27r /{70. The Floquet multiplier associated with [dz(s, e)/dsJ is 1 and the corresponding exponent is zero for € in the range (-EO, EO)' In the following, we use a theorem of Stokes [1964].
§2.4. Stability of the periodic solution
139
Theorem. (Stokes [1964]) Suppose the characteristic exponent zero for the equation (2.4.10) has multiplicity one and that all other remaining possible exponents have negative real parts. Then the bifurcating periodic solution z(s, €) is asymptotically stable with asymptotic phase. Thus, we are led to a study of all the characteristic exponents .of (2.4.10). We know that when € = 0, (2.4.10) has zero as an exponent and (2.4.10) has two linearly independent periodic solutions of period 27!" j lTo namely sin lToS and cos lTOS. Therefore, for € = 0 there is a double Floquet exponent at the origin of the complex plane and this corresponds to the two periodic solutions say ql (s, 0) == sin lToS and q2( s, 0) = cos lTOS' We will have to find the other Floquet exponents when € = 0; for € = 0, (2.4.10) reduces to
dv(s, 0) ds
+ av(s,O) + bv(s -
'To, 0)
=0
2.4.14
and if TJ = TJ* is another exponent with an associated periodic solution €( s) of period 27!" j lTo, then using a Fo.urier harmonic decomposition of € we can look for solutions of the form
for integral constants k and some constants Ak. In such a case TJ* and k satisfy 2.4.15 But from the characteristic equation (2.2.2), we know that an equation of the form (2.4.15) has no roots with positive real parts which means that the real parts of TJ* cannot be positive. This verifies that all the Floquet exponents different from zero for € = 0 have negative real parts. By continuity it will follow that those exponents which have negative real parts for € = 0 have negative real parts (with perturbation) when € # 0 also for small Itl. Thus, we have to investigate the perturbation of only two exponents for € # 0 corresponding to the perturbation of the zero exponent. The question of stability then reduces by the theorem of Stokes to finding these two perturbed exponents. We have already found one of these to be zero corresponding to dz(s, €)jds. So we are left with the calculation of only one exponent say TJ( €) which is a perturbation from zero. By the theorem of Stokes, the real part of 1]( €) will decide the stability or instability of the periodic solution z(s, E).
§2.4. Stability of the periodic solution
140
In order to determine ry( e), we proceed to look for a solution v(s, e) of (2.4.10) in the form 2.4.16 v(S,e) = q(s,e)exp{7J(e)S} where q(s, e) is periodic in s of period 27r /0"0' Supplying (2.4.16) in (2.4.10) we obtain
dq(s,e) 0"( e) -;];-
+ [a - fx(z(s, f), z(s - O"T, e»)q(S, e) + [b- Jy(Z(S,f),Z(S -O"T,e»)q(s,€)exp[-7JO"T] + 0"( e)7J( e)q(s, e) = O.
2.4.17
At e = 0, the solution space of (2.4.17) is spanned by cp(s) and dcp/ds; hence, we shall look for a solution of (2.4.17) in the form r(e) dz(s,e) 2 q(s, e) = cp(s) + - . - d - + eQo(s) + e Q1(S, e) e s
2.4.18
where we assume that 7J is of the form 2.4.19 and 2.4.20 while ref) is a function to be determined in the process of obtaining q(s, e). We recall that 0"(e)=1+€20"2(f) }
+ e2T2( f) z(S,e) = f'P(S) + €2WO(s) +
2.4.21
T( e) = TO
€3
W1 (S,f).,
We supply (2.4.18), (2.4.19) and (2.4.21) in (2.4.17); then collect together terms respectively of the orders of 1, e, e2 • This leads first to
d~~s) + acp(s) + bcp(s -
TO) = 0
2.4.22
which automatically holds by the choice of cp( s). The next equation to be solved is
dQo(s) 2 -;];-+aQo(s) + bQo(s - TO) = fxx(O, O)cp (s)
+ fyy(O, O)'P2(S -
TO) + bTo7Jl'P(S -
- 7Jl cp( s) - 7Jl r(O)cp' (s).
+ 2Jxy(0, O)cp(s)cp(s - TO) TO) + bTo7Jlr(O)'P'(s - TO) 2.4.23
§2.4. Stability of the periodic 30lution
141
The two unknowns "ll and reO) on the right side of (2.4.23) are determined using the Fredholm alternative type solvability conditions:
('Pj( s), right side of (2.4.23») = 0, (2.4.24) leads to
.
j = 1,2.
2.4.24
2
{-aTo -1 +
= O.
2.4.25
Choose "ll = 0 and leave reO) for the present undetermined. Such a choice of simplifies (2.4.23) to
dQo(s)
~
+ aQo(s) + bQo(s -
TO)
= 2F1 (s)
T]1
2.4.26
which has a solution given by
Qo(s) = 2wo(s).
2.4.27
When e = 0, Ql (s, 0) is governed by
dQld~' 0) +aQl(s, 0) + bQl (s - TO, 0) = 3F2(s, 0) -
2.4.28
+ b"l2(0)Tor(0)'P' (s - TO) - "l2(0)r(0)'P' (s) - "l2(0)'P( s). The two unknowns "l2(0) and reO) in (2.4.28) are determined by the solvability conditions which lead to "l2(0)r(0)ToO"~ = (20"0/1l')('Pl(S),F2(s,0)}
"l2(0){1
+ aTo} -
"l2(0)TO
+"l2(0)r(0){1 + aTo} =
(2/7r)('P2( s), F2( s, 0»).
2.4.29
Solving for "l2(0) and reO) from (2.4.29) and simplifying,
"l2(0) = -20"~T2(0)/[0"~T5 reO)
+ (1 + aTO)2]
= (-1/0"5){ a + b2To + [(1 + aTo)2 + 0"5 T5]
2.4.30 2.4.31
Having obtained Ql (s, 0), "l(O) = m(O) and reO), we have to apply the implicit function theorem in order to complete the proof of the existence of a neighbourhood of € = 0 for the functions r(e),"l(e) and Ql(S,e). The application of the implicit function theorem is similar to that already done and we omit the details.
§2.4. Stability of the periodic .'3olution
142
The relation (2.4.30) is similar to a relation derived by Hopf (see Marsden and McCracken [1976]) for the case of bifurcation of periodic solutions in autonomous ordinary differential equations; in fact, we can rewrite (2.4.30) in the form
ry2(O) = -2 [ (~~ )
rJ
T2(O)
where ,X is any root of the characteristic equation (2.2.2). We note that if 72(0) > 0, then 1]2(0) < 0 and as a consequence of the continuity of 72(E) and 1]2(E) (from the implicit function theorem) for E near zero, it will follow that 1]2 ( E) < 0 and 72( €) > O. Similarly, if 72(0) < 0 then m( €) > 0 and 72( €) < 0 for € near zero. We know that the linear variational system (2.4.10) has a characteristic exponent 0 since dZ(S,E)/ds is a periodic solution with period 2n)C10 in s. If € is small, then 7 is near 70 and hence all other characteristic exponents which were negative for E = 0 will remain so for small nonzero E: however, for E = 0 we have an exponent 1](0) = 0 which has multiplicity two. For € =f 0 this double exponent becomes two different exponents with values zero and 1]( €) =f O. Thus, we conclude that the space of solutions corresponding to the zero exponent has dimension 1 and the result of Stokes will be applicable if 1]2 ( €) < 0 for which a sufficient condition is that 72(0) > O. Thus, 72(0) > 0 leads to orbital asymptotic stability of the bifurcating periodic solution; if 72(0) < 0, then 1]2 ( €) > 0 which will imply that the bifurcating periodic solution is not asymptotically stable. We conclude that supercritically bifurcating periodic solutions are asymptotically stable while subcritically bifurcating periodic solutions are not stable. The bifurcation diagrams for (2.1.5) will appear as follows:
N
steble steble
unsteble
N*i-------t-
stob1e
l'
1. Supercriticol blfurcot1on
§2.4. Stability of the periodic 30iution
N
stable
N*~------+
unstable
- - _.- - - - - - -
---unstable
2. Subcrltical bifurcation
2.5. An example
It this section we illustrate an algorithm of a method for obtaining an approximation of the bifurcating periodic solution. Among the many methods developed in the study of periodic solutions of ordinary differential equations, the method of Poincare-Lindstedt has been an attractive one from the view-point of practitioners of mathematics. The reason may be that this method supplies specific asymptotic expansions for the state variable besides providing information on the perturbed period. The success of the method depends on one's ability to introduce a "small" parameter in the system equations. In the case of Hopf-bifurcation, introducing a small parameter is easy in most cases. For example, let us consider the discrete delay-logistic equation
duet) dt
= ru(t)
(1 _u(t - h») }(
2.5.1
where r,}( are positive constants and h > 0 is the parameter in terms of which bifurcation is analysed. For h = 0 we have in (2.5.1), u(t) -+ ]( as t -+ 00 whenever u(O) > O. We let 2.5.2 u(t) == }([1 + x(t)] in (2.5.1) and derive
dx(t)
--;It = -rx(t - h) - rx(t)x(t - h).
2.5.3
144
§2.5. An example
The characteristic equation associated with the linear variational system in (2.5.3) about the steady state x(t) == 0 is 2.5.4 We leave it as an exercise to the reader to show that if 0 < rh < 7r /2, then all the roots of (2.5.4) have negative real parts and if rh = (7r /2), then (2.5.4) has a pair of pure imaginary roots A = ±ir. Furthermore, one can verify from (2.5.4) that
~e(dA) dh
>'=ir
=~>O 2
2.5.5
4 + 7r
and hence by the bifurcation Proposition 2.3.1, there exists a small amplitude periodic solution whose period depends on the parameter h. In order to calculate (approximately) the bifurcating periodic solution of the nonlinear equation (2.5.3) we proceed as follows. We let
t
= hs;
so that
d~~)
=
x(sh) = yes)
-G + 1')
and
rh
= -7r2 + J.l
[yeS -1) +y(s)y(s
2.5.6
-1)].
2.5.7
Once again we let
s = (1
+ a)r
and
y((l
+ a)r) =
v(r)
2.5.8
and derive from (2.5.7),
dvd(r) r
= _ (~+ J.l)(l + a) [v(r - _1_) + v(r)v(r - _1_)]. 2
.
l+a
l+a
2.5.9
We assume the following perturbation expansion in terms of a perturbation parameter €: fl
= J.l2E2 + J.l4€4 + .. .
2.5.10
(J"
=
(72€2
+ (74€4 + .. .
2.5.11
v(r)
=
EVI(r) + €2V2(r)
+ €3 v3 (r) + ...
2.5.12
§2.5. A n example
145
Using (2.5.10) - (2.5.12) in (2.5.9) we obtain
d~ {eVI(r) + e2v2(r) + e3v3(r) + ... } = - {
i+
.2 (1'2
+ 0-2
i) + ... }[ {
-
1 + 0-2.
2
+ ... )
+ e2V2(r - 1 + 0'2e2 + ... ) + e2V3(r - 1 + 0'2e2 + ... )} + {WI(r) + e2V2(r) + e3V3(r) + ... }{ eVI(r -
1 + 0'2e2
+ e2V2(r - 1 + 0'2f. Z + ... ) + e3V3(r - 1 + 0'2f. 2 + ... )}].
+ ... ) 2.5.13
By theorem II on page 186 of El'sgol'ts and Norkin [1973], we can expand the right side of (2.5.13) in powers of € by means of a Taylor series of the functions with perturbed arguments. We leave details of such expansions for the reader to carry out. Collecting and comparing the coefficients of the respective powers of f. in (2.5.13),
dVl( r) 7r --a;;:= -('2)vI (r -1)
2.5.14
dV2( r) --a;;:=
2.5.15
-(7r/2)v2(r - 1) - Crr/2)vl(r)vl(r - 1)
dV3(r) --a;;:= -(7r/2)v3(r -
1) - {JL2
+ 0'2(7r/2)}vI(r -1)
- (7r/2)0'2v~(r -1) - (7r/2)[vI(r)v2(r - 1) + v2(r)vI(r -1)].
2.5.16
One can verify that (2.5.14) has two periodic solutions
VI (r)
= cos( 7r /2)r
2.5.18
so that (2.5.15) becomes 2.5.19
§2.5. An example
146
Since the right side of (2.5.19) satisfies the solvability conditions namely 2.5.20 (2.5.19) has a nontrivial solution in terms of trigonometric polynomials. Such a solution can be found by the method of undetermined coefficients which leads to
V2Cr) = (1/10)[sin(7rr) + 2cos(7rr)].
2.5.21
Equation (2.5.16) can be simplified to the form
dV3(r)
----;{:;:- + (7r /2)V3( r -
1) = -[p2
.
+ 0"2 (7r /2)] sm[( 7r /2)r]
- (7r 2/4)0"2 cos[(7r/2)r] - (7r/2) [cos {( 7r /2)r }V2( r - 1)
+ sin{s(7r/2)r}v2(r)]
2.5.22
(say).
2.5.23
The solvability conditions of (2.5.23) lead to two algebraic linear equations in 0"2 and '" given by
14
F(r, 0'2,/'2) sin[(,,-/2)r] = 0
14 F( r,
0'2,
/'2) Cos[(,,- /2)r] = O.
2.5.24
2.5.25
Explicit evaluation of the left sides of (2.5.24) and (2.5.25) and subsequent solution and 0"2 leads to of (2.5.24) and (2.5.25) for
"'2
P2
= [(37r -
2)/40].
2.5.26
We now have from 2.5.27 that €
~
40 ({rh _(7r/2)}--) 37r - 2
.1. 2
2.5.28
§2.5. An example
and hence v(r)~
147
40)! .
(
{rh-(1r/2)}31r_2
+ ~{rh 10
cos{(1r/2)r}
(1r/2)} (~){Sin1rr + 2cos1rr}
2.5.29
31r - 2
in which t
r=---
h(l + (7) t
2.5.30
~ h(l + [_1 ] [Th-1I/2 40) . 1071" 371"-2 J
Thus, an approximation to the bifurcating periodic solution, where rh (rh - 1r/2) is small, can be obtained in the form
u(t)
~ K[l
+ vCr)],
>
f
and
2.5.31
rand t being related by (2.5.30). The calculations leading to (2.5.29) and (2.5.30) can also be done by other methods, for example, by the centre manifold theory and by the method of averaging. The calculations based on centre manifold theory for (2.5.7) (see Hassard et al. [1981 J) leads to the same result obtained in (2.5.29) - (2.5.30). The method we have illustrated for (2.5.1) can be applied without any significant modification to integrodifferential equations also; for instance, we ask the reader to repeat the relevant calculations of (2.5.1) for the following integrodifferentia! equation 2.5.32 in which c is a positive constant and
t
~
o.
2.5.33
The bifurcation parameter in (2.5.32) is h ;?: O. One can verify that the steady state u(t) = c in (2.5.32) - (2.5.33) is locally asymptotically stable for 0 < h < 2;
§2.5. An example
148
for h > 2 and h - 2 small, there exists a bifurcating periodic solution which is locally aSYIIlPtotically stable. The details of the necessary computations are left to the reader. 2.6. Coupled oscillators Several authors have used coupled oscillators as mathematical models to explain and understand a variety of phenomena in biology, biochemistry and electronics. In biology, interacting and coupled oscillator systems have been used in modelling cellular behavior of pattern formation and circadian rhythms (Ashkenazi and Othmer [1978], Kawata and Suzuki [1980], Winfree [1980] and Kawata et aL [1982]). In physics, especially in mechanics, model systems of coupled oscillators are numerous. In chemistry, examples of oscillator systems are the Brusselator of Lefever and Prigogine [1969] and the reaction of Belousov - Zhabotinski (Field and Noyes [1974)). For a number of investigations of coupled oscillator systems and their applications we refer to Pavlidis [1973], Smale [1974], Cohen and Neu [1978], Howard [1979], Chandra and Scott [1983], Kuramoto [1984], Alexander [1986 a, b], Alexander and Auchmuty [1986], and Morita [1983-85, 1987-88]. The following discussion is based on Gopalsamy and Rai [1988]. In this section we examine the onset of synchronous (in-phase) oscillations and their stability in a coupled system of integrodifferential equations of the form
d~~t) dy(t)
-;It i( ex, x(·)) = T,
= x(t)[j(a,x(.))
+ p(y(t) -
x(t))] }
= y(t)[j(a,y(·))
+ p(x(t) -
yet))]
with
,,[1- ax(t) - b,,' [=(t - s
)e-a(t-,)
1;
x(s )ds
2.6.1
t > 0,
2.6.2
b, a, p are positive parameters while a is a nonnegative parameter.
It is possible to show (see Worz-Busekros [1978]) that for p = 0 and 0 < b < 8a, the system (2.6.1) - (2.6.2) is mathematically "dead" in the sense of Smale [1974]; that is (2.6.1) - (2.6.2) has a globally asymptotically stable uniform steady state. We show that for b > 8a, there exists a value of a = a* > 0 such that in a neighbourhood of a*, synchronous oscillations appear through a Hopf-bifurcation in the uncoupled system. We show furthermore that such synchronous oscillations are destabilized when the oscillators are "weakly" coupled if the parameter a is either zero or sufficiently small and positive.
§2.6. Coupled oscillators
149
An ecological interpretation of (2.6.1) is the following: x(t) and yet) denote the densities of a population of the same species in two identical patches (islands) and interpatch dispersal is allowed so as to reduce crowding on each island; it is assumed that the transit is instantaneous. There are, of course, several other ways of modelling interpatch dispersal p.rocesses (Gopalsamy [1983c]). Let us first consider the scalar integrodifferential equation
d~;t) =
ru(t)
[1 - au(t) - 00' /.= se-a'u(t - s)d+
t>0
2.6.3
together with an initial condition of the form
u(s) =
sE(-oo,O);
cp(O»O
2.6.4
where
dv(t) - = -arx*[ev(t) -1]
&
1
00
-rbx* a 2
se-as(ev(t-s) -l)ds
2.6.6
0
which we rewrite compactly as follows
dv(t) d j = F(a,v(-»;
t>
°
2.6.7
where
F(a, v(·)) =
1°= [- arx'6(O) + brx'a'Oeaoj [e·(HO) -
IjdO,
2.6.8
b( 8) denoting the "Dirac" delta ftulction. The steady state x* of (2.6.3) is asymptotically stable if and only if the trivial solution of (2.6.7) - (2.6.8), vet) == is
°
§2.6. Coupled oscillators
150
asymptotically stable. The linear variational system corresponding to the trivial steady state v == 0 of (2.6.7) - (2.6.8) is
d~~t)
=
1°= [-
arx*6(B)
+ brx*",2 Be06] z(t + 8)d8.
2.6.9
The characteristic equation associated with (2.6.9) for a -> 0 is 2.6.10 which we write as follows 2.6.11
It can be verified that the Routh-Hurwitz condition for all the roots of (2.6.11) to have negative real parts fails if
and this is equivalent to 2.6.12 where
:: } = ~x* [b -
4a
± ~b(b -
2.6.13
8a)].
Thus, if b 2:: 8a, the steady state v == 0 of (2.6.6) is unstable for a E [a*, a*]. For a = a*, the roots ).1,).2,).3 of (2.6.11) are given by say.
2.6.14
By direct verification, it is found that 2.6.15 It follows from the above discussion of the characteristic equation that the sufficient
conditions of "Hopf-bifurcation theorem" are satisfied and hence there exists a periodic solution of (2.6.3) with period near 27r jwo when a is near a*. We shall perform a stability analysis of the bifurcating periodic solution (2.6.3).
§2.6. Coupled oscillators We note that if
151
0 in (2.6.9), the associated characteristic equation is
a =
2.6.16 with bx* = 1 and for a = 0'* = r /2, (2.6.16) has a pair of pure imaginary roots ±wo , Wo = (1'/2) while the remaining root is negative. It can be verified that for ,\ in (2.6.16), one has
31e(~~)
<
0 for
,,= ",_
2.6.17
As before, when a = 0 a periodic solution of (2.6.3) appears by means of a Hopfbifurcation when a is near 0'*. The Hopf-bifurcation of an "in phase" periodic solution of the coupled system (2.6.1) is shown as follows: (see also Landman [1980]) we introduce a change of the variable t in (2.6.2) and (2.6.3) by the relation s = wi and let v( t) = v( s /w) = xC s ) where w is a real number such that 27r /w is the unknown period of the bifurcating periodic solution of (2.6.3). In terms of x, the system (2.6.3) becomes
dx(s)
W
d;- = F(a,x w ('))
2.6.18
where 2.6.19 We look for a periodic solution of (2.6.18) of period 27r in s such that
+ C2Y2(S) + C3Y3(S) + ... = W(E) = Wo +WIC +W2C2 + .. . = a(E) = 0'* + alC + a2E2 + .. .
xes) = CYl(S) w a
2.6.20 2.6.21 2.6.22
where € is a perturbation parameter and Yi (i = 1,2,3, ... ) are periodic in s of period 27r satisfying the orthogonality conditions
1 2
•
Yl(S)Yj(s)ds = 0;
j
= 2,3, ...
2.6.23
Supplying (2.6.20) - (2.6.22) in (2.6.18) and expanding Fin (2.6.18) around (Q*, 0) we have
§2.6. Coupled oscillators
152
[Wo
] [d Yl 2 dY2 3 dY3 1 + W1€+W2€ 2 +... € ds + € ds + € ds + .. . = F( a*, 0) + Fa(a*, 0)( a1 € + a2€2 + ... ) + Fx(a*,OI€Yl,w(') + €2 y2 ,w(') + €3 y3 ,w(') + ... ) 1 2 2 +Faa (a*,O)"2(a 1 €+a 2€ + ... )
1
2
+ "2Fxx(a*,OI€Yl,W(') + € Y2,w(') 1
+ "2Fxa(a*, 01€Yl,w(')
+
2 €
Y2,w(')
+ .. ·1€YI,w(·) + € 2 Y2,w(') + ... ) + .. .)(al€ + a2€ 2 + ... )
1
+ 6Fxxx(a*, 01€Yl,w(') + .. ·1€Yl,w(·) + .. ·1€Yl,w(·) ... )
+...
2.6.24
where Fx(a*,Olu) denotes the Frechet derivative of F(a*,x(·») with respect to x at x = 0 in the direction of u; higher order Frechet derivatives are respectively denoted by Fxx(a*,Olulv) and Fxxx(a*,Olulvlw). Comparing the coefficients of similar powers of € in (2.6.24) we have
dYI (s) Wo ~
= Fx(ax,OIYl,wo('»
2.6.25
dY2( s) Wo ~ = Fx(a x, 0IY2,wo(')) -WI
dYI
1
ds + '2 Fxx (a.,OIYl,w o(·)IYl,W o(·»
+ Fax(a.,OIYl,wo(·)aI + Fx( C¥*, 01· Y;,wo ('»wl'
2.6.26
We consider a function space P21r of complex valued periodic functions of period 27r defined on (-00,00) in which a scalar product is defined by (u, V h1r where
(u, V)21r
= -27r1 J.21r u( s )v( s) ds. 0
2.6.27
We seek real numbers WI, al, W2, a2, . .. and real valued 27r-periodic functions YI, Y2, Y3,· .. satisfying (2.6.25) and (2.6.26) such that 2.6.28
15S
§2.6. Coupled oscillators It is easy to see that
2.6.29 is a solution of (2.6.25) where (1 is any fixed number such that (1 (1 = 1 and for such a choice of Yl, the solvability condition for (2.6.26) leads to al = O,Wl = and hence (2.6.26) simplifies as follows:
°
2.6.30 We choose a solution of (2.6.30) in the form with (2
= 12
(2 F (a O/eiwo·/eiwo·)
1 xx· *, . 2iwo - Fx(a*,0/e 2•wo ·)
7]=-(1(1'
2.6.31 2.6.32 2.6.33
The governing equation for Y3 is obtained from (2.6.24) in the form 2.6.34 where
dY1 I () H(s)=-w2Ts+w2Fx(a*,OI'Y1,wo .) + Fxx( a*, 0/Y1,wo(')/Y2,wo( .» + a2 Fax(a*, O/Y1,wo('» 1
+ 6Fxxx( a*, 0lY1,wo(' )/Y1,wo(' )/Yl,wo (.».
2.6.35
The solvability condition for (2.6.34) is 2.6.36 where
Ce is
is any 27r-periodic solution of the adjoint equation associated with 2.6.37
§2.6. Coupled oscillators
154
The condition (2.6.36) simplifies to
iW2[1- Fx(G*,OI· e iwQ ')](l{ - G2 Fax(G*,0Ie iwQ ')(1{* = Fxx( G*, 0Ie-iwQ'le2iwQ')(2(1 {*
+ Fxx( G*, 0!e iwQ '!n)(1 {* + ~Fxxx( G*, O!e-iWQ'leiwQ'!eiwQ')(;Cl{*'
2.6.38
The real numbers W2 and G2 are detennined from (2.6.38). We shall examine (2.6.38) more closely; we choose the arbitrary nonzero eigenvectors (1 and as follows: 2.6.39
e
By the standard methods of bifurcation theory, one can derive (see Sattinger [1973]) that 2.6.40 and hence (2.6.38) can be simplified to
iW2 -
"2(~~t = A
where A denotes the right side of (2.6.38). For a = simplifies to iW2 -
"2
2.6.41
°it is found that (2.6.41)
(~~) o. = -wo(3 + 11i)/60
2.6.42
which implies 2.6.43 It follows from (2.6.43) that the bifurcating periodic solution of (2.6.18) is asymptotically stable. By continuity of both sides of (2.6.38) on a, (2.6.43) holds for sufficiently small positive a. Thus, (2.6.43) holds also for b > 8a and small positive a. It follows from the above that when f1- = 0, the bifurcating periodic solutions of (2.6.1) are asymptotically stable. It is not obvious that when f1- I- (that is when the oscillators are coupled), such synchronous or in-phase oscillations are stable for the coupled system (2.6.1) with f1- > 0. We shall examine the stability of the synchronous oscillations of the system (2.6.1) when the coupling is weak
°
§2.6. Coupled oscillators in the sense p, = O( e2 ) where uncoupled system. We let
X(t)
lei
155
denotes the amplitude of the oscillations of the
= log[x(t)/x*J;
Yet) = log[y(t)/x*J;
s =wt
2.6.44
in (2.6.1) and derive that
2.6.45
where 2.6.46 Let p(., e) denote the 27r-periodic solution of
dx(s)
w ~ =F(a, x w ('»
2.6.4 7
bifurcating near a* when b > 8a. We have seen that p(-, e) is an asymptotically stable solution of (2.6.47). One can see that the pair (p(., e),p(', e» is a 27r-periodic solution of the coupled system (2.6.45) and this pair denotes the in-phase or synchronous oscillations of the coupled system (2.6.45) when p, =1= O. It is not known whether this in-phase solution is asymptotically stable for (2.6.45) with p, =1= 0; we examine this aspect in the following. The linear variational system associated with (2.6.45) and the in-phase solution (p(., e),p(" e» is obtained as follows: we let
x (s) = p( s, e) + u( s, e) Yes) = pes, e) + v(s, e) in (2.6.46) and derive after neglecting the nonlinear terms in the perturbations
u,v w
~:
= Fx(a,pw(-)Iu w('»
+ p,x*eP(S,€) {v(s, e) -
w
~:
= Fx(a,pw(')lv w('»
+ p,x*eP(s,€)
u(s, e)} } 2.6.48
{u(s, e) - v(s, e)}.
§2.6. Coupled oscillators
156
The linear system (2.6.48) is periodic with period 27r and we are interested in an analysis of the nature of the Floquet exponents associated with (2.6.48). This analysis can be simplified if we let
U(s, €) = u(s, €)
+ v(s,
e)}
V(s, €) = u(s, €) - v(s, e)
2.6.49
in (2.6.48), and note that U, V are governed by dU
W
Ts
W
ds = Fx(a,pw(')IVw('» -
2.6.50
= Fx(a,pw(')IUw('»
dV
2J1x*eP (s,€)V(s, f).
2.6.51
We have already seen that the in-phase solution of the uncoupled system is asymptotically stable; this will mean that if € is small, one of the Floquet exponents of (2.6.50) is zero while the other is negative. Thus, we are left with an investigation of the Floquet exponents of (2.6.51) and for this we seek a solution of (2.6.51) in the form V(s, €) = Q(8, €)e us / w 2.6.52 where Q is 27r-periodic in sand 0" = 0"(J1) is a Floquet exponent of (2.6.51) such that 0"(J1) ~ 0 or (3( €) as J1 ~ 0+,
(3( €) being the negative exponent of (2.6.50). It is found from (2.6.51) and (2.6.52) that W
~~
= - (0"
+ 2J1x*eP (s'€)Q(s, €) + Fx( a,pw(' )leu'Qw(-, €».
2.6.53
In general, (2.6.53) may not have 27r-periodic solutions and we want to find those real numbers 0" depending on J1 for which (2.6.53) will have 27r-periodic solutions. We regard (2.6.53) as a perturbation of (2.6.51) due to the coupling. Assuming that the coupling is weak (i.e. J1 is small such that J1 = D( (.2», we shall calculate 0" as follows: we let as before
+ a2(.2 + .. . Wo + W2€2 + .. .
a = a* W
=
J1
= (J12/2x*)€2 + .. .
2.6.54
§2.6. Coupled oscillators
157
in (2.6.53) and note that a2 denotes a perturbation of the otherwise zero Floquet exponent. Thus, we are led to the determination of real numbers a and 27r-periodic solutions Qo, Q1, Q2, ... so that (2.6.54) can solve (2.6.53) with Qo :t O. We assume that our perturbation expansions can be justified by an appropriate application of the implicit function theroem. Supplying (2.6.54) in (2.6.53) and comparing the coefficients of the respective powers of € we derive, 2.6.55
2.6.56 2.6.57
in which G(s) = -W2
dQo
ds -
(a2
+ J-L2)QO(S)
+ Fx(cx*, 01· W2Q~,wo(') +. a2Qo,wo('»
+ CX2Fax(cx*,0IQo,wo('» + Fxx(cx*, °IYl,wo(·)IQI,wo(·» + Fxx(cx*, °IY2,wo(·)/Qo,wo(·» 1
+ 2Fxxx( CX*, 0IYl,wo(- )IYl,wo(' )IQo,wo('»'
2.6.58
We choose solutions of (2.6.55) and (2.6.56) in the form
Qo(s) = K(le is Q1(S) = 2K(2e2is
+ K(le- is
+ 2K(2e-2is + (K + K)TJ
2.6.59 2.6.60
where
+ (Ie-is = (2e2is + (2 e- 2is + TJ,
YI(S) = (leis Y2(S)
K being a nonzero (complex) constant. The solvability condition for (2.6.57) is of the fonn
(G(s),e*e"h, =
2~ [ ' G(s){'e-"ds = 0
2.6.61
§2.6. Coupled oscillators
158
where ~* eis is a 27r-periodic solution of the adjoint equation associated with (2.6.55). A simplification of (2.6.61) leads to
- w2i{1 - Fx( CY., 01· eiWQ ')}K(l{* - (Td1 - Fx(CY*, 01· eiWQ ')}K(l{*
- J.-L2K(1~*
+ CY2Fax(CY*, 0Ieiwo')K(1~* + Fxx(cy*,0Ieiwo'11)K(117{* + 2Fxx( CY*, 0Ieiwo'le2iwQ')(1(2{* K + Fxx( CY*, Ollie iwo' )17(1 ~* K + Fxxx( CY*, 0IeiWO'le-iwO'leiWQ')(1(1(1~* K + Fxx( CY*, 0le iWQ 'll )(117~* K + Fxx( CY*, 0Ie2iwo'leiwo')(2(1~* K
= 0.
2.6.62
+ ~Fxxx( CY*, 0leiwO'leiwO'le-iWQ')(l (1(1(* K Simplifying (2.6.62) further and using (2.6.38), we obtain
with A
= Fxx( CY*, 0le-iwo'le2iwQ')(1 (2~* + Fxx( CY*, 0Ieiwo'11)(117~* + ~Fxxx( CY*, Oleiwo'leiwo'le-iwo')(;(l~*'
2.6.64
A necessary and sufficient condition for the existence of a nonzero K in (2.6.63) is the following: (TiLB(1~*12 - (T2(2~e[A - J.-L2(1~*)j3(1C)
+ J.-L~1(1{*12
- 2J.-L2~e(A(1C) =
°
2.6.65
where 2.6.66 By our choice of (1 and a = 0,
C, we have for
!IIe[A(,Cl = !lie [ -
a
= 0, C = 1/(2 - i) and (1 = 1; hence for
~~ (3+ 11i)/(2 -
i)] > O.
2.6.67
§2.6. Coupled oscillators
159
It follows from (2.6.65) and (2.6.67) that if J1.z is sufficiently small (that is, if the coupling is "weak"), then the two roots of the quadratic eq:uation (2.6.65) are real and are of opposite signs. This implies that the bifurcating "synchronous" (or inphase) oscillations of the coupled system are destabilized by the "weak" difference coupling as in (2.6.1). By continuity arguments, the inequality (2.6.67) holds also for small positive a with b - 8a > O. One of the interpretations of the above result is the following: if two oscillating populations in two identical patches with stable oscillations are subjected to difference coupling with a "small" coupling coefficient, then such a coupled system cannot neutralise inhomogeneous interpatch differences in spite of the coupling as t ~ 00. If the coupling is not "weak", then instability need not arise; this aspect requires further investigation.
160
EXERCISES II
1. In the delay logistic equation
dN(t)
&
=
r
N(t)
[1 _N (tK- 2 r)]' 2
let rand K be positive constants and r be the delay parameter. Find the value of the delay for which the steady state N (t) == K becomes unstable and discuss the delay induced bifurcation to periodicity. Calculate an approximation to the bifurcating periodic solution. 2. Consider the integrodifferential equation i
= 1,2,
where r and K are positive constants; (i) K1(s) = f e- sr (ii) K2(S) = ~ se- sr If r is a positive valued parameter, determine for what values of r stable oscillatory solutions can exist. Compute an approximation of the periodic solution. 3. In the integrodifferential equation
d~~t) = x(t+ assume that
€,
ax(t) - ')'
l=
F(t - s)x(s) dS]
a, , E (0,00) and F(t) = a 2 te- at , a E (0,00). Prove that I
< 8a => xCi)
~
€
--
a+,
as
t
~ 00.
4. Show that the time delayed two species competition model
dx(t) = x(t) { rl ---;It
-
allx(t - r) - a12y(t) }
dy(t) = yet) { r2 ---;It
-
a21x(t) - a22y(t - r) }
Exercises II
161
where Tj, aij (i,j = 1,2) are positive constants has a stable nontrivial steady state (x*,y*), x* > 0, y* > 0 if all
rl
a12
a21
r2
a22
->->for small T > 0; determine the value of oscillations can arise if that is possible.
T
at which bifurcation to stable
5. Consider a competition model with continuously distributed delays
jt
dx(t) -;It
= x(t) { Tl
- a11
dy(t) -;jt
= yet) { TZ
- aZlx(t) - aZ2y(t) } .
-00
Rr (t - s )x( s )ds - alzy(t) }
Assume that the interaction coefficients T i, aij (i, j = 1, 2) satisfy the conditions of problem 4. Prove or disprove the following: (a) if K (s) = ~ e -sr, then for all T > 0 the two competing species can coexist (i.e. positive equilibrium is asymptotically stable). (b) if K(s) = ~se-sr, then the nonoscillatory coexistence is lost and an oscillatory coexistence arises for a suitable value of T. 6. Obtain sufficient conditions on the positive constants T, K, a, b, j3 for the existence of a stable nontrivial steady state in the Herbivore-Carnivore model
T)} _o:H(t)C(t)
dH(t) dt
= rH(t){l _
d~;t)
= C(t){ _ b + j3H(t)}
H(t K
and show that for a suitable value of T, the steady state becomes unstable leading to stable oscillations. Do the same with H (t - T) replaced by sUPsE[t-r,t]
H(s).
7. Examine the existence of delay induced stable periodic oscillations in the following prey-predator system
d~~t) = rx(t){ 1 _ x~) } dy(t) at = /3x(t -
- lXX(t)y(t)
T)y(t - T) - by(t)
Exercises II
162
in which r,a,/3,b are positive constants and
is the delay parameter.
7
8. Discuss the delay induced bifurcation characteristics of the multiplicative delay logistic equation
9. If r, a, b, /3, 7 are positive parameters show that for a suitable value of integrodifferential system
d~;t) = r N(t) [1 d~~t)
= -bP(t)
1 K(
7
the
00
s )N(t - s )dS]- aP(t)N(t)
+ /3P(t)N(t)
with
K(s)
= ~se-8/T s > 72 ,-
°
has a delay induced bifurcation to stable oscillations. Examine the stability of the periodic solution. 10. Examine whether delay induced bifurcation to periodic oscillations can arise in the harvesting models
dN(t) - = rN(t dt
d~?) dN(t) dt
r
2
7) - -N (t) - exN(t) ]{
(i)
= rN(t)
[1-
r)]_ H
(ii)
= rN(t)
[K+-eNN(t(t -- 7)]H 7)
(iii)
N(t;
1
in which f, K, ex, 7, H are positive constants; assume r > a and show the existence of nonnegative solutions for t > if N(s) > for s E [-7,0].
°
°
11. Prove that for a suitable value of the delay parameter 7, a delay induced bifurcation to stable oscillations can occur in the following population model
dN(t)
-;It
=
B[a -,N(t - 7)]N(t -
7) - /3N(t),
163
Exercises II
8,a,{3" being positive constants (here 7 corresponds to a maturation delay in reaching reproductive capacity); assume N(s) > a/, for s E [-7,0]. 12. Discuss the delay induced bifurcation of periodic solutions in the scalar system
dx(t)
-;It = -ax(t) + bexp{ -cx(t - 7)} where a, b, c, 7 are positive parameters. Discuss the local stability of the bifurcating periodic solution if such a solution exists. Can you generalize your analysis to an equation of the form d (t)
:t
n
= -ax(t)
+ ?=exP{Cjx(t -
Ij)}
}=1
having a, bj , Cj, Ij, (j = 1,2, ... ,n) as positive parameters? What can you say about an integrodifferential equation of the form
dx(t) -dt
= -ax(t) +
jt
k(t -s)exp{-bx(s)}ds
-00
where k is a suitable delay kernel and
I
E [O,oo)?
13. Discuss the existence of delay induced oscillations (periodic solutions) in the following scalar equations: (i) d~~t)
= ax(t-/)exp[-bx(t-/)]-cx(t)
(a,b,c,1 are positive parameters).
(ii) d~~t) = -cx(t) + Ej=l ajx(t - 7j)exp[-bj x(t - 7j)] 1, 2, ... n, are positive parameters).
(aj,bj,lj,c; j
=
(iii) d~~t) = -cx(t) + J~oo k(t - s)x 2 (s)exp[-cx(s)]ds. (k is a suitable nonnegative delay kernel and c is a positive constant). 14. Discuss the possibility of Hopf-type bifurcation in the following:
(1)
d~~t) = -[(11'/2) + J.l]x(t - 1)[1
(2)
d~~t) =-(3:!J+u)[x(t-1)+x(t-2)]{1-x(t)}.
(3)
d~~t)
= -,[I + x(t)] J~oo k(t -
+ x(t)]
s)x(s)ds.
Exerci3e3 II
164
= ,x(t){l- (x(t;r))B}.
(4)
d~~t)
(5)
d~~t) = o-x(t - 7)[1- x 2(t)].
(6)
d~~t)
(7)
(It
(8)
d:~~t) + (a/)..)d~~t) + (b/)..)sin[x(t - )..)]
(9)
d~~t)
dx(t)
= -ax(t) = -ax
(t)
bx(t - 7) + cx 2(t - 7)
(11)
dx(t) dt --
rx (i - 71 )
(12)
dx( t) (It
=
[
(13)
dx(t)
= [
(14)
8 dx(t) - [f3 x(t)X n{t-T) (t)]. dt f3+xn(t-r) ,x ,
(15)
dx(t) -
(16)
dx(t)
dt
dt
x(t)J2 [ J~oo x(t + 8)k(8)dB)]
[1 -
r
f3 a+xn(t-T) -,x(t)
PX(t-T)
l+xn(t-r)
- rx
-
'
= O.
X(t-r,)] K'
;
8 = 1,3,5, .. etc.
X(i)j'-
,
(i) [ l+cx(t-T) K -X(t-T)
r. ,
j.
= Tx(i) [ K -x( t-m ) l+cx t-m) ,
(17)
15. Let
= O.
+ a[1 + x(t)][ J~oo x(t + 8)k(8)d8] = O.
d~~t) + a[1 -
(18)
7).
btX(t-T)+ b+b(x(t-T»n .
(10)
dt
+ dx 3(t -
x(t) = SUPSE[t-T,tj xes); dx(t) _
dt
- rx
(i) [ K -x(At)
j.
l+cx(>'t) '
0<)..<1.
dXI at = all (t)Xl + alZ(t)x2 dX2 at = a21(t)x2 + a22(t)x2
165
Exercises II
be the variational system associated with an asymptotically stable periodic solution of a system of class C 2 ,
dx dt = h(t, x). IT a > 0 exists such that I h(t, x) Is
al x I, then prove that
the system
with DI > 0, D2 > 0 is asymptotically stable for a < min(Dl ,D2). Prove also that if I Dl - D2 I is sufficiently small, then the trivial solution of the linear system in YI, Y2 is asymptotically stable. Discuss also the asymptotic behavior of
16. Let
dXl
~ =/(Xl,X2,p)
dX2
~ =
g(XI,X2,p).
IT I, 9 E C 3 (R 3 ) and the system has an asymptotically stable periodic solution (Xl(t),X2(t)), then prove that the system ~
dXl
= I(x}, X2, p) + D 1(X3 -
xt)
dX2
= g(Xl,X2,p) + D 2(X4 -
X2)
~
dX3
dt
=
I(X3, X4, p) + DI(XI - X3)
~ =
g(X3, X4, p) + D Z(X2 - X4)
dX4
with Dl > 0, D2 > 0 has an asymptotically stable periodic solution if mine D I , D 2 ) < a or I Dl - Dzi < (3 where a, (3 are suitable constants. Prove
Exercises II
166 also that the system dXl
dt
= f(Xl,X2,1-£1)
+ D l (X3
- xI)
dX2
dt = g(Xl,X2,1-£1) + D2(X4 -
X2)
dX3 dt
X3 )
dX4
dt
= f(X3,- X4, 1-£2 )
+ Dl ( Xl
-
= g(X3,X4,1-£2)
+ D 2(X2
- X4)
I 1-£2 -
<8
has an asymptotically stable solution if 11-£1 - 1-£1
<8
,
1-£1
where 8 is a suitable constant.
Torre [1975]
17. Discuss the bifurcation of a stable periodic solution in the system
(where F(t)
= ae- at , a E (0,00)) by converting the system to dYl
dt = Yl(rl dY2
- alYl - a2Y2)
dt = Y2( -r2 dY3
dt = aCYl
b1 Y2
+ b2Y3)
- Y3).
18. Examine the possibility of delay induced bifurcation to periodicity in the following models of two species competition; also discuss the stability of the bifurcating periodic solution; i,j = 1,2, i =f j.
(1)
(2)
dN-(t) ' - = (r'N.(t)) _'_I dt Ki _
1
[ K. -N.(t)-OI. .. N.(t-j.) 1 1 1)) ).
Exercis es II
(3)
(4)
(9)
dNi(t) _ (riNi(t)) 1/2 & ~
-- -
dd?)
167
1
[K~/2 _ N~/2(t) _ CtjjNj1/2 (t - Tj ) . I
I
Kl/2 i
= (rii?») [Ki-Ni(t)-aijNj(t-Tj)-,BijNi(t-Ti)Nj(t-Tj)].
dNi(t) = (riNi(t)) [KiNi _ cxijNj.(t - Tj) dt Ki
- ,BijNi(t - Ti)Nj(t - Tj) - OiNl(t)].
(10)
dNi(t) = (riNi(t)) [Ki _ Ni(t) - cxijNj(t) dt Ki
- !3ij N i(t)Nj(t) - 8i NlCt - Ti)].
(11)
dNi(t) = (riNi(t)) [Ki _ NiCt) - cxijNj(t - Tj) dt Ki
- ,BijNi(t)Nj(t) - oiNl(t) - -yjNJ(t - Tj)].
Exercises II
168
(12) 19. Investigate the occurrence of bifurcation to periodicity in the following models of competition and cooperation (assume 81, 82 = 1,3,5, .. etc.)
20. Discuss the occurrence of Hopf-type bifurcation to periodicity in the following predation models.
d~y)
= rH(t)
dP(t)
---;It
d~?) dP(t)
---;It
[1 - H(t; r)]_ aH(t)P(t) I
= -bP(t) + f3H(t - 1')P(t - 1').
= rH(t)
[1-
= -bP(t) + f3H(t -
d~it) = rH(t) [1 - H(t) - ~ dP(t)
---;It
= -bP(t)
H(t;; rd]_ aH(t - r.)P(t)
+ {3
it
-00
l=
I
(1)
(2)
1'J)P(t - 1'1)'
G 1 (t - s)H(s)P(s)ds -
aH(t)P(t)]j
G2 (t - s)H(s)P(s)ds. (3)
Exercises II
d~?)
169
[1 _ Hit)] _ aP(t) [1 _ d~;t) = -bP(t) + fJP(t _ r) [1= rH(t)
e-CH(t)] )
(4)
e-CH(t-T)] .
[1 _H(t - r)] _ aP(t)H(t) )
dH(t) = rH(t) dt .
K
dP dt = bP(t) [1 -
P}
fJ
+ H ( t)
(5)
fJH(t _ r)"
(for details of the above models, the reader is referred to May [1973]).
d~?)
= H(t) [a _ bHo-1(t)P(t)]
d~~t) = pet) [-c + dH o- 1 (t = H(t)c
dH(t) dt
1
} (6)
r)P(t - r)] .
[1 _H(t)]_ aH(t)P(t) K b+ H(t)
d~;t) = c P(t) [1- P(t) l~ ~;~(S;) dS]
(7)
2
F(t)
= e- t
dN1 (t) = dt
N
r 1
;
F(t)
dXl(t)
•
[1 _NKl(t)] _ aN1(t)Nz(t) fJ+Nl(t) 1
1
dNz(t) _ N [ - d- 2 -r2 t ~ =
= te- t
+
)
kaN1(t - r) N ( )] Q N ( ) -, 2 t . . . . + It-r
(8)
Xl(t)(al - b1Xl(t - r) - CIX2(t))
dX2(t) -d- = X2(t)( -a2 - b2x2(t) + c2 t F(t) = ae- at •
d~~t)
1t_=
F(t - S)Xl(S) ds)
= x(t)g(x(t» _ ym(t _ r)p(x(t» }
dy(t)
-;It = -cy(t) + cym(t)p(x(t - r». dx(t) = rX(t)(l- X(t») _ mxn(t)y(t - r) ) dt K a + xn(t) dy(t) dt
(9)
= [c
mxn(t - r) a + xn(t - r)
n] yet).
(10)
(11)
Exercises II
170
(12)
21. Establish the occurrence or nonoccurrence of coupling induced instability of synchronous oscillations in the following coupled systems.
d~~t) = rx(t) [1 - x(t; r)] + 11 (yn(t) _ xn(t» d~~t)
= ry(t)
)
[1 - yet; r)] + 11 (xn(t) _ yn(t».
[1- x(t;;r)] +,,(y(t)-x(t))" ) d~~t) = ry(t) [1- yet;; r)] + ,,(x (i) _ yet))". d~~t)
d~~t) dy(t)
dt
=rx(t)
= x(t) [a + bx(t - r) - cx (t - r)] 2
= yet) [a
d~~t)
+ bx(t -
. r) - cx 2 (t - r)]
= x(t) [a - bx(t - r)]
+ 11
+" (x(t) -
+ 11 (x(t) - y(t».
[ey(t) - eX(t)]
~t
.
= x(t) [a - bx(t - r)]
dy(t)
dt = y(t)[a -
d~~t) = d ()
x(t) [[a - bx(t - r)]
~/ = yet)
[
[a - by(t - r)]
by(t - r)]
(2)
yet)) } (3)
}
dy(t) = yet) [a - by(t - r)] + 11 [eX(t) _ ey(t)] dt . d (t) , ~t = x(t) [a - bx(t - r)] + 11 [logy(t) -logx(t)] } dy(t) dt = yet) [a - bx(t - r)] + 11 ~og x(t) -log yet)] . d (t)
(1)
+ 11 [yet -
r) - x(t)] }
+ 11 [x(t -
r) - yet)] .
(4)
(5)
(6)
+ ,,{a(y(t) - x(t)) + f3(y([t]) - x(t)}] )
+" {a(x(i) -
y(i)) + f3(x([i]) - yet)} ].
(7)
Exerci8e8 II
dN1(t) -;u= N1(t) [rl dN2 (t) -;u=
171
allN1(t - T) - a12 N 2(t - T)]
. N2(t) [r2 - a21 N l(t - r) - a22 N 2(t - T)]
+ J1.1 [Xl(t) - N 1(t)J
+ J1.2 [X2(t) -
. N2(t)]
dXl (t) --;It" = Xl(t) [rl - anXl(t - T) - a12 x 2(t - T)] + J1.1 [Nl(t) - Xl(t)] dX2( t)
--;It"
= X2(t) [r2 -
. a21 x l(t - T) - a22 x 2(t - T)]
dx(t) = rx(t) dt . dy(t) = ry(t) dt x(t) = sup
[1 _X(t)] + J1. (yn(t) _ xn(t» K [1 _yet)] + J1. (xn(t) _ yn(t» K xes)
sE[t-r,t]
dx(t) dt
d~~t)
+ J1.2 [N2(t) - X2(t)]
,
yet) =
sup
(9)
yes).
sE[t-r,t]
= rx(t)
[1 - X(.\t)] + J1. [yn(t) _ xn(t)] K
= ry(t)
[1 _y~t)] + J1. [xn(t) _ yn(t)]
0<.\<1.
(8)
(10)
CHAPTER 3
METHODS OF LINEAR ANALYSIS
3.1. Preliminary remarks In this chapter we first introduce the fundamental types of ecological associations such as predation, competition, cooperation (or mutualism) and then present several methods for studying the dynamical characteristics of linear systems. We begin with some remarks on models of single species dynamics. Most of the differential equation models of population dynamics have been derived starting from the following simple format
dN(t) = { an indivi~ual's contri~utio~. } N(t) dt to population change III umt time
3.1.1
where N (t) denotes the density of a population (or biomass) of a single species at time t. Subsequently, one makes an assumption regarding the factor inside the brackets in (3.1.1). In particular, if one assumes that an individual's contribution to the change in population in unit time is denoted by a function say f(t, N) defined suitably for all t > 0, N 2 0, then one obtains from (3.1.1) the so called Kolmogorov formulation in the form
3.1.2 Various choices of f together with some ecologically plausible assumptions such as the temporal constancy of the environment and density dependent effects in f lead to several well known ordinary differential equations of population dynamics. For instance, if f(t, N) == r (a positive constant) one obtains the Malthusian formulation dN(t) = r N(t)
dt and if one assumes f( t, N) == r - (r / K)N for some positive constants r and K, one gets the familiar logistic equation
dN(t) = rN(t){l _ N(t)} dt K .
3.1.3
Since the above logistic equation implies monotonic approach as t - t 00 of the population density to the steady state N(t) == K, it has been desirable to look for
173
§9.1. Preliminary remarks
modifications of (3.1.3) in order to have fluctuating (nonmonotonic) solutions of model equations. IT one assumes
J(t, N) == r for some constant equation
1"
- ( ; )N(t
- r)
> 0, then (3.1.2) leads to Hutchinson's [1948) delay-logistic
= rN(t){l _
dN(t) dt
N(t K
r)}
3.1.4
which we have studied in Chapter 1. Instead of a single discrete delay as in (3.1.4), one can also assume
r
J(t,N) = r - I{N(t) -
t H(t i-oo
s)F(N(s»ds
3.1.5
where H is a suitable nonnegative scalar function and F is a nonnegative function of Nj when (3.1.5) is used in (3.1.2), we get Volterra's model of a population which pollutes its environment and the pollution itself has accumulative toxic effect; the resulting equation is of the form
d~it)
= N(t){r -(r/K)N(t)
-l=
H(t - S)F(N(S))dS}
which has been studied by a number of authors under various hypotheses on H and F (see for instance Cushing [1977]). One of the significant advantages of deriving various models from the prototype (3.1.2) is the following; when the initial conditions (initial population densities) are nonnegative, the nonnegativity of the population density N (t) for t > 0 follows from the fact, that any solution of (3.1.2) satisfies
N(t) = N(O) exp
[J.'
frs, N(s)) dS].
Our starting point to derive model equations with delays in production and destruction is the following balance equation, assuming there is no immigration or emigration;
d~~ t) = birth rate _
death rate.
3.1.6
For instance, if we consider a population of adult flies then the production or recruitment of adult flies, at time t depends on the population of adults at time
§9.1. Preliminary remarks
114
t-
where T is the time required for the larvae to become adults. If the birth and death rates are governed by density dependent factors, then we have from (3.1.6) T
3.1.7 where the functions b(·) and m(·) denote density dependent production (recruitment) and destruction (elimination or death) rates respectively. If the time delay in (3.1.7) is continuously distributed, then one can consider instead of (3.1.7) an equation of the form
or equivalently 3.1.8 in which the delay kernel H denotes a distribution of the intensity of the past or hereditary effects on the current birth rate. It is not obvious that suitable nonnegative initial conditions for (3.1.7) and (3.1.8) will imply the nonnegativity of solutions of (3.1.7) and (3.1.8). We show that in a class of population systems, delays in production (recruitment or birth rate) and destruction (consumption by predation or death rate) do not destabilize the systems if the self-regulating or resource-limiting negative feedback effects are sufficiently strong compared to the interspecific interaction effects and if the self-regulating negative feedback effects are realised with no time delays or with sufficiently small time delays. This point will be elaborated in section 4.4 of the next Chapter. When we say that time delays do no destabilize, we mean that delays do not render an otherwise (locally) asymptotically stable steady state unstable in such a manner, that the loss of stability either leads to a delay induced bifurcation to persistent and undamped oscillations (as in Chapter 2) or the relevant linear variational system has unbounded solutions; that is, instability in the sense of mathematical stability theory is not induced by the delays; in other words, the delays do not push the roots of the characteristic equation to the imaginary axis or to the right of that axis on the complex plane. However, delays can make an otherwise stable system less stable by which we mean that the rate of decay of perturbations can decrease with an increase in the values of delay parameters and
175
§9.1. Preliminary remarks
a system, which has been otherwise nonoscillatory can become oscillatory (perhaps violently) before converging to the steady state. Such a reduction in stability can happen even when all the roots of the characteristic equation associated with the linear variational system has negative real parts. For our purposes, such a reduced stability is still stability. In order to see the effects of delays on such less stable systems, we propose a method of estimating the rate of decay of local perturbations based on the corresponding variational system and the associated characteristic equation.
3.2. Delays in production We consider the dynamics of a single species population described by the logistic equation in which the birth rate depends on the entire past history of the population. In particular, we consider the integrodifferential equation
dx(t) -d-
t
=a
jt
-00
K(t - s)x(s) ds - bx 2 (t);
where a, b are positive constants and K : [0,00] on [0,00) and is normalised satisfying
/.00 K(s) ds = 1;
1-+
t>O
3.2.1
[0,00) is piecewise continuous
/.00 sK( s) < 00.
3.2.2
The system (3.2.1) - (3.2.2) has a steady state x* = alb and we show that x* is globally asymptotically stable (or attractive); that is, solutions of (3.2.1) - (3.2.2) corresponding to initial conditions of the form
xes) = 4>(05)
~
s E (-00,0];
0,
sup 4>( s) s~O
< 00,
4>(0)
>
°
3.2.3
(¢ being piecewise continuous on ( -00, OJ) are such that lim x(t) = x* = alb.
3.2.4
t-+oo
Let us first verify that all solutions of (3.2.1) - (3.2.3) will remain nonnegative for all t ~ 0. Supposex(t) becomes negative for some t > 0; then since x(O) > there exists a t* > such that
°
°
x(t) >
°
for
t E [0, t*)
and
x(t*) = 0.
§J.2. Delays in production
176
It will follow from (3.2.1) that
roo K(s)x(t* -
dx(t*)
--;u- = a Jo
s)ds > 0
and hence t* is not the first time at which x becomes zero. This contradiction shows that x(t) > 0 so long as x is defined. To show the global existence of x for all t > 0, it is enough to show that x cannot become unbounded for any finite t > o. Suppose x is not defined for all i > OJ then there exists a finite value of i say i} such that xCi) -+ 00 as i-+ i 1 - . In such a case, there exists a first instant of time say t = i2 < il such that for some constant m,
=m
X(t2)
2:: max.{ sup x(i), t~t2
alb}.
From (3.2.1) we derive that
jt2 ,
dx(t 2) -d- = a K(i2 - s)x(s)ds - bx2(t2) -00 t :::; x(t2)[a - bx(t 2)]
:::;0 which contradicts the definition of i 2 • Thus x remains finite for all t 2:: 0 from which the global existence of solutions of (3.2.1) - (3.2.3) for all t 2:: 0 will follow. To establish (3.2.4), we let
xCi) == x*
+ yet)
in (3.2.1) and rewrite (3.2.1) in the form
dy(t) = --;It
1
00
a
0
K(s)y(t - s)ds - 2ay(t) - by2(t).
Consider a Lyapunov functional vet)
= Vet, y(.)
vet) = Vet, y(.» = I yet) I + aJ.~ K(S){
3.2.5
defined by
t,l Y( u) Idu }ds
t > O.
3.2.6
Calculating the upper right derivative D+v(t) of vet) along the solutions of (3.2.5) and simplifying, we have
D+v(t) :::; -bl yet) I[ x*
+ y(t)]
3.2.7
177
§J.2. Delays in production
which with (3.2.6) implies that
Iy(t) I +b
1.' Iy(t) I[
x'
+ vis) Jds s viOl < 00,
3.2.8
One can now show as in Chapter 1, that (3.2.7) and (3.2.8) imply that lim ly(t)l[x*+y(t)] =0
3.2.9
t ...... oo
°
so that either I yet) I ~ as t ~ 00 or x* + yet) ~ 0. It is possible to show (the details are left as an exercise) that x* + y(i) cannot approach zero as i ~ 00 since otherwise (3.2.7) will lead to vet) ~ -00 as t ~ 00 which is not possible. We summarize the foregoing in the following. Theorem 3.2.1. Let a, b be positive constants and let K : [0,00) 1---+ [0,00) be piecewise continuous and normalised such that (3.2.2) holds. Then all solutions of the integrodifferential equation (3.2.1) corresponding to bounded and piecewise continuous and nonnegative (not identically zero) initial conditions on (-00,0] have the property, x(t) ~ alb as t ~ 00. Corollary 3.2.2. If aj, Tj (j solution of
=
1,2, ... , n) are positive constants, then every
dx(t) - = 6~ ajx(t dt .
2
Tj) - bx (i)
3.2.10
;=1
xes)
=
satisfies
xCi)
~
( L:i=1 a j ) bast
~
00.
3.2.11
[]
Proof of the corollary is left as an exercise.
Let us consider the nonoscillatory nature of solutions of scalar equations of the form
dx(t)
d:t
= x(t - T)[a - bx(t)];
t>o
3.2.12
with xes) =
§3.2. Delays in production
178
the form (3.2.1) will be similar. It is an elementary fact that if r = 0 in (3.2.12), then every solution x(t) of (3.2.12) with 0 < x(O) f (a/b) is such that
x(t) - (a/b)
f.
0
for all
t>0
and in this sense solutions of (3.2.12) for r = 0 are nonoscillatory for t E [0,(0). We recall that by definition, the system (3.2.12) with r > 0 is said to be oscillatory, if every solution x(t) with 0 < x(O) f (a/b) is such that, x(t) - (a/b) has infinitely many discrete zeros for t E [0,(0). The equation (3.2.12) is said to be nonoscillatory, if it has at least one solution x(t) with < x(O) (a/b) such that [x(t) - (a/b)] has only a finite number of zeros on [0,(0). The proof of the next result is quite elementary.
°
t=
Theorem 3.2.3. Let a, b be positive constants and let r be a nonnegative constant. Then the system (3.2.12) is nonoscillatory on [0,(0) about the steady state (a/b) for all r ~ 0. Proof. If we let x*
= alb, it is then found that solutions of (3.2.12) satisfy
x(t) - x* = [x(O) - x*] exp [ - b
J.' xes - r)] ds
from which the conclusion is immediate.
[]
We consider next, the following system with a delay in production:
dx(t)
dt
(3x(t-r)
1 + xn(t _ r) -,x(t)
where r, (3, "I E (0,00). If we assume exists and satisfies
1 . "I 1 + (x *) n = fi
j <
or
3.2.13
1 then the steady state x* of (3.2.13)
:y{3 =
1
+
(*)n x .
It is left as an exercise to show that solutions of (3.1.13) corresponding to positive initial values of the type
xes) are defined for all t :2:
=
°and satisfy x(t) > 0.
3.2.14
179
§3.2. Delays in production
To examine the local asymptotic stability of x* we proceed as follows: we let
3.2.15 so that y is governed by
dy(t)
-;It
(eY(t-T)-y(t») + (x*)"eny(t-T) - ,.
= (3 1
3.2.16
Note that x* is locally asymptotically stable, if the trivial solution of (3.2.16) is locally asymptotically stable. Linearizing (3.2.16) about y = 0,
dz(t) = -,z(t) - , [ dt n(l -, p)] - 1 z(t - r).
3.2.17
One can write the characteristic equation associated with (3.2.17) and then obtain sufficient conditions for all the roots of the characteristic equation to have negative real parts; we leave this aspect to the interested reader. We shall follow a different procedure so that our method can be used when the positive feedback (production) term in (3.2.13) is replaced by
(3x(t) 1 + [x(t)]"
or
(3x([t - mJ) 1 + x"([t - m))
where x(t) = SE&~T,tJX(S), t E Rand [t - m] = greatest integer in (t - m), t E lR , mEN. For a study of the potential chaotic behavior for equations of the form (3.2.13), we refer to Hale and Sternberg (1988]. Proposition 3.2.4. Let r, (3" ger satisfying
be positive numbers and n be a nonnegative inte-
,
p < 1,
3.2.18
Then the positive steady state x* of (3.2.13) is locally asymptotically stable. Proof. It is sufficient to show that all solutions of (3.2.17) satisfy
lim z(t) = O.
t-+oo
3.2.19
§3.2. Delays in production
180
Let z be an arbitrary solution of (3.2.17). Then for any fixed t E [0,(0), we have z(t):2: 0 or z(t) < O. We define J 1 and J2 such that [0,(0) = J 1 U J 2 where J 1 = {t E [O,oo)lz(t):2: O}
J2 = {t E [O,oo)lz(t)
< OJ .
. For any t E J1 , we have from (3.2.17)
~ Iz(t) I::; -1'1 z(t) 1+ l' [n(l- j) -1]1> I(t)
3.2.20
in which
Izl(t)=
Iz(s)/,
sup
t E [0,00).
sE[t-r,tJ
By a result of Halanay [1966, see also Lemma 1.4.6], there exist constants Cl > that ·3.2.21
o, 01 > 0 such If t E
h, then we have similarly,
~Iz(t I::; -1'1 z(t) 1+1' [n(l- j) -1]1> I(t), for which there exist positive numbers C2, 02 satisfying 3.2.22 From (3.2.21) and (3.2.22), we derive 3.2.23
t~O
where c = max( C1, C2), 0 proof is complete.
= mine 01 , 02).
The result follows from (3.2.23) and the []
We remark that the type of stability established in Proposition 3.2.4 is known as the "absolute" or "delay-independent" type. Such a type of stability can occur (with some exceptions) whenever there is a "dominant" negative feedback without delays. On the other hand, if there are delays in the negative feedback (destruction), the stability is usually delay dependent. This chapter will elaborate this aspect in considerable detail. Briefly let us now consider,
duet) --;u= -au(t -
it) + bu(t - 1'2);
t>O
3.2.24
181
§9.2. Delays in production
and rewrite (3.2.24) for t > 2(11
d~(t) t
+ 12) as follows:
= -au(t) +ajt
[-au(s -
II)
+ bu(s -
12)]ds
t-Tl
3.2.25
+ bu(t - 12) One derives from (3.2.25) that
d~~t)
::; -au(t) + [( a2 + al b DI + I b I] £l(t)
£l(t) =
sup
sE[ t-2( Tl +T2 ),t 1
u(s);
3.2.26
1"=11+1"2
from which one concludes
a > [(a 2 +albl)l+
Ibl] => u(t) ~ 0 as
t ~
00.
3.2.27
Note that if the delay II is large, then the condition (3.2.27) can fail. While the condition a > [( a2 + al b 1)1 + I b IJ in (3.2.27) is not necessary, for the conclusion of (3.2.27), it is possible by methods of Chapter 2 to show the loss of stability of the trivial solution for large enough delay I I in (3.2.24). Note also that if I I = 0 and a > 1b I, then the size of the delay 12 does not matter, as far as the convergence of u( t) ~ 0 is concerned. The rate at which u( t) ~ 0 as t ~ 00 is usually delay dependent. The following are some examples of systems with delays in production. It is left to the reader to show that solutions corresponding to positive initial values remain positive for all t > 0, and to examine the local asymptotic stability of the positive steady states.
[1< -
dN(t) = r N(t _ I) N(t)]. dt 1 + cN(t)
d~?) dN(t) dt
= -rN(t)
(1)
+ e--yN(t-T).
(2)
= -rN(t) + bN(t _ l)e--yN(t-r).
dN(t) = -rN(t) dt
+ aN(t)[N(t a + j3[N(t -
dN(t) _ -r N(t) dt -
1")Jm. I)Jm
a
+ a + [N (t -
I)] n
.
(3)
(4) (5)
§3.2. Delays in production
182
dx(t) dt
= ax([t _
dx(t) _ ([ t _ - -rx dt dx(t) dt
m
])
= -rx(t) + e-iX(t)
'
m)) _ bx 2 (t).
(K1 +-cx([t x([t - m])) - mJ) x(t) =
dx(t)
=
dx(t)
dx(t)
dt
= [a
= [a
sup
x( s).
a
1
+ bx(t -
+ bx([t -
(8)
(9)
mJ)]n
(1= K(s)x(t - s)ds) (a - bx(t)).
d:~t) = a dt
(7)
.
sE[t-r,t]
dt = -rx(t) + a + [x([t -
d~~t)
(6)
(10)
00
K(s)x(t - s)ds - bx 2 (t).
r)] [c - x(t)],
(11)
a,b,c,E (0,00).
m])] [c - x(t)x(t - r)],
mEN.
(12) (13)
3.3. Competition and cooperation
Let Xl (t) and x 2 (t) denote the population densities (or biomasses) of two species competing for a common pool of resources in a temporally uniform environment. For an extensive discussion of the processes of competition we refer to the article by Miller [1976]. Let bi and mi (i = 1,2) denote the respective density dependent birth and death rates so that in the absence of time delays the population densities are governed by
3.3.1
In order to make the system (3.3.1) denote a model of competition of the "interference type" (see Miller [1976], Brian [1956]), we make the following assumptions on the birth and death rates.
183
§3.3. Competition and cooperation
(i) bi, mi (i = 1,2) are continuous with continuous partial derivatives for all Xi 2: o(i = 1,2); also we assume
8b o· -8 >, i
for
Xi
Xi> 0;
i,j = 1,2;
3.3.2
3.3.3 (iii) for some
xi > 0, xi > 0 we have b1(xn - m1(x~,0) = 0 b2 ( xi) - m2(0, xi) = 0;
3.3.4
(iv) there exist positive constants b1, b2 such that
b1(bt) - m1(bl,X2) < 0
3.3.5
b2(b2) - m2(xl,b2) < 0 (v) there exist numbers
0:
> 0, 13 > 0 such that bi (0:) b2 (fJ) -
=0 m2( 0:,13) = O. m1
(0:,13)
3.3.6
The conditions on bi in (3.3.2) mean that the birth rates are positively density dependent and any crowding effects acting negatively on the birth rates are included in the death rates; the assumptions on mi indicate intraspecific and interspecific competition. The equations (3.3.3) imply that (0,0) is ~ trivial steady state of the system (3.3.1) as it is customary in all models of population ecology; (3.3.4) means that in the absence of anyone species, the other has a positive steady state. The inequalities (3.3.5) will imply, that each of the species cannot grow to unbounded levels since
dXi(t) < 0 dt - . The equations (3.3.6) will guarantee the existence of a positive steady state (0:,13) for the system (3.3.1).
§3.3. Competition and cooperation
184
A simple example of (3.3.1) - (3.3.6) is the familiar Volterra-Lotka model system described by
3.3.7
where
ri, aij
(i,j
= 1,2) are positive constants satisfying either or
all
Tl
al2
aZl
TZ
a22
-<-<-.
The following is one of the questions to be resolved for (3.3.1). If Xl(O) > 0, xz(O) > 0, will it follow that the corresponding solutions of (3.3.1) continue to remain nonnegative for all t 2: O. On the (Xl,XZ) state space of (3.3.1) we have
(0,0), (x;,O), (O,x;), (a,;3) as possible steady states and a plausible question is whether (3.3.3) will lead to the invariance of the boundaries Xl = 0 and X2 = of the state space; such an invariance will imply that if Xl (0) > 0, xz(O) > 0, then the population trajectory (Xl(t), xz(t)) of (3.3.1) cannot reach the outside of the nonnegative quadrant Xl 2: 0, X2 2: O. These questions are easily answered for (3.3.7); we will discuss these aspects for a system of the form (3.3.1) with time delays.
°
Let us first derive a set of sufficient conditions for the local asymptotic stability of the positive steady state (a, (3) of (3.3.1) and subsequently show that the same set of conditions are also sufficient to maintain such a stability even if there are time delays in production (or recruitment) and destruction by competing species. Local asymptotic stability of (a,;3) is easily examined by an analysis of the associated linear variational system in the perturbations Xl, X z where
such a linear variational system is found to be
3.3.8
185
§9.9. Competition and cooperation
where (.J;)" p.
__
[8b i OXj
_
8m i ]
evaluate d at
({.J a, fJ),
i,j = 1,2.
3.3.9
OXj
The steady state (a, (3) of (3.3.1) is locally asymptotically stable, if the trivial solution (0,0) of (3.3.8) is_asymptotically stable. We can derive the following result from elementary considerations;
"In the system (3.3.1)-(3.3.6), if the following holds 8m}
obI
am2
8m2
8bz
amI
> -ax!+ -ox} OX1
3.3.10
- >aX2 -+OX2 aX2 ' then the steady state (a, (3) of (3.3.1) is (locally) asymptotically stable." The proof of the above result is easy, if we note that the characteristic equation associated with the linear system (3.3.8) is given by
/311 -(j12 ) -0 -/321 ,\ - /322 -
det (,\ -
3.3.11
or equivalently 3.3.12 and the roots of (3.3.11) will have negative real parts implying the asymptotic stability of the trivial solution of (3.3.8). If we apply the condition (3.3.10) to the Volterra-Lotka model (3.3.7), then (3.3.10) leads to
which together with a > 0, (3
> 0 lead to
It is known, that if a steady state with positive components exists for (3.3.7), then allan -a12a21 > 0 is a sufficient condition for the global asymptotic stability of (a,{3) in (3.3.7). In fact, using a function V(Xl,X2) defined by
§9.9. Competition and cooperation
186
with suitable positive constants Cl, C2, it is possible to show that (0',13) will be globally asymptotically stable for (3.3.7) whenever alla22 - a12a21 > 0; details of this verification are left to the reader. The following is an interpretation of (3.3.10); the intraspecific negative feedback effects on the i-th species ~, (i = 1,2) dominate its own positive feedback as well as its influence on its competitor ~r;:.i i =1= j . A detailed discussion of this can be found in the article by the author (Gopalsamy (1984bJ). ]
,
We proceed to an examination of the dynamics of a system of two competing species with sufficiently strong intraspecific negative feedbacks and with delays in production (recruitment or birth rate) and destruction by competitor species. We have seen in Chapters 1 and 2 that delays in intraspecific negative feedbacks can render otherwise stable systems oscillatory. Delays in production are more general and common in most biological populations. We shall now formulate a competition model with delays in production and destruction. Let Tij (i,j = 1,2) be a set of nonnegative constants with r = max{rij li,j = 1, 2} and suppose that the two competing species display delayed reproduction and interspecific interaction while the intraspecific interactions involve no time delays. Such a competition system in a constant environment can be modelled by an autonomous delay differential system of the form
dx~y)
= bl (XI( t - TU)) - ml (Xl (t), X2(t - T'2))
dX~t(t)
= b2 (X2(t - Tn») -
3.3.13 m2
(Xl(t - T21), X2(t»)
in which the birth rates bI , b2 and the death rates ml, m2 satisfy the same conditions as in (3.3.1). Along with (3.3.13) we suppose that the initial population sizes are specified by the following:
Xi(S)
= (Pi(s) > 0,
¢iEC([-r,O],IR+),
S E [-r, 0]; ¢i~O
i
on
= 1" 2' [-r,O],
r =
max r"
1 ~i,j~2
I)
3.3.14
i=1,2.
Since (3.3.13) - (3.3.14) are not of the Kolmogorov-type, we have to verify that the solutions of (3.3.13) - (3.3.14) will remain nonnegative so long as such solutions are defined. Let {~~n>c s)} , s E (-r, 0], i = 1,2, (n = 1,2,3, ... ) be a sequence of strictly positive continuous functions such that (in a pointwise sense)
1im~;n)(s)=¢i(S) SE(-T,O] \
n--+oo
i=1,2.
3.3.15
181
§9.9. Competition and cooperation
Let {xin\t) ,x~n)(s)} be the solution of (3.3.13) corresponding to the initial condition Consider the solution
n
where 7* is the positive minimum of 7ij, i,j = 1,2. Suppose xi )(t) does not remain positive for all t E [0,7*]; then there exists a t* in (0, 7*J for which
It will follow from (3.3.13) and the positivity of the initial condition that
dx~n)(t*) dt
::::: b (x(n)(t* 1
1
7
11
») > 0
3.3.16
where we have used the properties of bI and ml; it is found that (3.3.16) contradicts the definition of t* and thus we have
Similarly,
We can repeat the above procedure for intervals of the form [7*,27*], [27*,37*] etc. n Thus, it will follow that so long as (xi )(t), x~n) (t) is defined, we have
If we consider the limit as n -+
00,
we get 3.3.17
with Xl(t) ~ 0,X2(t) ~ 0 where {Xl(t),X2(t)} is the solution of (3.3.13) - (3.3.14) and this is a consequence of the continuous dependence of solutions on ini tial conditions (Hale [1977], p.41). A second question for (3.3.13) - (3.3.14) is concerned with the existence of solutions of (3.3.13)-(3.3.14) defined for all t 2: O. Suppose now a solution of
§3.3. Competition and cooperation
188
(3.3.13) does not exist for all t 2: 0;
one of either
Xl
then there is a t1 > 0 such that for at least
or X2, we have lim Xi(t) =
i = 10r2.
00,
t-+tl-
3.3.18
To be specific let us suppose that limt_tl- Xl (t) = 00; then let t2 be the first time for which Xl(t2) = 51, t2 < t l . It will follow from (3.3.13) that
dXI(t Z) ( ) ----;u-=b 1 Xl(tz-Tll)
-ml
( Xl(t 2),X2(t z - T12) )
») - m, (XI(t2),X2(t2 - T!2»)
< h, (XI(t 2 < bl (51 )
-
ml
(5 ,xz(t 1
2 -
TIZ))
<0 which contradicts the definition of t 2 • Thus, solutions of (3.3.13) - (3.3.14) exist for all t 2: o. We are ready to examine the effects of time delays in production and interspecific competitive destruction, on the asymptotic stability of a positive steady state of (3.3.13). The steady state (a, 13) of (3.3.1) is again a steady state of (3.3.13); we let
in (3.3.13) and derive the linear variational system governing the perturbations Y1 , Y2 as follows: dYj (t)
~
dY:z(t)
~
= -mllY1(t) + bllY1(t -
Tll) - mlZ YZ(t - TIZ) 3.3.19
= -m22 Y2(t) - m2l Y l(t -
T21)
+ b22 Y2(t - Tn)
where
It is well known (Bellman and Cooke [1963], p.336) that the steady state (a, j3) is (locally) asymptotically stable for (3.3.13), if the trivial solution of the linear variational system (3.3.19) is asymptotically stable. The next result provides sufficient conditions for delay independent stability of (a, 13).
§3.3. Competition and cooperation
189
Theorem 3.3.1. Assume that the two species competition model (3.3.13) with delays in production and interspecific competitive destruction satisfy the conditions (3.3.2) - (3.3.6). Suppose that the conditions (3.3.10) hold. Then the positive steady state (0:, (3) is (locally) asymptotically stable for (3.3.13) whatever the delays Tij ~ 0 , i,j = 1,2. Proof. We consider a Lyapunov functional vet) = Vet, Yi, Yz ) defined by
vet) = Vet, Yi,:Y2) = IYi(t) I + IYz(t) I
+bnl~TH IY'(S)ldS+1>,21~T" 11'2(s) Ids + m21 l~T" 1Y, (s) 1ds + m121~r" 11'2(s) 1ds.
3.3.20
Calculating the right derivative D+v of v in (3.3.20) along the solutions of (3.3.19), we have (after some simplification)
which together with (3.3.20) and (3.3.10) leads to
IY,(t) 1+ 11'2(t) 1+ [mn - (b n + m2.)] + [m22 - (b 22 + m'2)]
J.' I y, J.'
(s) 1ds
1Y2(s) 1ds S v(O) < 00. 3.3.22
One can show that (3.3.22) implies (by Lemma 1.2.2)
IY1 (t) I + IYz(t) I ~ 0
as
t ~
00
and
IY1 (t) I + IY;(t) Is v(O),
from which the (local) asymptotic stability of (0:, (3) for (3.3.13) will follow and this completes the proof. [] Instead of discrete delays as in (3.3.13), it is also possible to have continuously distributed delays over an infinite interval; for instance, instead of (3.3.13) one can
§3.S'. Competition and cooperation
190
consider the integrodifferential system
dX~il)
= bl
([}11(1 - )XI(S)dS) S
- ml (XI(I), .
dx;?) = b,
ft~k12(1 -
kij
:
kij,
(0,00)
1-+
3.3.23
([}22(1 - s)x,(s) dS) - m, (f'~ k21 (I -
in which
s)X,(S)dS)
S)XI (s) ds, x,(
t))
i,j = 1,2 are as follows:
(0,00) ; k ij are piecewise continuous on (0,00) such that
One has to consider together with (3.3.23), initial conditions of the form {Xi( s) = ~i( s) ~ 0; ~i is bounded and piecewise continuous on (-00,0); ~i '¥= 0; i = 1,2.}. It is not difficult to see that (ex,,B) is again a steady state of (3.3.23). The following result asserts the local asymptotic stability of (ex, ,B) for (3.3.23) in the sense: there exists a b > 0 such that sup 8~O
{I Xl(S) - ex 1+1 X2(S) - ,BI} S b =}
I Xl(t) - ex I + I X2(t) -,8 l-t 0
as t
-t 00.
Theorem 3.3.2. Assume that (3.3.10) holds. Then (ex, (8) of (3.3.23) is (locally) asymptotically stable. Proof. The linear variational system corresponding to (ex,,B) in (3.3.23) is found to be dX 1 (t) --;u= -mllX 1 (t) + bll 0 kll(S)Xl(t - s)ds
T
dX2 (t)
--;u- =
/.00
- m12
/.00 k12(S)X2(t _ s)ds
-m22 X 2(t) -
roo m2110 k21 (S)X (t -
+ b22 /.00 k22(S)X2(t _
1
s)ds.
3.3.24
s)ds
§3.3. Competition and cooperation
191
A Lyapunov functional v for (3.3.24) is given by
vet)
= V(t,X 1 ,X2) = IX1 (t) I + IX2(t) I + bl l
/,= kll(s ){L,
[XI (u) [du } ds
+b22/,= k,,(S){L.[X,(U) [dU} ds + m21 + ml
t'
3.3.25
k21 (S){L,[ XI(u) [du} ds
'/,= k
12
(S){L,[X,(U) [du }ds.
The remaining details of proof are left to the reader as an exercise.
[]
We shall consider the dynamics of cooperation (or mutualism) between two species. Such an interaction can be modelled by a system of the form
3.3.26
where
II, h
are continuously differentiable such that
An example of such a model is
3.3.27
where ri,
Ki,
ai
E (0,00)
and
(¥i
> Ki ,i = 1,2.
3.3.28
Depending on the nature of Ki , (i = 1,2), the mutualism model (3.3.27) can be classified as facultative, obligate or a combination of both. For more details of mutualistic interactions we refer to Vandermeer and Boucher [1978], Boucher et al. [1982], Dean [1983], Wolin and Lawlor [1984] and Boucher [1985].
§J.J. Competition and cooperation
192
The system (3.3.27) has a unique positive equilibrium N*
= (Ni, N z) satis-
fying 3.3.29
It is possible to show by means of phase plane methods or by an application of Kamke's Theorem (Coppel [1965}i more details of Kamke's Theorem will appear in the next chapter where we will study the global behavior of the system (3.3.27» that solutions of (3.3.27) satisfy the following:
N2(t)
N;
-+
as
t
-+ 00.
3.3.30
We are interested in the study of the following time lagged model corresponding to (3.3.27);
dN 1 (t) _ dt
--- -
°
dNz(t) --dt
N ()
lIlt
= 12 N 2 (t )
[Kl + a1Nz(t 1 + N2(t -
[K2 + a2 N l(t 1 + NI (t -
TZ)
-
N 1 (t )]
T2)
3.3.31
Tl) - N 2 ( t )]
Td
where Tl 2:: 1 T2 2:: 0, T1 + T2 > 0. The system (3.3.31) means that the mutualistic or cooperative effects are not realized instantaneously but take place with time delays. We shall show that the steady state N* of (3.3.31) is linearly asymptotically stable irrespective of the sizes of the delays Tl and T2. We associate with (3.3.31) initial functions of the form 3.3.32 where
iEC([-Ti,O],R+) and i(O»O for
i=1,2.
Lemma 3.3.3. The initial value problem (3.3.31) - (3.3.32) has a unique solution which exists for all t 2: 0, is positive (componentwise) and is uniformly bounded for t 2: o.
Proof. The solutions of (3.3.31) and (3.3.32) exist uniquely on an interval of the form [0, T) for some T > 0 and remain positive. Let [0, T) be the maximal interval of such an existence. We have from (3.3.31),
liNi[]{i - Ni(t)] ~ Ni(t) ~ liNdai - Ni(t)] for
0~t
< T.
3.3.33
193
§3.3. Competition and cooperation
This implies that Ni(t) is nondecreasing as long as Ni(t) ~ Ki and nonincreasing as long as Ni(t) ~ ai. An implication of this is that Ni(t) is positive and bounded for all t E [0, T). Hence T = 00, and this completes the proof. [] Let us examine the linear asymptotic stability of N* of (3.3.31). We let
Ni(t) = Nt
+ Xi(t)
i
for
= 1,2
and linearise the system (3.3.31) around N* ; such a linear system is
3.3.34
The characteristic equation associated with (3.3.34) is
3.3.35
The steady state N* of (3.3.31) is said to be linearly and "absolutely" stable in the delays if the trivial solution of (3.3.34) is asymptotically stable for all 71 2:: 0) 72 ~ O. "Absolute" stability corresponds to the fact that delay induced stability switches (see section 3.7 below) cannot occur.
Theorem 3.3.4. Assume that (3.3.28) holds. Then the positive steady state N* of (3.3.27) is linearly asymptotically stable absolutely in the delays. Proof. It is sufficient to show that for all values of 71 2:: 0, istic equation does not have roots of the form
,.\ = a + i/3 We replace ,.\ in (3.3.35) by a leading to a 2 - {32
with
a
72
2:: 0, the character-
2:: O.
+ i{3 and then separate the real and imaginary parts
+ aQ + R = _SRe- ar COS{3r} 2a{3 + flQ = SRe- ar sinfl7
3.3.36
§3.3. Competition and cooperation
194
where 7"
= 7"1 + 72, Q = T"lN: +T"2N;, R = T"lr2N;N; S = (a1 - KJ)(a2 - K 2) . (1 + Ni)2(l + N;)2
Squaring both sides of (3.3.36 ) and then adding, we obtain
(a 2 + 13 2)2
+ a 2Q2 + R2(1- s2 e-2aT) + [2aQ + ri(N:)2 + T"2(N;?]f3 2 + 2a(a 2Q + aR + QR) = O.
3.3.37
We first claim 3.3.38
O<S<1. In view of (3.3.29) we have
3.3.39
which shows that (3.3.38) holds. It is not difficult to see that if a 2 0, then (3.3.37) can hold only when a
= 0, 13 = 0
and
S
= 1.
But this contradicts (3.3.38) and therefore every root of (3.3.37) has a negative real part and this is not conditional on the magnitude of the delays 71 and 72 . [] It will be shown in the next chapter that the above linear "absolute" stability of N* is in fact global with respect to all positive solutions. A few examples of models of cooperation and competition are listed below and the reader is asked to examine by phase plane methods, the nature of their positive steady states.
195
§9.9. Competition and cooperation
(B)
(C)
(D)
The following are models of interference competition between two species proposed by Ayala et al. (1973].
1
dNi "1 dt= (riNi) Ki- [ ]).·-N·-a··N· 1 I]], i
dN = (r i Ni ) dt logKi
[ log(R,.i ) -log(Ni )
-
aij log(Nj
1
dNi = (riNi) [K1/2 _ N /2 _ "'iiNi dt I{~/2 z 1' I(~/2' Z
= 1,2.
i,j
)
1
i,j = 1,2.
,
i,j= 1,2.
(2)
(3)
t
dNi [ R·-N·-a··N·-(J··N·N· ,1 ], - Ki dt= CiNi) 1 I]] I] 1 ]
2]
i ( -r i Ni ) [ J\.·-N·-a dN= , .. N·-(J·N· dt Ki I I. I]] I I i ( r i Ni ) -dN dt = -I{i-
(1)
i.j
,
(5)
i,j = 1,2.
(6)
2]
i,j = 1,2.
'N')]
i dN ,.I -Ki- [ R·-N·-a .. N·-(J·1 ( l-e - l' dt= (riNi) 1 I]]
•
(4)
i,j=1,2.
[ It" , - N· - a' ·N· - (J·N· liZ]] ]]'
dNi = CiNi) [KO' - N°' - "'iiNi] dt I{~i I 1 K~-()i' I I
= 1,2.
,
i,j = 1,2.
(7)
(8)
§3.3. Competition and cooperation
196
dNi (riNi) [R.·-N·-a ~ - Ki .. N·-j3··N·N·-8·N.2]
dt=
1
1
1))
1)
1)
1
1
,
i,j = 1,2.
(9)
= 1,2.
(10)
i,j
i,j = 1,2. (11)
The following models of two species competition have been proposed by Schoener [1976] (see also Winsor [1934] and Hutchinson [1978])
For a discussion of exploitative and interference models of competition we refer to Gopalsamy [1986b] and the literature cited therein.
3.4. Prey-predator systems
In this section we study systems modelling the dynamics of one predator species feeding exclusively on one prey species. For an elaborate discussion of the prey-predator association we refer to the article by Murdoch and Oaten [1975]. Assuming that the environment is temporally uniform, let Yl(t) and Y2(t) respectively denote the population densi ties (or biomasses) of a prey species and a predator
197
§9.4. Prey-predator systems
species. As before, we consider a system of the following form governing the rates of change of the two population densities:
3.4.1
where Ii and 9i (i = 1,2), respectively denote the production (or birth) and destruction (or death) rates of the two species; in order to make (3.4.1) denote the dynamics of a prey-predator system we assume that Ii and 9i satisfy the following conditions: (i) fi and 9i are continuous with continuous partial derivatives for Yi 2: 0 (i 1,2); also we require 8fI O' a> , Y1
812 >08Y1 - ,
891 > O' 8Y1 - ,
891 > 0 8Y2 for Y1 2: 0,
812 >0' 8Y2 - ,
Y2 2: 0
=
3.4.2
892 > 0 8Y2 -
(ii) fiCO) = OJ
91(0,yZ) == 0
12(0, Y2) == 0;
12(y,O) == 0;
(iii) there exists a number
3.4.3 92(0) = 0
Y; > 0 such that 3.4.4
(iv) there exists a pair of constants a*, /3* > 0 such that
!I(a*) - 91(a*, /3*) = 0
3.4.5
f2( a*, (3*) - 92((3*) = OJ (v) there exist positive constants
T}t, T}z such that
fI(T}t) - 91(T}t, Y2) < 0; fI(T}!, T}2) - 92(T}2) < O.
Yz
2: 0,
3.4.6
§3.4. Prey-predator systems
198
The conditions (3.4.2) mean that the system (3.4.1) is of the prey-predator typej (3.4.3) means that (0,0) is a trivial steady state of (3.4.1)j (3.4.4) implies that, in the absence of the predator, the prey species has a nontrivial (positive) steady state; (3.4.5) shows that (3.4.1) has a nontrivial steady state in the interior of the positive quadrant of the state space of (3.4.1); (3.4.6) will imply (see below) that no species can grow unbotmdedly. An example of (3.4.1) - (3.4.6) is the familiar Volterra-Lotka prey-predator system
where ri, aij (i,j state to exist)
= 1,2) are positive constants such that (fora nontrivial steady
As in the case of our analysis of two species competition, we have to verify that when Yl(O) > 0, Y2(0) > 0, the corresponding solutions of (3.4.1) will be nonnegative for those t 2:: 0 for which they exist. Since the coordinate axes Yl = 0 and Y2 = 0 are invariant sets for (3.4.1) by virtue of (3.4.3), it will follow that the solutions of (3.4.1) starting from the interior of the positive quadrant cannot enter the outside of that quadrant; this is a consequence of the invariance of the coordinate axes for (3.4.1). In the following, we suppose that Yl (0) and Y2(0) satisfy 0 < Yl (0) < 1]1 and 1]2; (on the otherhand) if ei ther one of Y1 (0) 2:: 1]1 and Y2 (0) 2:: 1]2 or both hold, it will follow that the corresponding Yi(t) will decrease as t increases by virtue of (3.4.6) at least for small t > O. Suppose Y1(t) is not defined for all t 2:: OJ then for some tl < 00, t~~ _ Y1 (t) = 00 and let t2 > 0 be the first time for which Yl (t2) = 1]1; we will. have from (3.4.6) and (3.4.1) that
o < Y2 (0) <
which contradicts the definition of t2 showing that Y1 (t) is defined for all t ? 0; if Y2(t) is not defined for all t ? 0, then for some t3 < 00 we will have t~~- Y2(t) = 00;
now let
t4
be the first instant for which Y2(t 4 ) =
1]2;
then from the properties
§9.4. Prey-predator systems of
h
199
we have
dY2(t 4 )
~ =
h(Yl(t4),T/2) - 92(TJ2)
::; h( TJl, TJ2) - 92( TJ2)
< 0 (by (3.4.6) assuming 0 < Yl (0) < TJi,
i
= 1,2
which again contradicts the definition of t 4 • Thus, Y2(t) is defined for all t 2:
o.
Let us now derive a set of sufficient conditions for the (local) asymptotic stability of the steady state (a*, (3*). To examine the asymptotic stability of (a*,{3*) in (3.4.1), we let
Y2(t)
= (3* + U2(t)
in (3.4.1) and derive the linear variational system in the perturbations UI, U2 as follows:
dUl(t) = oh U1(t) _ 091 U1 (t) _ 091 U2 (t) dt OYl OYl OY2 2 dU (t)_oh () oh () 092 () - - - - UI t +- U2t - - U2t & OYl O~ o~
t > 0;
3.4.7
where all the partial derivatives in (3.4.7) are evaluated at (a*, (3*). The steady state (a*, (3*) of (3.4.1) is asymptotically stable, if the trivial solution of (3.4.7) is asymptotically stable and this will be the case, if the roots of the characteristic equation associated with (3.4.7) given by
det. (A - (Ill - 911) - 121
912 ) - 0 A - (122 - 922) -
3.4.8
where
09i 9ij = -0 ;
at
(a*,{3*)
Yj
have negative real parts. If AI, A2 are the roots of (3.4.8), then
Al
+ A2 = (Ill
- 911) + (h2 - 922)
AIA2 = (ill - 911)(122 - 922) + h2I2l . The following result is an immediate consequence of (3.4.7) - (3.4.8).
3.4.9
200
§J.4. Prey-predator systems
Theorem 3.4.1. In the prey-predator system (3.4.1) - (3.4.6) assume that the
partial derivatives satisfy the following additional conditions:
evaluated at
(a*, (3*);
3.4.10
then the steady state (a*, (3*) is (locally) asymptotically stable for the preypredator system (3.4.1) The literature on prey-predator model systems involving time delays is quite extensive. Most of the models have been derived from the Kolmogorov-type systems with time delays incorporated in the average growth rates of the prey and predator species. Volterra [1931] has proposed the following system of integradifferential equations for a prey-predator model system
3.4.11
where TI, r2, 51 ,52 are positive constants and FI , F2 are nonnegative continuous delay kernels suitably defined on [0,(0). The equations (3.4.11) do not contain negative effects of predator crowding. Brelot [1931] has cosidered a modified format of (3.4.11) in the form
3.4.12
which incorporates crowding effects typified by the positive constants (AI, A2)' Under suitable conditions on the various parameters in (3.4.12), one can show that (3.4.12) has a (locally) asymptotically stable steady state. We will consider in detail a system more general than (3.4.12) in the next chapter. A number of integrodifferential equation models of the type (3.4.11) and (3.4.12) have been investigated by Cushing [1977].
§9.4. Prey-predator systems
201
Starting from Hutchinson's [1948) delay logistic equation, May [1973) has proposed the following system
[1 _NI(tK- T)]_ aNI (t)N2(t)
dNt(t) = rNI(t) dt dN2 (t) -;u- = -bN2(t)
3.4.13
+ f3N2(t)NI(t)
where r, T, k, a, (3, b are positive constants; (3.4.13) contains a single discrete delay; one can modify (3.4.13) and incorporate a continuously distributed delay in it so that 3.4.14
The first model of a prey-predator system which departs from the Kolmogorov-type formulation is due to Wangersky and Cunningham [1957] who have proposed 3.4.15 where aI,a2,b l ,q,c2,T are positive constants; (3.4.15) means that a duration of time units elapses when an individual prey is killed and the moment when the corresponding increase in the predator population is realised. Cushing [1979J has considered a model of the form
T
dNI(t) = rNI (t)[l- N 1 (t) - aN2(t)] dt K
+
dN (t) = -6N (t) 2
+ bN1 (t)
1= 0
3.4.16
(3(a)N2(t - a)e- 6a da
where r, k, a, 6, b are positive constants and (3 is related to age dependent fecundity of the predator species. With the models (3.4.11) - (3.4.16) in the background, let us consider a time delayed model in the spirit of (3.4.1);
dNI(t) = It ( N 1 (t -;u-
T11) ) - gl ( NI(t),N2(t - T12) ) 3.4.17
dN2 (t) = -;u-
fz ( N 1(t - T21),N2(t - T22) ) - g2 ( N2(t) )
§3.4. Prey-predator systems
202
where tij are nonnegative constants with 7 = 1;:a.:X:<27ij; the production (birth) _ ,J_ and destruction (death) rates represented by It, h, gI, g2 satisfy the same conditions as in (3.4.1) - (3.4.6). Along with (3.4.17), we have to specify initial conditions as follows: 3.4.18 We assume that in addition to the conditions corresponding to (3.4.2), we have for (3.4.17),
8h 8h 8N > 0, 8N > 0, h(NI,N2 ) > I
2
large enough
N1
°
for
N2
>
°
and
3.4.19
> 0.
It is not difficult to show by the methods used before for the competition system
°
that whenever NI(s) > 0,N2 (s) > for s E [-7,0], solutions of (3.4.18) exist for all 7 ~ and remain nonnegative for an7 ~ 0. Also (a*, (3*) of (3.4.5) is a steady state of (3.4.18). The proof of the following is similar to that of theorem 3.3.1 and hence we omit the details of proof.
°
Theorem 3.4.2. In the prey-predator model (3.4.17), let the conditions (3.4.2)(3.4.6), (3.4.18), (3.4.19) hold for the birth and death rates. Furthermore, if the conditions (3.4.10) hold, then for all nonnegative delays tij in (3.4.17), the nontrivial steady state (a* ,(3*) of (3.4.17) is (locally) asymptotically stable. An alternative to the system (3.4.17) is an integrodifferential system of the form
3.4.20
under appropriate conditions on the nonnegative delay kernels kij (i,j Details of further analysis of (3.4.20) are left to the reader.
=
1,2).
203
§9.4. Prey-predator systems
The prey-predator model systems (A) to (D) listed below have been investigated by Nunney [19S5a, b, c]; N denotes the predator density and R denotes the resource (or prey) density:
d~it)
= N(t)F(R(t)) - N(t)M(R(t))
d~;t) ~ B(R(t)) -
d~;t) = d~;t)
(A) D(R(t)) - N(t)G(R(t)).
N(t) [F(R(t)) - M(R(t))]
d~(t)
I
.
(B)
= B(R(t - T)) - D(R(t)) - N(t)G(R(t)).
d~?) = N(t _ T)F(R(t d~;t)
}
T)) - N(t) M(R(t)) }
(C) = B(R(t)) _ D(R(t)) - N(t)G(R(t)). = N(t _ T)F(R(t - T)) - N(t)M(R(t))
}
(D)
t
d~;t) = B(R(t -
T)) - D(R(t)) - N(t)G(R(t)).
The following are examples of models of one prey and one predator systems in the absence of delays; the interested reader should formulate appropriate models with various time delays (such as discrete, continuous, piecewise constant etc.).
dH(t) = rH(l- H) - aHP dt I{
d~;t)
= -bP
I
+ j3 H P.
d~;t) =rH(l- ; ) _aP(l_e- CH ) dP(t) dt
(1)
I
(2)
= -bP + j3P (1 _ e- cH ) .
dH( t) = r H dt
(1 - H) - +HI
d~; t)
[1 - ~].
= bP
K
aP j3 H
(3)
204
§S'.4. Prey-predator systems
(4)
(5)
I
dH(t) = rH[K - H]_ aHP dt 1 + cH j3 + H
(6)
dP(t) =p[-f3 ~-bP]. dt + f3 + H dH(t) = rH[K - H]_ aHP dt 1 + cH
d~~t) dH(t) dt dP(t)
---;It
=
I (7)
p[ _a +bH - CP]_
= aH -
I
bHP - €H 2
1 + aH
bHP
2
(8)
= -cP + 1 + aH - T}P .
3.5. Delays in production and destruction One of the techniques for the analysis of local asymptotic stability of steady states in autonomous delay-differential equations is based on an examination of the roots of the characteristic equation associated with the corresponding linear variational systems. As one can see from the following, that such a method based on the characteristic equation is quite difficult and often is an analytically almost impossible task if the system has several delay parameters; a reward for such a task is, however, that one can derive necessary and sufficient conditions for local asymptotic stability. In the case of ordinary differential equations, a stability analysis based on the characteristic equation is almost trivial due to the availability of the Routh-Hurwitz criterion. We have already considered several techniques based on Lyapunov functionals and we will consider other related techniques in the next section.
§3.5. Delays in production and destruction
205
For purposes of our illustration we first consider the system
dx(t)
dt = x(t)f{x(t), yet - T)} dy(t)
at =
3.5.1
y(t)g{ x(t - r), yet)}
in which T is a nonnegative constant, f and 9 are continuously differentiable in their arguments. Suppose there exists a point (x*, y*) , x* > 0, y* > 0 such that
f(x*,y*) = 0 = g(x*,y*). The local asymptotic stability of the steady state (x*, y*) of (3.5.1) is studied by an analysis of the asymptotic behavior of the related variational system obtained from (3.5.1) by setting
x(t) = x*
+ X(t),
yet) = y*
+ Yet),
and neglecting the nonlinear tenns in the perturbations X and Y so that
dX(t) -;It = x* fxX(t) dY(t) -;It
+ x* fyY(t -
r)
3.5.2
= y*gxX(t -
T)
+ y*gyY(t)
where the partial derivatives fx,fy,gx,gy are evaluated at (x*,y*). We formulate our result as follows:
Theorem 3.5.1. If the coefficients of the system (3.5.2) are such that
fx(x*,y*) < 0, Ifx( x*, Y*)1 > /gx( x*, Y*)I,
gy(x*, y*) < 0, Igy(x*,y*)1 > Ify(x*,y*)/,
3.5.3
tben for any r ?: 0, tbe trivial steady state (0,0) of (3.5.2) is asymptotically stable. Proof. First let r = 0 in (3.5.2); one can show that (3.5.3) will imply that the steady state (0,0) of (3.5.2) is asymptotically stable. Now, assume r > 0 be fixed and for convenience let -a = fx( x*, Y*)j
-b=fy(x*,y*)
-c = gx(x*, Y*)j
-d = gy(x*, y*).
§3.5. Delays in production and destruction
206
Consider any solution of (3.5.2) in the form
[X(t)] Yet)
=
[A]
zt
B e
where A, B, z are constants (not necessarily real), satisfying the system of equations (z + ax*)A + bx*e- zr B = 0 3.5.4 cy*e- zr A + (z + y*d)B = o. A necessary and sufficient condition for the existence of nontrivial solutions of (3.5.4) is that the constant z in (3.5.4) satisfies the characteristic equation z + ax* det. [ cy*e- ZT
bx*e- ZT ] z + y*d
=0
3.5.5
or equivalently Z2
If we let Z
+ z(az* + dy*) + adx*y* -
bcx*y*e- 2zr
= O.
3.5.6
= 2Z7 in (3.5.6), we can rewrite (3.5.6) in the form (Z2 +pZ + q)e z
+r =
O.
3.5.7
with
+ Y*d)} q = adx*y*47 2
p = 27(ax*
r = -bcx*y*47
2
3.5.8
•
To investigate the nature of the real parts of the roots of (3.5.7) we use Theorem 13.7 from Bellman and Cooke [1963, pp. 443-444]. In order to apply this theorem we let 3.5.9 H(Z) = (Z2 + pZ + q)e z + r and note that a necessary and sufficient condition for all the zeros of H(Z) to have negative real parts is that 3.5.10 F(w)G'(w) > 0 at all the roots of G( w)
H(iw)
= 0 where = F(w) + iG(w)
and
wE (-00,00).
3.5.11
207
§9.5. Delays in production and destruction
From (3.5.9) and (3.5.11) we derive,
F( w) = (q - w 2 ) cos W
-
pw sin w + r
3.5.12
G(w) = (q - w )sinw + pwcosw. 2
It is known (see Bellman and Cooke [1963], p.447) t-hat all the roots of G(w) = 0 are real. Let Wj (j ~ 0,1,2, ... ) denote the zeros of G(w) with Wo = 0. For Wo, (3.5.10) demands that
F(O)G'(O) = (r + q)(p + q) >0.
3.5.13
With a simple computation, we obtain that the nonzero roots of G( w) roots of cot w = (w 2 - q)/wp
=
°are the 3.5.14
and hence for such nonzero roots of G( w) = 0, we have
F(w) =
r_(s~:) [(W' _q)' + w'p']
G'(w) = -
(S~pw) [(w' - q)' + w'(p' + p) + pq]
3.5.15
from which it will follow that the sign of F( w )G' (w) is the same as that of
L(w) = (S~pw)' [(w' _p)' + w'p'] _r (S~pw)-
3.5.16
(3.5.14) and (3.5.16) together imply that
3.5.17 Since
Irl <
q, p2 ~ 2q and since all the roots of G(w) =
°
are real, we have L( w) > 0. Thus by theorem 13.7 of Bellman and Cooke [1963], a necessary and sufficient condition for all the roots of (3.5.9) to have negative real parts is,
Irl < q,
p
> 0,
q ~ 0.
It is easily seen from (3.5.3) that
Ir/- q = (Ibcl- ad)4r 2 x*y* < 0,
208
§3.5. Delays in production and destruction
and therefore the trivial solution of the variational system (3.5.2) is asymptotically stable and the proof is complete. 0 For more details related to the result of Theorem 3.5.1 and an estimation of the rate of convergence of solutions o~ (3.5.2) to the trivial solution we refer to Gopalsamy [1983a] where examples can be found. For a mathematical analysis of physiological models with time delays in production and destruction, we refer to the articles of an der Heiden [1979J and an cler Heiden and Mackey [1982]. Let us consider a system somewhat more general than (3.5.1); let 1,2) be a set of nonnegative constants and consider the system dx(t) = -;u-
x(t)i ( x(t -
7'11),
yet -
7'12)
7'ij
(i,j ::::
)
3.5.18 dy(t) = y(t)g ( x(t ---;It
) 7'2J), yet - 7'22)
in which i and g satisfy the same conditions as in Theorem 3.5.1. Note that our analysis above corresponds to (3.5.18) with 7'11 = 0 and 7'22 = 0 and 7'12 = 7'21. Hence, let us suppose at least one of 7'11 , 7'22 is not zero. IT we let
x(t) :::: x*[l
+ X(t)]
yet) :::: y*[l + Yet)]
in (3.5.18), then the linear variational system in X, Y is of the form dX ( t) :::: allx * X ( t ~
-
7'11
) + a12Y * Y ( t -
7'12
)
3.5.19
dY(t) ~
::::
* ( a21x X t -
) *y( t --7'22 ) 7'21 + a22Y
where all, al2, a21, a22 denote the partial derivatives ix, i y , gx, gy respectively evaluated at (x*, y*). The following questions are of interest for (3.5.19); (i) ifthe trivial solution of (3.5.19) is asymptotically stable in the absence of delays, will it continue to be so for all delays; (ii) is there a threshold value for the delay parameters so that (3.5.19) can become unstable, if (3.5.19) is stable in the absence of delays; that is, can an estimate on the delay parameter be obtained for stability to hold; (iii) if the system (3.5.19) is unstable in the absence of delays, will it remain unstable for all delays or it will switch to stability; (iv) will the system exhibit stability switches, i.e. switch from stability to instability and back to stability and so on? In the next two sections we investigate certain aspects of the above questions for
§9.5. Delays in production and destruction
209
a general linear system with a single as well as several different delays. Usually linear analyses of models with delays in production and destruction lead to equations of the form (3.5.19) and their integrodifferential analogues; a characteristic of such systems is that they need not necessarily have terms without delays. The following are some examples of models with delays in production and destruction:
d~~t)
= rN(t _ r1)
[1 - N(t)Nl! - r z )].
dN(t) = rN(t) [K - N(t - 7"z)]. dt 1 + eN (t - r2)
d~it)
= J.=X(S)N(t-s)d+-N(t) J.= H(S)N(t-s)dsj.
dx(t) _ f3x(t - r) _ () ( _ ) ( ) 'Yx txt r. 1 + xn t - 7" dt dN1(t) dt dNz(t) dt
= N 1(t-rd []{1 +a1N z(t- r2) -Nl(t)]) 1 + N 2 (t - rz) . = Nz(t _ 7"4) [Kz + aZN1(t - 7"5) - Nz(t)]. 1+N1 (t-7"s)
d~?) = rN(t -
r)[l- Nit) - eUCt)])
duet)
aT =-au(t) + bN(t -1]).
d~~t) = x(t dy(t)
di
= yet - 7") [-Kz
d~~t) = x(t -
x(t) - ay(t - r) J }
7")[]{1 -
+ f3x(t -
r)J.
e-.(t-rl ) -
X(t)])
= yet - 7")[K2(1-
e-x(t-r)
yet)].
dxd(t) = -,x(t) + ae-/Jx(t) , t
x(t) =
d~~t)
r)[K1 (1-
-
sup
xes).
sE[t-r,t]
dx(t) J.OO -;tt=-'Yx(t)+aexp [- 0 K(s)x(t-s)dsJ.
§3.5. Delays in production and destruction
210
d~~t) = -,x(t) + ax(t _ r),e-PX(t-T). dx(t) = -,x(t) + axn([tDe-Px([tj). dt dx(t)
---;It = xCi) [a - blog[x(t)] - clog[x(t - 7)J]. It should have become clear from the foregoing, that local analyses of various models with time delays lead to investigations of linear delay differential equations. In the next section and in the remainder of this chapter, we consider the asymptotic and oscillatory behavior of linear vector - matrix systems using certain algebraic facts related to matrices and vectors.
3.6. X(t) = AX(t)
+ BX(t -
r)
Let us first consider the delay differential system
dx(t) dt
= Bx(t _
7)
3.6.1
where x(t) E IRn and B is a real constant n X n matrix. The following result shows that if the trivial solution of (3.6.1) is asymptotically stable for 7 = 0, then it will remain so for 7E [0,70); we also obtain an estimate on 70 (for more details see Goel et al. [1971]). Theorem 3.6.1. Let the eigenvalues of the matrix B be denoted by
Suppose that the trivial solution of the non delay system
dy(t) = By(t) dt
3.6.2
is asymptotically stable implying that 3.6.3
If j
= 1,2, ... , n,
3.6.4
§9.6. X(t)
= AX(t) + BX(t -
211
T)
then the trivial solution of (3.6.1) is asymptotically stable.
Proof. The characteristic equation corresponding to (3.6.1) is 3.6.5 Since
n
II (A +
det. [AI - Be- Ar ) = 0 =}
Qje-
Ar
)
= 0,
3.6.6
j=l
it will follow that the roots of the characteristic equation (3.6.5) are the roots of j = 1,2,3, ... ,n.
If we let AT =
Z
3.6.7
in (3.6.7), we can rewrite (3.6.7) in the form 3.6.8
j = 1,2,3, ... ,no
Equations of the form (3.6.8) with real Qj have been discussed in the literature on delay differential equations; since Q j can be complex, we provide a complete discussion of (3.6.8). For convenience, let us consider a fixed j and let the corresponding Q j be denoted by Q with ~e( Q) > O. We let (J,O
being real,
and introduce the substitution L
H(L)
=z -
(J>
0,
I B I < 7r /2
iO so that (3.6.8) becomes (for the fixed j)
= Le L + iOe L + (JT = O.
3.6.9
We note that ~e(L) = ~e(z) and hence ~e(L) < 0 will imply ~e(z) < 0 and conversely. In order to use Theorem 13.7 of Bellman and Cooke [1963), we proceed by letting L = iy (y real) in (3.6.9) so that where = F(y) + iG(y) F(y) = (JT - (y + B) sin y G(y) = (y + 0) cos y.
H(iy)
3.6.10 3.6.11
3.6.12
By the above Theorem of Bellman and Cooke [1963], a necessary condition for all the roots of H(L) = 0 to have negative real parts is that
212
§S.6. X(t) = AX(t) + BX(t - 7)
(i) the zeros of F(y) and G(y) are real, simple and they alternate; (ii) G'(y)F(y) - G(y)F'(y) > 0 for y E Rj 3.6.13 a set of sufficient conditions for H(L) = 0 to have roots only with negative real parts is that (a) all the zeros of G(y) are real and for each such zero, (b) the relation (3.6.13) holds.
The roots of G(y) = 0 are given by yO and Yn where
Yn
= ±( n + 1/2)7r;
n
= 0,1,2,3, ...
At the roots of G(y) = 0, we have and
3.6.14
It is readily verified that G'(Yn)F(Yn) > 0 when (7r/2) -IBI- (77 > 0; this result translated back to (3.6.8) implies that all the roots of (3.6.7) and hence of (3.6.6) will have negative real parts whenever (3.6.4) holds and this completes the proof.
,,;
We note that if ~e(a) 2:: 0 then the necessary condition G'(y)F(yO) > 0 is violated implying that the system (3.6.1) cannot switch from the instability to stability with an increase in T. We ask the reader to investigate this in detail. Let us consider the linear vector - matrix delay-differential system
dX(t) --;It
= AX(t) + BX(t -
7)
3.6.16
and examine the following: if the trivial solution of (3.6.16) is asymptotically stable when 7 = 0, for what positive values of 7 such a stability is maintained. There are several possible ways of answering the above question each leading to a different estimate of 7. The following result is due to Rozhkov and Popov [1971] (see also Tsalyuk [1973], Gosiewski and Olbrot [1980]). Theorem 3.6.2. Let A and B be real n x n constant matrices such that the trivial solution of
d~;t) = (A + B)Y(t)
3.6.17
§9.6. X(t)
= AX(t) + BX(t -
219
r)
is asymptotically stable and let M,o: be positive constants satisfying 3.6.18
If r is small and
MIIBllr(IIAIl + IIBI!) < 1, 3.6.19 a then the trivial solution of (3.6.16) is asymptotically stable. Furthermore, if X(t) denotes any solution of (3.6.16), then IIX(t)1I
~ M{
sup
SE[-T,T]
IIX(s)lI}e-/3(t-T);
3.6.20
in which f3 is the unique root of
T MIIBII( e/3 - 1) (IlAlI
f3 1- - =
0:
+ liB II e/3
T )
-----~----.::..-
3.6.21
/30:
Proof. We rewrite (3.6.16) in the form
X(t) = (A + B)X(t) - B L/(s)ds; = (A + B)X(t) - B
t;:: r
l~J AX(s) + BX(s -
r)) ds;
t;:: r
leading to
and hence
/lX(t)/I ::;
/lX/I.Me-a('-T)
+ M/lB/I[dsU>-a(.-s) (/I A /III X(u) /I 3.6.22
+IIBIIIIX(u-r)ll)du}i t?r where II X 11*
=
sup II X(t)
tE[-T,T]
II·
§3.6. X(t) = AX(t)
214
+ BX(t -
r)
Define t ~-r
3.6.23
and note that since f3 is a root of (3.6.21),
Z(t)
=
Mil X lI,e- o ('-r) + Mil B II
l' e-
O ('-'){
Lr
(II A IIIIZ(u)1I
+ IIBIIIIZ(u -
r)1I
)dU} ds 3.6.24
for t
~
r. We have from (3.6.22) - (3.6.24) that
3.6.25 where
Wet) = IIX(t)lI- Z(t).
3.6.26
From the definition of W in (3.6.26), Wet) < 0 for t E [-r, r] and Wet) is continuous for t ~ O. If IIBII 0, then we have from (3.6.25) that Wet) < 0 for t E (r, r + t) for some possibly small t> O. We shall show that Wet) < 0 for all t > rj for instance, if Wet) i- 0 for all t > r, then there exists a finite number t* such that t* = inf {t > r + tj Wet) ~ O}
t-
so that W(t*)
=0
and
Wet) < 0 for
t E [-r, t*),
t
=f r.
But in such a case we have from (3.6.25) that W(t*) < 0 and this is a contradiction. The result follows. [] The next result due to Khusainov and Yun'kova [1981] provides an alternative estimate on the delay parameter r in (3.6.16) for maintaining the asymptotic stability of the trivial solution of (3.6.17). Theorem 3.6.3. Assume that the trivial solution of (3.6.17) is asymptotically stable. Let C denote the real symmetric positive definite matrix satisfying
3.6.27
§9.6. X(t) = AX(t) where I is the n x n identity matrix. Let 70
= ( 2(IIAII + IIBII)IICBII )
+ BX(t -
70
215
7)
be the positive constant defined by
-1 (
Amin(C)/Amax(C)
)1/2
3.6.28
where Amin(C) and Amax(C) respectively denote the smallest and largest eigenvalues of C. Then the trivial solution of (3.6.16) is asymptotically stable for all 7
<
70·
Proof. The existence of a positive definite real symmetric C in (3.6.27) follows from the asymptotic stability of the trivial solution of (3.6.17). Consider a Lyapunov function veX) defined by
veX) = (XT, CX)
3.6.29
where (XT, Y) denotes the scalar product in Rn. For each real constant a > 0, the equation veX) = a defines a closed surface in IRn which we denote by avo. We let 3.6.30 Vo = {X E IRnlv(X) :::; a}. Let us first verify the following observation; suppose that for t an a > 0, and that
X(t) E avo Then for every
X(t - 7)/1 <
E
and
Xes) E Vo
> 0, there exists a
70
for
t - 27 :::;
such that for
S :::;
~ 7
there exists
t.
3.6.31
< 70 we will have IIX(t) -
7
EIIX(t)lI·
One can derive using the estimates of Lyapunov functions in Barbashin [1970] that (3.6.31) implies, 1/2
IIX(s)lI:::; ( Amax(C)/Amin(C) ) t-2r::;s9 sup
But we also have
IIX(t) - X(t - r)1I =
II
Lr (
AX(s) + BX(s -
IIX(t)lI.
r») dsll
:s r (IIAII + IIBII) t-,s;f,,,;, IIX(s )11 =
r(IIAII + IIBII) {Amax(C)/Amin(C)} 1/2 1IX (t)11
=
EIIX(t)/I
3.6.32
§3.6. X(t)
216
= AX(t) + BX(t -
r)
provided r
< ro = ,(II A I + IIBII) - I (A min(C)/Amax(C)),'2,
For arbitrary a > 0 and r > 0 we find a number 8(0', r) > 0 such that
IIX(t)11 < 8(0', r) for t E [-r,O] =* X(t) E
Vcr
t E [-r, r].
for
For instance, we have from
X(t) = X(O) +
l
(AX(S)
+ BX(s -
r)) ds;
t E [O,r]
that
If we choose 8 such that
then we have X(t) E above relation so that
Vcr
8( a r)
for t E (-r, r]. Thus, we are led to choose 8 from the
= e- IIAllr
[1 + /lBllr]-1 [0'/ Amax( C)]1/2 .
With these preparations we consider the rate of change of v along the solutions of (3.6.16).
~ V ( X(t»)
= _(XT(t) , CX(t»
«(
+ X(t -
r) - X(t)) T B T , CX(t))
+ (XT(t) , CB[X(t -
r) - X(t)]).
If X(t) E avcr and Xes) E Vcr for t - 2r ~ s ~ t, then we have that
~ v ( X(t») ~ -/lX(t)/l2 + 2€IICBIIIIX(t)1l 2
X (2e Il CBII-1).
:<; II (i)1I 2
§9.6. X(t)
= AX(t) + BX(t -
7)
211
Thus, for € < 211~BII' dV(~(t») < 0 and hence lIX(t)1I is decreasing in t. v «X(t») will be decreasing for all t > O. As a consequence v (X(t» - t 0 as t - t 00 implying that IIX(t)!! - t 0 as t - t 00 and this completes the proof. [J We remark that an estimation procedure similar to that in Theorem 3.6.3 has been extensively used by Burton (1983) for investigating stability of integrodifferential systems. A difficulty of this type of technique is the following: while the existence of the matrix C in (3.6.27) is easy to verify, it is not easy to determine C for given A and B; since the result in (3.6.28) depends on the maximum and minimum eigenvalues of the real symmetric matrix C, it will be usually difficult to apply this technique for specific systems except in low dimensional cases. If the trivial solution of (3.6.16) is asymptotically stable for 7 = 0, then we have seen that at least for small 7 > 0, the trivial solution of (3.6.16) is asymptotically stable. One can now ask the following question; under what conditions on A and B in (3.6.16), the trivial solution of (3.6.16) is asymptotically stable for all 7 2: o. In the following two results, we answer the above question by providing a set of sufficient conditions for such "delay-independent" (also known as "absolute") asymptotic stability of the trivial solution of (3.6.16). Theorem 3.6.4. (Gromova and Pelevina [1976]) Suppose that all the eigenvalues of A have negative real parts. Let H be the real symmetric positive definite n x n matrix satisfying 3.6.33
(1 being the n x n identity matrix). Define
V (Z(t» = ZT(t)HZ(t)
3.6.34
and let Z = KY where K is a constant n x n matrix which renders (3.6.34) in the form v(Y) = V(I
= yT [KT AT(1{-I)T + K- I AK] Y + 2y T K- 1 BI
3.6.35 3.6.36
(where UI , U2 are defined by (3.6.42) below) are negative definite, then the system (3.6.16) is asymptotically stable for all 7 > O. Proof. We calculate the magnitude of the rate of change of V along the solutions of (3.6.16) on the set of solutions satisfying V«Z(e» < V(Z(t»,e ~ t, t 2: 0
§9.6. X(t)
218
= AX(t) + BX(t -
7)
(such V is known as a Razumikhin function). First we transform (3.6.34) by means of a nonsingular linear transformation Z = KY so that we have v(Y) = V(KY) = yT(t)Y(t).
3.6.37
Then Razumikhin's condition becomes
yT(t - 7)Y(t - 7) ::; yT(t)Y(t);
t
~
o.
3.6.38
The fact that the largest value of ~~ along the solutions of (3.6.16) under condition (3.6.38) is negative, will ensure the asymptotic stability of the trivial solution of (3.6.16). The largest value of
in the region defined by (3.6.38) does not exceed the largest value of ~~ in the region determined by the inequality n
IYi(t - 7)1 ::; LIYj(t)I,·
i = 1,2, ... , n;
y = (Y!'Y2'" ,Yn).
3.6.40
j=1
Since the maximum of the right side of (3.6.39) in the region (3.6.40) is attained on the boundary of the region, majorants of the right side of (3.6.39) are obtained for i = 1,2,' .. ,n from (3.6.40) with
Viet - 7)
= Viet) ± LYk(t);
Viet - 7) = -Viet) ± LYk(t).
k=;ei
3.6.41
k=;ei
Define the vectors U1 , U2 as follows:
Y1(t) ± LYk(t)
1
:
;
k=;eI
U1
=
[ Yn(t) ± LYk(t) k=;en
-YI (t)
± LYk(t) k=;el
.
1 .
3.6.42
-Yn(t) ± LYk(t) k=;en
A set of majorants of the right side of (3.6.39) under (3.6.40) are given by WI and W 2 of (3.6.35) and (3.6.36) respectively. Thus, by Razumikhin's Theorem (Theorem 5, Razumikhin [1960]) a set of sufficient conditions for the quadratic forms (3.6.35) and (3.6.36) to be negative definite will provide sufficient conditions
§3.6. X(t)
= AX(t) + BX(t -
r)
219
for the asymptotic stability of the trivial solution of (3.6.16) and the proof is [] complete. We remark that one can employ Sylvester's conditions (see Gantmacher [1959]) for a set of sufficient conditions for the positive definiteness of the quadratic . forms - WI and -W2 • The next result illustrates an intuitive idea that if certain systems without delays are stable, then "small" perturbations involving delays can maintain stability whatever the size of the delay; the result is formulated in terms of a matrix measure, details of which can be found in section. 3.8 below. Theorem 3.6.5. Suppose f1(A) < 0 where f1(A) denotes a matrix measure of A. Assume that B is "small" so that
IIBII < -f1(A).
3.6.43
Then the trivial solution of (3.6.16) is asymptotically stable whatever the size of delay. Proof. The proof is accomplished by means of a Lyapunov functional v defined by 3.6.44
where B = {b ij }. Details are left to the reader as an exercise. The results of Theorems 3.6.4 and 3.6.5 motivate the following question; if the hypotheses of those theorems hold, can one obtain an estimate for the rate of convergence of the solutions of (3.6.16) to the trivial solution. An answer to this question might also indicate the dependence of the rate of convergence on the delay parameter. Note that even though the asymptotic stability of Theorems 3.6.4 and 3.6.5 is delay independent, the rate of approach to the trivial solution· can be delay dependent. The next result is extracted from the work of Huang Zhen Xun and Lin Xiao Biao [1982]. Theorem 3.6.6. Assume that the hypotheses of Theorem 3.6.5 hold. Then every solution of (3.6.16) satisfies the relation
IIX(t)1I ::;
Me-u(t-r)
for
t 2: r
3.6.45
§9.6. X(t) = AX(t)
220
+ BX(t -
where M is a positive constant depending on X( s), (j
=
Proof. Let X(t) , t 2:
I)
s E [-I, I] and
IIL(A)I-II B II
3.6.46
--..,--,.,--.....;.;.......~---=-~---,-.:---:-:~
(1
+ IIBII)(l + I) + 1(/tt(A)I-IlBII)
-I
be any arbitrary solution of (3.6.16). Define
IIX(llllr = t. [IXi(lll + Lpi(SlldS]
V(I,X(.)) and note that
=
t.[IXi(lll + t,lbii{rIXi(SlldS]
IIX(t)11 ~ V(I, X(.)) ~ (1 + IIBlIllIX(tlllr'
3.6.47
Calculating the right derivative D+v of v along the solutions of (3.6.16), 3.6.48 Define 0'.1, 0'.2, {3 as follows:
IIL(A)I-IIBII 0'.1(/) = (1 + 1)(1 + IIBII) + 1(ltt(A)I-IIBII) 0'.2(/) = (1 + IIBII) + (IIL(A)I-II B I/)I (3(s) = IIL(A)I-IIBII s + {ltt(A)I-II B Ii }/. 1+/
{3(0) =
3.6.50
3.6.51
1+/
We have from (3.6.51) that
{3( -I) = 0;
3.6.49
r(II'(All- IIBII) 1+I
{3' (0) = {3(0)
I
3.6.52
> O.
I
Define V as follows
t>1
3.6.53
= AX(t) + BX(t -
§9.6. X(t)
7)
221
and derive that D+V(t,X(.)) = D+v (t, X(.))
+ P(O)IIX(t)/I
- P( -7)IIX(t - 7)11
- L/(S - tlllX(slllds ~ -(IJ.t(All-Ii B II- .B(Ol)IIX (t lll -
.B~Ol
Lr
IIX(slllds
3.6.54
::; _ (lfl(A;~-}BII) IIX(t)lIr < _{ Ifl(A)I-IIBII} V(t,X(.)) 1+7
-
1 + !lBIl
+ 1'(0)'
3.6.55
which on integration leads to
V(t,X(.)) ~ V
(r,x(.)) exp[-q(t - rl],
3.6.56
where (j
={
Ifl(A)1 -/lBI!} { 1 } 1+7 1 + IIBII + ,8(0)
from which the result will follow on using (3.6.47) and (3.6.51).
[]
Theorem 3.6.7. Assume that the hypotheses of Theorem 3.6.5 hold. Then, any solution X of (3.6.16) satisfies 1/ X(t)
/I ::; Me-at
3.6.57
where M is a positive number (depending on the initial values of X(.)) and a satisfies 3.6.58
Proof. It is sufficient to show that the real parts of the roots of the characteristic equation 3.6.59
= AX(t) + BX(t -
§3.6. X(t)
222
associated with (3.6.16) satisfy
~e(A)
G(z) = F(z - 0:)
where Al
= 0:1 + A,
r)
< -0:. Define G such that
= (z -
0:)1 - A - Be-T(z-a)
= z1 -
Al - B1e- TZ
3.6.60
Bl = BeaT. By Theorem 3.6.5, the solutions of the equation
det G(z) = 0 satisfy
~e
(z) < 0 or equivalently 1\ Bl 1\
~e
3.6.61
(A) < -0: if
+ /-L(Ad = II B Ile aT + fleA) + 0: < 0
3.6.62
[]
and this completes the proof.
We remark that the least upper bound of the decay rate 0: in (3.6.62) can be obtained as the unique solution of 3.6.63 For more details of the decay rate of solutions of delay differential equations and integro-differential equations we refer to Mori et. al. [1982] and Gopalsamy [1983b]. We proceed to develop results related to comparison of solutions of linear systems of ordinary differential equations based on the work of Chew [1976]. We recall the following componentwise order relations in IRn and IR nxn : if y E IR n
and
z E IR n ,
then y ::; z
~
Yi ::; Zi, i = 1,2, ... nj
if Y E IR n
and
z E IR n ,
then y < z
~
Yi
then A Theorem 3.6.8. Let A = satisfying aii aij ::;
[aij]
~
B
~ aij ~
and B = [b ij
]
<
,i = 1,2 ... ni
Zj
bij , i, j = 1,2, ... n.
denote
n
x
n
> 0, bi ; > 0 i = 1,2, ... n.
0, bij::; OJ aij ~
i i=j, i,j = 1,2" ...
bij ; i,j = 1,2, ... ,no
constant matrices
§$.6. X(t) Define Ta , Tb : R x R
t---+
Rnxn
= AX(t) + BX(t -
r)
as follows:
TA(t, s)= diag
[e -a,,(t-.) , e-a,,(,-.) ... , e:c ann (t-.)]
TB( t, s) = diag
[e-'" «-.) , e-.,,(,-.) ... ,e-'nn('-.)].
Assume that 'x, y : [0,00)
~
223
3.6.64
Rn are continuous satisfying
x(t) :::; TA(t, O)e yet) 2: TB(t, O)e -
1.' 1.'
TA(t, s)[ A - AD Jx(s) ds
3.6.65
TB(t, s)[ E - ED Jy(s) ds
3.6.66
where AD = diag (all, aZ2, . .. ,ann) , BD of (3.6.65) or (3.6.66) is strict; then
x(t) < yet)
= diag (bll , bzz , . .. ,bnn ) and at least one for all
t
~
0.
3.6.67
Proof. We first observe, by the continuity of x and y, the sets
Si = {tE [0,00 )IXi(t)
~
Yi(t)} , i = 1,2, ... ,n
are closed and therefore the set S = U~l Si is also closed. It is sufficient to show that the set S is an empty set. Let us suppose that S is not an empty set. We can then define to as follows: to = inf{t E [O,oo)lt E S}. We note x(O):S; c and y ~ c with at least one being strict; also we have x(O) < y(O); these facts imply to > 0. Since S is closed, to E S and therefore there exists an i E {I, 2, ... ,n} such that
x(t) :; yet),
°:; t :; to
3.6.68
with
Xi(tO) = Yi(tO)'
3.6.69
But we have from (3.6.65) - (3.6.66),
x(to) :::; TA(to, O)e - [ ' TA(to, s) [A - AD Jx(s) ds
:::; TB(to, O)e - [ ' TB(to, s)[ E - ED Jy( s) ds :S;y(to).
3.6.70
Since at least one of the inequalities in (3.6.65) or (3.6.66) is strict, it follows that x(to) < veto) and this contradicts (3.6.69). Thus, the set S is empty and hence x(t) < yet) for all t ~ 0. The proof is complete. []
§J.6. X(t) = AX(t)
224
+ BX(t - 7")
Corollary 3.6.9. In tbe linear system of autonomous ordinary differential equa-
tions
dx(t)
-dt- + Ax(t) = let the constant matrix A
= (aij)
0 t
> O'
" be real with
x(O) aij :::;
x(O) = Xo > 0 E Rn =:::;. x(t) > 0 E Rn
= Xo 0, i
3.6.71
ERn
of j.
for
Then
t
~
O.
3.6.72
Proof. We can rewrite (3.6.71) in the fonn
x(t) = TA(t, O)xo Since Xo > 0,
x(t) > -
J.'
J.'
TA(t, s)[ A - AD Jx(s) ds.
3.6.73
TA(t, s)[ A - AD Jx(s) ds.
3.6.74
By Theorem 3.6.8 we obtain,
x(t) > yet) for t
~
3.6.75
0
where yet) is a solution of
y(t) = -
J.'
TA(t, s)[ A - AD Jy(s) ds.
3.6.76
But (3.6.71) has the unique solution yet) == 0 on [0,00) and hence the result follows [] from (3.6.75). Corollary 3.6.10. Let the constant matrices A = (aij), B = (bij) be such that (i) aii > 0, i = 1,2, ... ,n (ii) bi ; > 0, i = 1,2, ... ,n j bij :::; 0, i of j , i,j = 1,2, ... ,n (iii) 3.6.77 If
dx(t)
= 0,
t
~ 0;
x(O) = Xo
d~~t) + By(t) = 0,
t
~ OJ
y( 0) = yo > I x 0
--;It + Ax(t)
I
§S.6'. X(t) = AX(t) + BX(t - r)
225
o.
3.6.78
I xCt) I < yet) Proof.
By Corollary 3.6.9, yet)
yet)
for all
t 2:
> O. But y satisfies
= Ta(t,O)yo -
J.'
Ta(t, s) [B - BD 1yes) ds,
3.6.79
while x(t) satisfies
I x(t) I < TA(t, 0)1 Xo 1Since TB(t, O)Yo
J.'
TA(t, s)[ A - AD 1x(s) ds.
3.6.80
> TA(t, 0)/ Xo I, we have
J.' 1- J.'
yet) > TA(t, 0)1 "0 1-
~ TA(t, 0)1 xo
Ta(t, s)[ B - BD 1y(s) ds TA(t, s) [A
- AD 1y(s) ds.
An application of Theorem 3.6.8 to (3.6.80) and 3.6.81) leads to (3.6.78).
3.6.81 []
The following results are concerned with comparison and convergence characteristics of systems of delay differential equations and inequalities of the form
du.(t) n n dt < - "" ~ a"u I) ) ·(t) + "" 6 b··u I) I·(t - r:I}·(t»
_ I_
i=l
i = 1,2,"', n.
,
i=l
Proposition 3.6.11. (Tokumaru et. al. [1975]) Consider the systems
dx(t)
-;It ::; Ax(t) + Bi(t) , t > 0
3.6.82
dz(t)
--;J,t = Az(t) + Bi(t) , t > 0
3.6.83
where
z(t) z(s) 2: xes),
s E [-r, 0],
= {Zl(t) r
... , Zn(t)}T
E (0,00)
3.6.84
+ BX(t -
§3.6. X(t) = AX(t)
226
x(t) = {
sup
Xl(S),...
sE[t-r,t]
i(t) = {
Xn(S)}T
sup
3.6.85
sE[t-r,t]
Zl(S),...
sup
r)
sE[t-r,t]
Zn(S)}T.
sup sE[t-r,t]
Suppose further iij,
i,j=1,2, ... ,n
i,j=1,2, ... ,n.
bij~O,
Then
xCi)
~
z(t)
for
t
~
0.
Proof. Our strategy of proof is to show first that every solution y of
d~~t) > Ay(t) + By(t), y(s»x(s),
t > 0, yet) E R n
3.6.86
sE[-r,O]
°
satisfies yet) > x(t) for all t ~ and then apply a limiting process. Suppose there exists a positive number fJ and an integer j such that for the j -th component of xCi) and yet), Xj(fJ) = yiCfJ)· Then there exists fJo such that for some j}.
fJo = inf{fJlxj(fJ) = yj(fJ) But fJo
3.6.87
°
> since xes) < yes) for s E [-r,O]. For this fJo
and there exists a k, 1 ~ k ~ n for which Xk(fJO)
Xk(fJO) ~
~
= Yk(fJo)i hence
n
n
m=l n
m=l n
I: akmXm(fJO) + L
bkmxm(fJo)
I: akmYm(fJO) + I: bkmYm(fJO)
m=l
m=l
3.6.88 But by the definition of fJo, we have Yk(fJO) ~ Xk(fJO) which contradicts (3.6.88) and therefore we have yet) > x(t) for all t ~ 0.
§3.6. X(t)
= AX(t) + BX(t -
227
r)
To complete the proof we let €* E Rn denote a vector each of whose components is equal to an arbitrary positive number €. Let ze(t) denote the solution of
Ze(t) = AZe(t) + BiE(t) + €*
> AZc(t) + BiE(t),
t>0
3.6.89
with the initial condition
Zc(S)
= x(s) + €* >
x(s), s E [-r, 0).
By the above discussion,
Z€(t) > x(t) for t > 0. Since Z€ depends continuously on
€,
3.6.90
we can conclude
Z(t) = lim z€(t) ;::: x(t) for t E--O
~
° []
and the proof is complete.
For the convenience of the reader we recall from Chapter 1 the following result on a scalar differential inequality due to Halanay [1966]: Proposition 3.6.12. (Halanay [1966]) Let to be a real number and r be a nonnegative number. If f : [to - r, (0) I---? IR+ satisfies
dfd(t) t and if a >
~ -af(t) + (3[
(3 > 0, then there exist I
sup
f(S)];
3.6.91
SE[t-T,t}
°
°
> and ~ > such that 3.6.92
The above result of Halanay has been generalized to a class of vector-matrix systems of differential inequalities by Tokumaru et.al. [1975]. In preparation to present their result we note a few properties of M -matrices (see below for a definition) formulated for convenience in the form of the following two propositions.
228
§9.6. X(t)
= AX(t) + BX(t -
r)
Proposition 3.6.13. (Araki and Kondo [1972]) Let P = (Pij) be an n X n matrix with Pij :::; 0 for i '1= j. Then the following conditions are mutually equivalent. 1. There exists a positive vector x such that Px > O. 2. The matrix P is nonsingular and p- 1 2:: 0 (elementwise). 3. All the successive principal minors of P are positive; i.e. Pll
det [PH P21
PH
det
[
> 0, P12] P22
P12
> 0,
PIn
1
~~~ .. ~2.2...........~~~ > O. Pnl
Pn2
••.
Pnn
4. The real parts of all the eigenvalues of P are positive.
P can be put in the form P
= pI -
A where A =
(aij) , aij
2:: O. The following
facts about M matrices can also be found in Araki and Kondo [1972]. Proposition 3.6.14. Let A = [aij] be a real n x n matrix. LetB = (pI - A] where I denotes the n x n identity matrix. Then the following hold. 1. If we increase some elements of an M -matrix so that no element changes sign, then the new matrix is an M -matrix. 2. If we multiply a row or column of an M -matrix by a positive number, then the new matrix is an M -matrix. 3. The matrix pI - A is an M -matrix if and only if p > AA where AA denotes the nonnegative eigenvalue of A. 4. An M -matrix has a positive eigenvalue AA such that, if f3 is the maximum
element on the main diagonal, then f3 2:: AA and for any eigenvalue W A of A,
5. If A is an M-matrix, then A - III is an M-matrix, if and only if Il < AA' Definition. A matrix Q = [qij] with qij :::; 0, i 1= j is said to be an M-matrix if anyone of the four equivalent conditions of Proposition 3.6.13 bolds.
§S.6. XCi)
= AX(i) + BX(i -
r)
229
Theorem 3.6.15. (Tokumaru et. al. [1975)) Let A, B be real n x n matrices and let xCi) E Rn denote a solution of the system of differential inequalities
xCi)
~
-Ax(t) + Bi(t),
t>O
3.6.93
°
where i is defined by (3.6.85). If B ~ and if A - Bis an irreducible M -matrix, then there exist a number b > and a vector XO E Rn with positive components satisfying for t ~ 0. 3.6.94
°
Proof. We shall first show that the system
i(t) = -Az(t) + Bz(t), z(s)~
k,
S
t>o
E [-r, OJ,
kE
3.6.95
h~s a solution of the form
O
3.6.96
and then apply Proposition 3.6.11 to complete the proof. If z(t) = ke- 6t is a solution of (3.6.95), then i(t) = -bke- 6t and = ke- 6t e 6T and therefore -bk = -(A - BehT)k. Thus, 8 is an eigenvalue of the matrix (A - Be 6T ) and k is the corresponding eigenvector. Conversely, if there exist a positive kERn and a number 8 > satisfying 3.6.97
z
°
then z(t) = ke- ot is a solution of (3.6.95) with initial value z( s) = ke- 6s , s E [-r,O}. Define a map F(.) : [0,00)
t--t
Rnxn as follows:
F(O")
=A-
Be UT
•
3.6.98
Let -\(0") denote the minimum of the absolute values of the eigenvalues of F( 0"). We first verify that -\( (J") is an eigenvalue of F( 0"). We can write
F(ul=aI- [aI-(A-Be OTl ] =aI-P(O")
3.6.99
§3.6. X(t) = AX(t) + BX(t - r)
230
and observe that the matrix P( a) 2: 0 where Q is the maximum of the diagonal elements of A-B. From the properties of M- matrices (Araki and Kondo [1972]), F(a) is an M- matrix, ifand only if one of the following hold: where [F(a)]-l E Rnxn (i) [F(a)J-l ~
°
(ii) Q> p(p'(a»)
where
p(P(a») denotes the spectral radius of pea).
If al < a2, then irreducibility of F(at) will imply that of F(a2) and furthermore, p[P(al)] S p[P(a2») since peal) S P(a2). By hypothesis, F(O) is irreducible; hence F(a) is irreducible for a ~ 0. If F(a) is an M- matrix, (F(a)]-1 is an irreducible nonnegative matrix by (i) above. The well known Perron-Frobenius theorem (Gantmacher (1959]) guarantees that p([F(a)]-l) is an eigenvalue of
[F(a)]-l and the associated eigenvector k(a) is positive (componentwise). It is clear that >.(a) = IIp( [F(a)]-l) and >.(a) is an eigenvalue of F(a) with k(a) as the corresponding eigenvector of F(a). We have from >.(a) = Q - p[F(a)J that -\(a2) S -\(a1) for a1 S a2 and -\(0) > since F(O) = A - B is an Mmatrix. From the properties of M -matrices, F( a) cannot be an M - matrix for a sufficiently large a > 0. Hence A( a) S for large enough a > 0. It follows from all these facts, that A( a) > so long as F( a) is an M - matrix and A( a) continuously approaches zero. Therefore, the equation A( a) = a has a positive root ao and the corresponding eigenvector k( ao) is positive. It follows now that z(t) = k(ao)e- tTot is a solution of i(t) = -Az(t) + Bi(t). For any continuous initial value xes), s E [-r,O), xes) E IR+., one can find a f3 > such that x(o) S f3k( ao) == k and z(t) = ke- 6t , 8 = ao is a solution of i(t) = -Az(t)+Bi(t). The result follows by an application of proposition 3.6.11.
°
° °
°
n
In order to illustrate the applicability of the result of Proposition 3.6.13, we consider the linear system dx .(t) 3 -dt '- = """ ~ a"x t} } ·(t j=1
3
-
·(t»
T't}
+ """ b··x '(ft ~
I}
}
m t}.. ]).,
i = 1,2,3
3.6.100
j=l
where x(t) = {Xl(t),X2(t),X3(t)} E R 3 ;aij,bij E lR;i,j = 1,2,3;rij : [0,00) 1--+ [0, roJ; mij EN, {i,j = 1,2, 3}, (PJ denotes the greatest integer contained in pER and Xi(t) denotes the right derivative of Xi at t. Except for notational complexity and inconvenience, there is no difficulty in extending the following analysis of (3.6.100) to vector systems with any finite number of
§3.6. X(t)
= AX(t) + BX(t -
1')
231
components. We assume i,j = 1,2,3.-
For t 2: 21'0
+ 2(m + 1), we can write (3.6.100) in the form
3.6.101
3.6.102
where i,j = 1,2,3.
For any fixed t 2: to = [21'0 + 2(m + 1)], the possible sign pattern of the components XI(t), X2(t), X3(t) of the vector x(t) E R3 is as follows: we can without loss of generality assume that Xl(t) 2: 0 since otherwise, we can multiply the corresponding equation governing Xl in (3.6.100) by (-1) and restore XI(t) 2:: O. With this choice for Xl, we have the following sign pattern for x(t) for any fixed value of t:
{+,+,+}, {+,+,-}, {+,-,+}, {+,-,-} (If x(t) E IRn, then we will have 2 n- 1 possibilities of sign combinations for the components of x(t)). We write
§3.6. X(t) = AX(t) + BX(t - T)
232 where
J1
= {t ~ toIXi(t) ~ O,i =
1,2,3}
J 2 = {t ~ toIXl(t) ~ 0,X2(t) ~ 0,X3(t)
< O}
J 3 = {t ~ toIXl(t) ~ 0,X2(t) < 0,X3(t) ~ O} J 4 = {t ~ to/Xl (t) ~ 0, X2(t)
< 0, X3(t) < OJ.
For any t E J1 , we have from (3.6.102),
where
I x(t) 1= {l x l(t)I, IX2(t)l, IX3(t)I}T !AI2=IAlxIAl,
IAI=(!aijj)
IB 12 = IB I x / B I,
1B I =
I X ICt) = I Xj 1Ct)
(I bij I)
{Ixd(t), IX2!Ct), IX31(t)}T
=
sup
Ix j ( S ) I
sE[t-2(ro+m+l),t)
C1
=
al1+bll max(O, a21 + b21 ) [ max(O, a31 + b3J)
max(0,aI2+ b12) max(0,aI3+b13)] a22 + bzz maxCO, a23 + b23 ) . max(O, a32 + b32 ) a33 + b33
We rewrite (3.6.103) so that
~ Ix(t)1 :0; -
[-
Cd x I(t) - {I A I'ro +21 A II B I(ro +m+ 1)+ I B I'(m+ 1) } I x I(t)]. 3.6.104
If we assume now that
-{ C.
+ I A I'ro + 21 A II B I(ro + m+ 1) + I B I'(m + 1)}
is an M-matrix, then by Proposition 3.6.13, it will follow that there exist 81 > 0 and a positive vector kl' such that 3.6.105 Now let t E
Jz .
Define T2 as follows:
Tz =
[~ ~ ~].
°°
-1
§3.6. X(t)
= AX(t) + BX(t -
r)
233
It is easy to see that T2X(t) = I x I(t) so that
:t 1x I(t) :::; T2(A + B)T;-l [T2X(t)] +
{I A 12ro + 21 A II B 1(/6 + m +
1)+
I B 12(m + 1)}1 x I(t) :::; C21x I(t) + {I A 12/0 + 21 A II B I(ro + m + 1) + I B 12(m +
3.6.106
1)}1 x I(t)
where
C2 =
all + bl l max{O, (a2l + b21 )} [ max{O, -(a31 + b31 )}
max{O,a12 + b12 } a22 + b22 max{O, -(a32 + b32 )}
max{O, -(a13 + bI3 )}] max{O, -( a23 + b23 ) • a33 + b33 3.6.107
Again if we suppose that the matrix
is an M -matrix, then it will follow as before, that there exist 82 > 0 and a positive vector k2 such that t E J2 • 3.6.108 Suppose now t E J3 j define a matrix T3 so that
Ta =
[~ ~l ~]
One can derive again, that
d dt I x I(t) :::; T3(A
+ B)T3- 1 x I(t) + {I A 12/0 + 21 A II B 1(/0 + m + 1)+ IB 12(m + 1)}1 x I(t) 3.6.109 2r :::; C3 1x I(t) + [I A 1 o + 21 A II B I(ro + m + 1) I B 12(m + 1)]1 x I(t) 3.6.110 1
where max{O, -(aI2 + bI2 )} a22 + b22 max{O, -(a32 + b32 )}
max{0,aI3 + b13 } ] max{O, -(a23 + b23 )} a33
+ b33
•
3.6.111
§3.6. X(t)
234
= AX(t) + BX(t -
T)
If the matrix
is an M-matrix, then there will exist 83
>
°and positive vector k3 E IR~ such that 3.6.112
Finally if t E J 4 , one considers the matrix
and derives that
~ 1x I(t) ::; T4(A + B)T4- I I x I(t) + {I A 12To + 21 A II B I(To + m + 1)+ 1B 12(m + I)}I x I(t) 3.6.113 2T ::; C4 x I(t) + {I A O + 21 A II B I(To + m + 1) 3.6.114 + I B 12(m + I)}I x I(t) 1
1
where max{O, -( al3 + b13 )}] max{O, a23 + b23 )} [ max{O,a32 + b32 } a33 + b33 3.6.115 from which one can conclude that there exist a positive vector k4 E 1R3 and 84 > 0 such that C4
=
all + bl l max{O, -( a2I + b2 J)} max{O, -(a31 + b31 )}
+ bI2 )} + b22
max{O, -( al2 a22
We can summarize the above analysis in the form of the following:
Proposition 3.6.16. If the following matrices
- (C j
+ [IA 12 TO + 21 A I B 1(To + m + 1)+ IB 12(m+ 1)]) j = 1,2,3,4
3.6.116
§3.6. X(t) = AX(t)
+ BX(t -
r)
235
are M -matrices, then the trivial solution of (3.6.100) is asymptotically stable (in fact exponentially asymptotically stable). Proof follows immediately from our discussion and
I x(t) I :::; ke- 6t
where
3.6.117
As an example of a linear system of differential equations with unbounded delays, we consider a linear system of the form
x(t) = Bx(t)
x(t) E IR n ,
+ AX(At), t>O A> 0,
3.6.118
A, BE IRnxn.
The linear system (3.6.118) has been investigated in detail by Lim [1976] from where we have extracted the following result.
Theorem 3.6.17. Let
Let A
= [aij]
°<
A < 1. Let B
= diag (bI, b2 , ••• bn )
with
E IRnxn and let a be defined by 3.6.119
Then every solution of (3.6.118) satisfies
x(t) = OCtO')
as
t
- t 00.
3.6.120
Proof. We write (3.6.118) as follows: n
Xi(t) =
L aijXj(At) + biXi(t);
i = 1,2,3, ... ,n
j=l
and let
t = e\
c = log A < O.
3.6.121
§3.6. X(t)
236
= AX(t) + BX(t - T)
Then (3.6.121) becomes
1
atQ-1wi(S)
+ tQWi(S)( -) = I:: )..QtQajjwj(s + c) t . n
;=1
+ tQbjwj(s) or
n
Wi(S)
+ (a -
b;eS)w;(s)
= L )..QajjeSwj(s + c); i = 1,2, .. , n.
3.6.122
j=l
The rest of the proof is similar to the corresponding scalar case .and for more details of the proof we refer to Theorem 1.2.23 of Chapter 1. We omit the rest of the proof. (J We now consider the quasilinear system
dx(t)
dt
= A(t)x(t)
+ I(t, x(t -
T(t»)
3.6.123
with an unbounded delay in which x is an n-vector, A is an n x n matrix of continuous functions for t ~ to > 0, I is an n-vector, continuous in all its arguments in a neighbourhood of II x II = 0; T(t) ~ 0 is a continuous function which can increase unboundedly for t ~ 00 and is such that
T(t) ::; )..t,
o < ).. < 1, t ~ to.
3.6.124
A continuous initial function
Eo =
{I I
1 = s - r( s) :5 10 ; s
~ 10 }.
For applications of equations of the type (3.6.123) we refer to Fox et al. (1971]. Let X(t) be the fundamental solution of
d~;t)
= A(t)X(t),
3.6.125
X(to) = I.
We assume that X satisfies k e -a(t-s) ,
a
> 0,
k
~
1,
to::;
S ::;
t<
00
3.6.126
and in a neighbourhood of 1/ xl/ = 0,
II I(t, x) II ::;
bllxW',
b> 0, v > 1,
t > to.
3.6.127
§9.6. X(t) = AX(t)
+ BX(t -
We also assume mo
= max II
ml
297
7)
a) "~l < (2bk
I
3.6.128
Define
f3 =
3.6.129
-log[v] , log[l - 0'0]
Theorem 3.6.18. (Rozhkov and Popov [1968]) Assume that (3.6.125) - (3.6.129) are satisfied. Then eve.ry solution of (3.6.123) satisfies 3.6.130
> tl and consider the scalar comparison equation
Proof. Suppose first that to
dy(t) = -ay(t) -dt
) + (af- 3 -P-totl-P
Y v (t - O'ot)
3.6.131
which has the particular solution
y( t)
= exp { -
C:)1
3.6.132
We prove that yet) in (3.6.132) majorizes the solution x(t) of (3.6.123). We have from (3.6.123),
x(t) = X(t)
+
it
X(t)X-l( s)1 (s, x( s - 7( S ))) ds.
3.6.133
to
We have also 3.6.134
We let 3.6.135
§3.6. X(t) = AX(t)
238
+ BX(t -
7)
in (3.6.133) and obtain, 3.6.136 leading to
II Xl (t) II < e-a(t-tO)-1 + ~
2
1t
e-a(t-s)
II Xl( S -
7( S))
II v ds.
3.6.137
to
From (3.6.137) and (3.6.134) ,
\I X, (t) 1\- yet) <
1:
e-·(I-»
[~I\ XI(S -
res)) II'
- (a - _13 ) yVes - aos)dsj. to
3.6.138
f3 s l -f3
We replace yes - aos) by yes - 7(S)) and the coefficient of yVby obtain
II XI(t) 11- yet) < ~
1:
e-·(H)
(II X, (s -
res)) II"
-
~
in (3.6.138) and
yVes - r(s))) ds.
3.6.139
By virtue of (3.6.128), II Xl(t) 11- yet) < 0 on Eo. Since II Xl(t o) 11- y(to) < 0, by continuity, IIXI (t)1I - yet) is negative in some neighbourhood of to. We show that II XI(t) 11- yet) < 0 everywhere for t > to. Assume the contrary and let t* > to be the first point where
II XI(t*) II = y(t*). By assumption, II Xl(t) II Hence, (3.6.139) implies
< yet) for t < t* and so IIxI(t)IIV < yV(t) for t < t*. 0=
II XI(t*) 11- y(t*) < 0
and this contradiction proves the assertion that
II Xl(t) 11- yet) < 0
for
t > to·
Thus,
t 2:: to·
3.6.140
§3.6. X(t) = AX(t)
+ BX(t -
239
r)
Now let tl > to. For t E [to, tIl, we can use the following majorizing equation for
Xl(t), 3.6.141 which has a particular solution yet) = ~. For t ?:: t l , we use the majorizing equation (3.6.131) and proceed as before to ·obtain 3.6.142 Comparing (3.6.142) and (3.6.140) one obtains (3.6.130).
[]
3.7. Stability switches
A primary purpose of this section is to develop techniques to perform a comparative study of linear systems with and without delays. In particular, we are interested in finding conditions for delays not to destabilize an otherwise stable linear system. It is also of interest to examine whether delays can stabilize otherwise unstable systems. Furthermore, it is possible for a system to be stable for a small delay; if the delay is longer, the system can become unstable and for still longer delays the system can regain stability; this sequence of switching from stability to instability and back to stability can repeat if the delays progressively become longer (see Cooke and Grossman [1982]). Our analysis below will be restricted to systems with one delay; a number of generalizations are indicated in the exercises. The results of this section and the relevant exercises are selected from the works of Cai Sui Lin [1959J, Qin Yuan-Xun et 81. [1960], Chin Yuan Shun [1962J, Wang Lian [1962] and Chang Hsueh Ming [1962J. Consider the linear delay differential system
dx~?) =
t
(ajk;k(t)
+ bjkXk(t -
r») ;
j=1,2, ... ,n
3.7.1
k=l
where ajk, bjk are real constants and r ?:: O. Let the characteristic equation associated with (3.7.1) be denoted by 3.7.2 The following result provides conditions for the absence of delay-induced switch from stability to instability.
§3.7. Stability switches
240
Theorem 3.7.1. A set of necessary and sufficient conditions for the trivial tion of (3.7.1) to be asymptotically stable for all r ~ 0 is the following:
solu~
(i) the real parts of all the roots of 3.7.3 are negative. (ii) for any real y and any r
~
0, the following holds: 3.7.4
in which i = yCI. Proof. It is easy to verify the necessity of the conditions (3.7.3) and (3.7.4). For instance, if (i) does not hold then the trivial solution of (3.7.1) is not asymptotically stable for r = O. If there exist a real number y and some r ~ 0 such that
D(iy, r) = 0
3.7.5
then for such r, the characteristic equation (3.7.2) will have a pair of pure imaginary roots and hence the trivial solution of (3.7.1) is not asymptotically stable. To prove the sufficiency part of the result, we have to show that when the conditions (i) and (ii) hold, all the roots of (3.7.2) have negative real parts. We note that we can rewrite (3.7.2) in the form 3.7.6 where
AI,'"
ajk, bjk
(j, k = 1,2, ... , n) are known constants and
An are polynomials in
ajk,
bjk
and
e-
Ar
.
We assume that
3.7.7 Thus, for ~e( -\) ~ 0 and r ~ 0, the coefficients Aj (j == 1,2, ... ,n) in (3.7.6) are bounded in absolute value. Let 3.7.8
241
§3.7. Stability switches
and define
M = max For IA I 2:: M and
~e( A)
(1, (n + l)N) > O.
3.7.9
2:: 0 we will then have
I( _l)n An + A1A n- 1 + ... + Ani;::: 1),[" [1- I~II -... - I~II] ~Mn[l- (n:~)N] which implies that in the domain IAI 2:: M and no roots and this is valid for all r 2:: o.
~e(A)
> 0,
3.7.10
::::: 0, the equation (3.7.2) has
Now let us examine the region IAI < M, ~e(A) 2:: 0 and show that under (i) and (ii), this region also cannot have roots of (3.7.2). By condition (i) we know that for r = 0, the roots of (3.7.2) are all in the half-plane ~e(A) < O. Now for r t= 0, the only possibility that the roots of (3.7.2) can fall in the region ~e(A) > 0 is that for some r > 0, one or more roots of (3.7.2) lie on the imaginary axis of the complex A plane between -M and Mj but (3.7.4) will not permit any of the roots of (3.7.2) to lie on the imaginary axis of the A- plane for any r 2:: and therefore all the roots of (3.7.2) will be such that ~e(A) < when (i) and (ii) simultaneously hold. (]
°
°
The result of Theorem 3.7.1 provides conditions for the absence of a delay induced switch from stability to instability in (3.7.1). The next result gives sufficient conditions for the absence of a delay-induced switch from instability to stability in (3.7.1).
Theorem 3.7.2. Suppose we have in (3.7.2) that (-ltD(O,r)
= (-l)nD(O, O) < 0
and furthennore, that D(A,O) = 0 has an odd number of roots with positive real parts. Then (3.7.2) has at least one root with a positive real part for all r 2:: O. Proof. Define a continuous real valued function arbitrarily fixed r 2:: 0, we let
f (0:) = (-1 t
D( 0:, r);
f : [0,00)
T
2::
o.
-7
(-00,00). For an
3.7.11
§3.7. Stability switches
242
We note
f(O) = (-lr D(O, r) < 0
(by hypothesis)
and
f(a)
-+
00
as
a
-+
00
r 2 0.
for all
It will follow that there exists a real number say a* such that a* > 0 and
f(a*) = (-ltD(a*,r) = 0 for any
r 2 0
showing that the trivial solution of (3.7.1) is unstable for all r 2
o.
[J
We remark that the additional assumption in Theorem 3.7.2 regarding the existence of odd number of roots with positive real parts is necessary, although, this assumption has not been used in the proof. A counter-example showing the necessity of this assumption is formulated in Exercise 30 at the end of this chapter. The condition (ii) of Theorem 3.7.1 is not in a form convenient for applications. So, let us examine this condition further by considering two possibilities: y = 0 and y =f 0. For y = 0, the condition (3.7.4) becomes
D(O, r)
= det[ajk + bjk ] =f
°
3.7.12
and (3.7.12) is valid for all r 2 0. For y =f 0, let us suppose that r varies on the interval [0,211" Ilyll implying that IrYI will vary in [0,211"]. This will mean that iry e will vary over the unit circle. Thus, for y =f 0, we can let ry be another independent variable say (J (i.e. (J = -ry), and derive the following two conditions in the place of (ii) in Theorem 3.7.1:
( iii) {
det[ajk
+ bjk ] =f 0
H(y, (J)
= det[ajk + bjkeiu -
for nonzero real
iy8j k]
=f 0
3.7.13
y and any real (J.
Since (J and yare regarded as two real independent variables, we can write
H(y,(J) where
= F(y,(J) +iG(y,(J)
3.7.14
§3.7. Stability switches The equation H(y,a)
243
= 0 will lead to two equations
O}
F(y,a) = G(y, a) = 0
3.7.15
from which by eliminating either of y or a, we can derive an equation of the form
u (y) = 0 in terms of y
or
U ( cos 0' , sin (7)
=0
3.7.16
i- 0 or cos 0' and sin a.
Now if U(y) = 0 has no real nonzero roots y, then the second of (3.7.13) holds or if U(y) = 0 has a real nonzero root and if for such a y i- 0, the two equations in (3.7.15) have no common real root a, then also the second of (3.7.13) holds. Thus, we have to check only these facts to prove the validity of the second of (3.7.13). We note that checking these aspects will involve only algebraic equations and not the solving of transcendental equations (see the examples below). For convenience, we summarize the above discussion. Theorem 3.7.3. A necessary and sufficient condition for the trivial solution of (3.7.1) to be asymptotically stable for all 7 ~ 0 is that tbe following bold. (a) all the roots of 3.7.17
have negative real parts. ((3) the equation U(y) = 0 of (3.7.16) either has no real root or if U(y) = 0 of (3.7.16) bas real root, then for such a real root y, the two equations (3.7.15) bave no common real root a. An alternative to (f3) above is provided as follows: (fi') the equation U(cos a, sin a) = 0 of (3.7.16) mayor may not have real roots 0'; if such roots exist, then for those roots, the two equations F(y, 0') = 0, G(y,O') = 0 of (3. 7.15) have no common real nonzel'O roots y. Let us now consider a few illustrative examples; first we discuss the scalar equation
dx(t)
-;It
= ax(t)
+ bx(t - 7).
3.7.18
§3.7. Stability switches
244
Suppose a + b > 0; then the trivial solution of (3.7.18) is not stable for If A is a root of the characteristic equation of (3.7.18) for 7 > 0 satisfying a
+ be-AT -
A = 0,
7
== O.
3.7.19
then look at the roots of
D(A,7) = ae AT For real A ~ 0 and
7
+b -
Ae AT = O.
3.7.20
> 0 we find, D(O, 7) =
a
+b> 0
and lim D(A, 7) = lim (a - A)e AT ')'''''''00
+b=
-00.
A-+OO
°
It follows that there is at least one real root A = >'(7) > satisfying (3.7.20) implying that for all 7 > 0, the trivial solution of (3.7.18) remains unstable. No switch from instability to stability can take place due to increase in 7.
Let us suppose that a+b < 0 in (3.7.18) so that the trivial solution of (3.7.18) is asymptotically stable for 7 = 0. We want to find additional conditions on a and b (if any) so that the trivial solution of (3.7.18) will be asymptotically stable for all 7 > O. We have to verify the condition (ii) of Theorem 3.7.1. We note H(y,a)
implying that
= (a + bcos a) + i( -y + bsin a) = F(y,a) + iG(y,a) = 0
F( y, a) = a + b cos a G(y,a)
=0
= -y + bsin a
= 0,
and hence A necessary and sufficient condition for U (y) =
Now if U(y)
= 0 has nonzero real roots, then
°
not to have nonzero real roots is
§3.7. Stability switches
245
Thus for b =1= 0, we have from F(y, 0') = 0 = G(y, a), that
tan a =
_l!... a
For any real y, the above equation has real roots a which will simultaneously satisfy F(y, a) = 0 = G(y, a). Thus, a set of necessary and sufficient conditions for the trivial solution of (3.7.18) to be asymptotically stable for all r 2:: 0 is given by the following: a + b < 0 and b2 - a 2 ~ 0 which can also be written as a
+ b < 0,
b - a 2::
o.
In the next example, we consider the second order equation
d?x(t) -;Ji2
dx(t)
+ a-;Jt + bx(t) + cx(t -
r)
= O.
The characteristic equation associated with the above equation is
D()", r)
= )..2 + a).. + b + ce-,xT = O.
Condition (i) of Theorem 3.7.1 requires that the real parts of the roots of
D()",O) =
)..2
+ a).. + (b + c) =
0
are negative and this will be the case, if and only if
a> 0,
b + c > O.
Condition (ii) of Theorem 3.7.1 leads to an analysis of the roots of
H(y, a)
= F(y, a) + iG(y, a) = 0 = (_y2 + b + c cos a) + i( ay + c sin a) =
or equivalently
+ b + c cos a = ay +c sin a = O.
F(y, a) = _y2 G(y,a) =
0
0
§3.7. Stability switches
246 If we let A
G(y,O')
=
a2
-
=
2b and B
b2
-
c2 , then we have from F(y, 0')
o and
= 0 that
whose roots are given by
y=± [
-A ± {A2 - 4BP/2jl/2 2
Hence, a necessary and sufficient condition for the nonexistence of nonzero real roots of fey) = 0 is
A2 - 4B < 0;
either
A
or
2
-
4B
=0
and
A 2:: 0
or
A2 -4B > 0,
B > 0,
A> 0
or
A2 - 4B > 0,
B=O,
A 2:: 0.
One can further simplify the above conditions to obtain, either A 2:: 0, B 2:: 0 or A < 0, A2 - 4B < 0. If fey) = 0 has nonzero real roots, we can get from
G(y, 0')
= ay + c sin a =
°
real values of 0' which will also satisfy F(y, a) = 0. Thus, a set of necessary and sufficient conditions for the asymptotic stability of the trivial solution of the second order equation is
(i)
a> 0,
A 2:: 0
(ii) either or where A
=a
2
-
2b
and
b+c>O and
B 2:: 0
A2 -4B < 0 A < 0, 2 2 B =b - c .
Let us consider a third example provided by the following prey-predator system with mutually interfering predators;
3.7.21
§3.7. Stability switches
247
where a, b, c, K are positive constants and T 2:: 0 while 0 < m < 1; x(t) and yet) respectively denote the biomasses of the prey and predator populations. The above system has a positive steady state (prove this) E* : (x*, y*) satisfying
,(1- :*) =
ay*m
bx*(y*)m = cy*
(=?
bx*(y*)m-l =
c).
3.7.22
The linear variational system associated with E* is
dX(t) dt dY(t)
= _1x* X(t)
_ amx*(y*)m-lY(t)
K
3.7.23
--;It = b(y*)m X(t - T) + bm(y*)m-lx*Y(t - T) - cY(t) which has the characteristic equation given by
or
D()..,T)
=)..2
+ )..{c+ 1x* -
bm(y*)m-lx*e- AT }
K
+ 1x*c _Ix*bm(y*)m-lx*e- AT K
K
+ amx*(y*)m-1b(y*)me- AT
3.7.24
=0. When
T
= 0, D(A,O)
=)..2
+ )..(c+ ~x* -
bm(y*)m-l x*)
+ Ix* c - 1x*m(y*)m-l x* K
K
+ amx*(y*)m-l b(y*)m
=0.
It is easy to check, bm(y*)m-l X * = me < c < e + Ix* K =}
c + 1x* - bmx*(y*)m-l K
We have from
, * = , - a ( y *)m -x K
> O.
3.7.25
§3.7. Stability switches
248 that
Ix* c K
= b - a(y*)m]c = b - a(y*)mJbx*(y*)m-l = ,bx*(y*)m-l _ a(y*)mbx*(y*)m-l > ,b x* x*(y"*)m-l _ abx*(y*)m(y*)m-l K
and hence
Thus, by the Routh-Hurwitz criteria, all the roots of D()..,O) = 0 have negative real parts. We check whether a delay induced switching to instability can take place.' We let A = iy in D(A, 7) = 0 and derive
D(iy, r) = -y'
[c
+ iy + ~x* -
+ ~x*c = O.
bm(y*)m-'x*e- iYT 1
1
[;x*bm(y*)m-. x* - amx*(y*)m-'b(y*)m e- iYT 3.7.26
Separating the real and imaginary parts in (3.7.26), y2 _lx*c
= -ybm(y*)m-l x * sin(Y7)
K
-
[~x*bm(y*)m-.x* -
1
amx*(y*)m-'b(y*)m cos(yr)
-y( c + ~ )x* = -ybm(y*)m-· x* cos(yr) +
3.7.27
[ (;) x*bm(y*)m-· x*
- amx*(y*)m-'b(y*)m] sin(yr).
3.7.28
Square and add both sides of the above two equations;
y' + y' { c'
+ (;x*)' + 2C~X* _ 2;x* c _
+ (;x*c)' -
( bm(y*)m-. x*) '}
{;x*bm(y*)m-. x* _ amx*(y*)m-'b(y*)m}'
= O.
3.7.29
§9.7. Stability 8witche8
249
A sufficient condition for the nonexistence of delay induced instability is that the quartic in (3.7.29) has no real roots. Consider next a linear system of the form
duet) -at = allu(t) + a12v(t) + bllu(t dv(t) -;]t
711) + b 12 V(t
-
712) 3.7.30
= a21 u(t) + a22v(t) + bZ1 u(t -
72d
+ b Z2 v(t -
722)
where aij, bij (i,j = 1,2) are real numbers and tij (i,j = 1,2) are nonnegative real numbers. The characteristic equation associated with (3.7.30) is
which on expansion becomes,
+ a22)A + (all a22 - a'2 a 21) - A [b ll e-'Tn + b22 e-'T" 1 + a22bUe-.l.Tll + allb22 e-.l. Tll - a21b12e-.l.T12 - a12 b 21 e -.l. + b l1 b22 e-.l.(Tll+T22) - b12b21e-.l.(T12+T2d = o. 3.7.31
PiA) = A2 - (all
T21
We assume that lently,
aij,
bij in (3.7.30) are such that P(O)
t=
0 in (3.7.31) or equiva-
3.7.33
For convenience in the following we let al =
-(au
+a22)
i31
= -bl l
i3z = -b 22
81
= b11 b22
82 = -b Zl b12
/'11
= a2Z bll
/'21
= -a12 b21
/'12
= -a21 b12
/'22
=
aU b22
The condition in (3.7.32) becomes 2
n
az
+
L i,j=l
lij
+ L 8i t= O. i=1
3.7.34
§9.7. Stability switches
250 We let>.
=
iw (w being a real number) in (3.7.31) and derive that
= (iw)z - (all + azz)(iw) + (aUa22 - a12aZl)
P(iw)
+ b11(-iw)e-iwTll + anbzze-iwTll
+ bzz(-iw)e-iwT22
+ allbzze-iwT22 + bllbzze-iw(Tl1+T22)
- aZ1 b1Ze-iwT12 - a1ZbZ1 e- iWT21 - b21 blze-iw(T12+T21) = 0
or equivalently [_w z + (allan - a12az1)] - i[w(all +w
[b
ll
+ an)]
e-i(WTll +rr/Z)
+ bzze-i(WT22+7r/Z)1
+ aZZb ll e- iWTll + all bzze-iwTn - aZ1bIze-iwT12 - alzbzle-iwT21
+ bll bzze-iw(Tll +T22) 3.7.35
Separating the real and imaginary parts in (3.7.35), w 2 - (alla2Z - a12a2J) = w{b l1 COS(WTll
+ anbll
COSWTll
+ 7r/2) + b22 COS(WT22 + 7r/2)} + allb22 COSWTZ2
+ bllbn COS{W(Tll
+ T22} -
a21b12 COSWT12
- alZb21 COSWT21 - bIZ b21 cos{w(rIz
w(all
+ a22) =
-
[w{b ll sin(wTl1
+ T2J)}
+ "/2) + 1>,2 sin(WTn + ,,/2)}
+ an b11 sinwT11
+ an bzz
sinwTz2
+ bl1 bZ2 sinw( 1'11
+ TZ2) -
a21 bI2 sinwT12
- a1Z b21 sinwT21 - b12 b21 sin{w( 1'12
+ T21)}]'
Squaring and adding the respective sides of (3.7.36) and (3.7.37),
{w 2-(alla22 - a12aZl)}2 +wZ[all
3.7.36
+ a221 2
= w 2{b ll Z + bzz Z + 2bl1 bZ2 COS(WTll + 7r /2 - WTZZ + (anbll)2 + (allb 22 )z + (a21blz)2 + (b ll bn )2 + (a21 blZ)z + (a 1z b2d 2 + (b 12 b2d z
- 7r /2)}
3.7.37
251
§3.7. Stability switches
+ 2w [{ blla22bll + bl l bl l b22
cos{WTll
- bu a,Z "'" COs(WTll
+ {b 22 a 22 b ll
-
7r /2 + w( T11
WTll)
+ bllallb22
+ T22)}
COS(WT22
WTll - WT22) -
b 22 a 12 b 2I COS( WT22
+ 7r /2 -
WT21) -
+2[a22bllallb22 COSW(Tll -
T22)
COSW(Tll -
T12) -
-
a22bllb12b2I
COS{W(Tll -
T12 -
-
all b 22 a 21 b 12 cos{w( T22 - T12)} -
-
allb22b12b21 COS{W(T22 -
+ T22 -
- bll b 22 a 12 b 21
cosw( Tll
+ a21b12a12b21
COSW(T12 -
+ a12b21b12b21
COS{W(T21 -
T21)
T11 -
COSW(Tll -
T2t)
bllb22a21b12
bll b22b12b21
+ a21b12b12b21
T12 -
cosw( T11
WT12) WT2d}
T22)}
T11 - T22)}
T2d}
COSW(Tll
COS{W(TlZ -
T2J)}].
+ 7r /2 -
COS{W(T22 -
all b 22 a lZ b 21 cos{w( T22 -
T2d -
WT12)
WT22)
+ 7r /2 -WT12 -
COS{W(Tll -
+ all bZ2 b ll b 22
T12 - T21)} -
WT22)
+ 7r /2 -
+ IT/2 -
Cos(WT22
b22b12b21 COS( WT22
a22 b ll a I2 b 21 T2d
COS(WT22
b 22 a2I bI2
+ a22bllbllb22
- a2zblla2Ib12
+ 7r/2 -
COs(WTu - WT,Z - WT21) }
+ 7r/2 ~ WTll) + b22allb22
+ IT /2 -
cos(WTl1
bl l a2I b I2 cos(WTll
-
+ 7f /2 - WTZ d -b ll b,Z ""1
cos(WT22
+ b 22 bll b 22 -
+ 7r/2 -
COS(WTll
+ T22 -
+ T22 -
T12 -
T12)
T12 -
T21)
T21)}
3.7.38
Let the right side of (3.7.38) be denoted by few). For arbitrary real w, we have from (3.7.38),
few) ~ w 2 {lb ll l + Ib22 1}2
+ [Iazzbll l + lallbzzl + Ibubzzl + laZ,b,zl + la'Z"",1 + Ib12b21rf + 21wI [(Ibn l+ Ibzz l) (lazlbIZi + la,zbzd) + Ibubzzllall - azzl + (Ib u l+ I""zl) (Ib u bzzl + Ihz"",l) ]. If we denote the right side of (3.7.39) by M, then
3.7.39
1
§3.7. Stability switches
252
where
+ Ib22 ( = lazzblll + lall bZ2 + la21 b12 + la12 b211 5 = Ib ll b2Z ! + jb 12 b21 I·
f3 = Ibul
1
1
}
1
3.7.41
A sufficient condition for the. nonexistence of a real number w satisfying (3.7.31) can now be obtained from (3.7.38) - (3.7.41) in the form
w' + ("'; - 2"'2 -
fP)w2 -
21w1 [,9 (la21 b12 I + la12b,d)
+ Ibl l b,2l1all - a221 + ,98] + "'~ -
(7 + 8)2 >. 0
3.7.42
in which
The inequality (3.7.42) is of the form w4
+ awZ -
bw + c > 0
3.7.43
where
3.7.44
If a and c in (3.7.43) are positive numbers, then we can write (3.7.43) as follows:
w4
+ a(w -
bj2a)2
+ (c -
bZ j4a) > O.
3.7.45
One can see from (3.7.45) that a set of sufficient conditions for the nonexistence of a real number w satisfying P( iw) = 0 in (3.7.31) are given by a > 0 and
c - b2 j4a > O.
3.7.46
Thus, we conclude from (3.7.46) that a set of sufficient conditions for the nonoccurrence of stability switching in (3.7.30) are given by 3.7.47
253
§3.7. Stability switches
{ (bul+lb:22I) [laZlbl21 + lalzb2l 1+ Ibllbzzliall + (I blll + Ibzz l)(lb11 b22 + Ib12 bzll)] 1
r
azzl
< [ail + a~z + 2alZaZl - (Ibll l + Ibzzl) Z] [ (all azz _ a12 aZl ) z - (la21 blll + la21 bI2 1 + la12 b211
+ lall b22
1
+ Ibubzzl + Ib12bzll + Ibubz,l + Ib12bzll) ']. 3.7.48 It is not difficult to apply the above technique for the derivation of sufficient conditions for the nonoccurrence of stability switching in systems with arbitrary delays such as
?= b1ju(t - + ?= b2jv(t - r2j) dv(t) -;It = a21 u(t) + a22v(t) + ?= CljU(t - 6j) + ?= C2jV(t - e2j). duet)
-;It = auu(t) + a12v(t) +
n
n
rlj)
J=l
J=l
n
n
J=l
J=l
3.7.49
The interested reader can examine (3.7.49) with respect to stability switching as well as stability (for more details see Freedman and Gopalsamy [1988]). 3.B. Oscillations in linear systems
In this section we consider delay induced oscillations (not necessarily leading to periodicity) in linear vector - matrix systems. In particular, we obtain a set of sufficient conditions for all bounded solutions of a linear system of differentialdifference equations of first order to be oscillatory ( defined below) when the system has a one or more delays. For results related to this section we refer to Gopalsamy [1984c, 1986a, 1987]. We first consider systems of the form
t > 0, i
= 1,2 ...
,n
3.8.1
where aij and r are real constants with r > O. If we denote the colunm vector {Xl(t), ... ,xn(t)}T by x(t), then we can rewrite (3.8.1) in vector matrix notation
254
§3.8. Oscillations in linear systems
as follows: dx(t) A Xt-7; -( ) ---;u-=
t>O
3.8.2
where A denotes the n x n matrix of constants {aij, i,j = 1,2 ... ,n}. If (3.8.2) is supplemented with initial conditions of the form 3.8.3 where cjJ : (-7,0] I-t Rn, cjJ is continuous, then one can show that solutions of (3.8.2) - (3.8.3) exist on [-7, (0); in fact, we have from (3.8.2) - (3.8.3), x(t)
=
it
+A
xes - 7)ds
t:
t
>0
r
x(ry)d(ry)
and hence
from which by Gronwall's inequality it will follow
showing that solutions of (3.8.2) - (3.8.3) are of exponential order. Thus, one can use methods of Laplace transform for the study of equations of the form (3.8.2). The literature on oscillations of nonscalar systems of delay - differential equations is sparse; we shall adopt the following definition:
Definition. A nontrivial vector x(t) = {Xl(t), ... ,Xn(t)}T defined on [0, (0) is
x
said to be oscillatory, if and only if at least one component of has arbitrarily large zeros on [0, (0). The vector is said to be nonoscillatory if all components are nonoscillatory.
x
We remark that the above definition of oscillatory solution vectors of nonscalar systems is one of several possible generalizations of the corresponding notion of scalar equations; however, our definition of oscillatory vectors reduces to that of
255
§3.8. Oscillations in linear systems
the familiar scalar functions on [0,00) if the vector has trivially one component only. Let X(A) denote the Laplace transform of a solution vector X(t) of (3.8.1) (3.8.3) defined by X(A) = {Xl(A), ... , xn(A)} T
1 Ae-Af'
3.8.4
00
Xj(A) =
Xj(t)e->.t dt.
It will follow from elementary properties of Laplace transforms that
X(A) = [A1+
[
+A
t
_(adj. [AI - Ae- AT ]) [
3.8.5
H(A) X
n identity matrix and H (A) is defined by
H(A) = del. [AI - Ae- AT ].
3.8.6
By the inversion theorem on Laplace transforms we have from (3.8.4) - (3.8.6) that any solution of (3.8.1) is given by the integral representation, 1 X(t) - -2' -+
_
7rZ
10'+ioo ((adj. [AI-Ae- AT ]) [i(O)+AJ~Ti(S)dS]) H(A)
. 0'-100
At
e dA
3.8.7 where ais a real number greater than the real parts of roots of H(A) = OJ the existence of such a real number a is well known (Hale [1977]). The integral in (3.8.7) can be evaluated using residue calculus
x(t)
= LPj(t)eAjt,
t >0
3.8.8
j
in which the polynomial (in t) vector Pj(t) is determined by
_. _.
p}(t) - resIdue of
(ad [AI j
AT Ae- ] [
+ J~T
AI) e
3.8.9
at a root Aj of H(A) = O. The convergence of the series representation of the type in (3.8.8) has been established by Banks and Manitius [1975]. With these preparations we formulate our first result.
§9.8. Oscillations in linear system8
256
Theorem 3.8.1. Suppose that the matrix A of real numbers 1,2, ... ,n) in (3.8.1) is such that (i) det A =f. 0 (ii) the eigenvalues 0'1,0'2, ••• , an (real or complex) of A satisfy
lajlre> 1;
aij
(i,j
=
3.8.10
3.8.11
j = 1,2 ... ,no
Then all bounded solutions of (3.8.1) are oscillatory. Proof. Since solutions of (3.8.1) are representable in the form (3.8.8), it will follow that a necessary and sufficient condition for all bounded solutions of (3.8.1) to be oscillatory is, that the characteristic equation H(>..) = 0 has no real nonpositive roots. (An independent proof of this can be extracted from the article of Arino and Gyori [1989]). Since, 0'1,0'2, •.• , an are the eigenvalues of A, we have immediately that 3.8.12 Thus, we are led to an investigation of the nature of the roots of j
= 1,2, ...
, n.
3.8.13
Suppose now that there exists a bounded nonoscillatory solution of (3.8.1); that is, there exists a real nonpositive root say>.. * such that 3.8.14 Since det A j:. 0, aj =f 0, (j = 1,2, ... , n) and hence >..* then follow from (3.8.14), that
=f
0; thus >..*
< o. It will
l=rlajl(e l >'*IT/I>"*lr) forsomej E {1,2, ... ,n} ~
rlO'jle
for somej E {1,2, ... ,n}.
3.8.15
But (3.8.15) contradicts (3.8.11) and hence (3.8.1) cannot have a bounded nonoscillatory solution when (3.8.10) - (3.8.11) hold and the proof is complete. [] Let us consider next, a linear delay-differential system of the form
d~~t) = Bx(t) + Ax(t -
r);
t>O
3.8.16
257
§3.8. Oscillations in linear systems
where A and B denote real constant n X n matrices with elements aij , bjj (i, j = 1,2, ... , n) respectively and r > is a constant. We adopt the following norms of vectors and matrices:
°
n
/lx(t)/I
n
n
IXi(t)li., IIAII = m~ L /aijli
= L i=1
J
IIBII = m~x L Ibjjl·
i=1
J
i=1
The measure fJ(B) of the matrix B is defined by
fJ
(B) = lim III +BBII-l 8-0+ 8
which for the chosen norms reduces to
p(B)
=
l~j'In [b
jj
+
t.lbijl]. i¥:j
(For more details of the measure of a matrix we refer to Vidyasagar [1978]).
Theorem 3.8.2. Assume the following for the system (3.8.16);
°
(i) detA # 3.8.17 (ii) fJ( B) + HAil # 3.8.18 (iii) (IIAllre) exp ( - rlfJ( B)I) > 1 Then all bounded solutions of (3.8.16) corresponding to continuous initial conditions on [-r,O] are oscillatory on [0,00).
°
Proof. Let us rewrite (3.8.16) in component form
dx.(t) ---;it = LbijXj(i) + Lajjxj(t-r), n
n
j=1
j=1
i
= 1,2, ... ,n
3.8.19
and suppose that there exists a solution say yet) = {Yl (t), . .. Yn(t)}T of (3.8.19) which is bounded and nonoscillatory on [0,00). It will then follow that there exists a t* > 0 such that no component of yet) has a zero for t > t* + r and as a consequence we will have for t ~ t* + 2r, 3.8.20
§9.8. Oscillations in linear systems
258
and hence
duet)
at
~
p(B)u(t) + 1\ A Ilu(t - 7),
t
2:: t* + 27
where u(t) == L:~=1 IYi(t) I; by the above preparation, we have u(t) t* + 7. Consider the scalar delay differential equation
dv(t)
dt = p(B)v(t) + II A Ilv(t - 7), with v(s) = u(s),s E [t*,t*
+ 7].
u(t)
~
t 2:: t*
3.8.21
> 0 for t 2::
+ 27
3.8.22
It is left as an exercise to show that
vet)
for
t 2:: t*
+ 27.
3.8.23
We claim that all bounded solutions of (3.8.22) are oscillatory on [t* + 2r, 00). Suppose this is not the case; then the characteristic equation associated with (3.8.22) given by 3.8.24 A = pCB) + II A II e- AT will have a real nonpositive root say). **. It will follow from (3.8.17) that A** Thus A** < 0 and we have from (3.8.24)
IA** 12:: IIAII elA**rl -lp(B)I·
=/: O.
3.8.25
It is now a consequence of (3.8.25), that
1~
(II A II .1'''1 T) I{W'I + IJl(B) I}
1}
H
~ {II A II .-1 p( 8) ITT} { exp [ (I >." I + IJl(B) I) T I [{I >." I + II 1'( B) I
3.8.26 The last inequality contradicts (3.8.18). Hence, our claim regarding the oscillatory nature of v on [0,(0) is valid; now since v has arbitrarily large zeros, u will have arbitrarily large zeros which means that I:~=1 IYi(t) I is oscillatory implying that yet) is oscillatory; but this is absurd since ]jis a nonoscillatory vector. Thus, there cannot exist a bounded nonoscillatory solution of (3.8.19) when the conditions of the theorem hold and the proof is complete. [] The following result deals with oscillations in linear systems of equations with a multiplicity of delays.
259
§3.8. Oscillations in linear systems
Theorem 3.8.3. Let aij , Tij (i,j = 1,2, ... , n) denote real constants such that aij =I- O,Tij > O(i = 1,2, ... ,n) andT;j ~ 0, (i,j = 1,2, ... ,n; i =l-j) and consider the system
t > O.
3.8.27
If aj and Tij of (3.8.27) satisfy n
and
det A = det(aij) =I- 0,
!aii ITii e > 1 + e .2: !aij !Tije,
3.8.28
j=l
j#i
then all bounded solutions of (3.8.27) corresponding to continuous initial condiTi]' are oscillatory on [0,00). tions denned on [-T, 0], T = l<~a.x< _',]_n
Proof. The characteristic equation corresponding to (3.8.27) is det
[)..Ii; - ai; e->TO;] = O.
3.8.29
Suppose (3.8.27) has a bounded nonoscillatory solution. Then (3.8.29) has a real nonpositive root say 8 such that
Since det( aij) =I- 0 it follows that 8 < 0 is an eigenvalue of the matrix with entries aijexp{-8Tij} (i,j = 1,2, ... n). By Gershgorin's theorem (Franklin [1968]),8 satisfies n
18 -
aii e- 6Tii I
:::;
.2: Iaij le-
6Tij
for some
E {I, 2 . .. , n}.
j=l
j#i
From (3.8.30) and
181 = laii e- 6Tii + 8- aji e- 6Tii I ~ Iaii le- 6T;; - 18 - aii e- 6T ;; 1 n
> Ia"le Ti ;l6 1 - L....t ~ Ia··1 e Tij f]
-
H
j=l
j#i
1
6
1,
3.8.30
§.9.8. 08cillation.3 in linear .3Y.3tem8
260
we derive that
n
!81 + L
!aii!eTiilol2:: laiileTiiO.
j=l
J#-i
Rearranging terms in the above,
and this leads to (1+e
t
lij! aij
I) 2:: I
ajj
llii e
for somei E {I, 2, ...
,n} .
3.8.31
)=1
i#-i
But (3.8.31) contradicts (3.8.28). Thus (3.8.27) cannot have a bounded nonoscillatory solution when the conditions of the theorem hold. [] The following corollaries are of interest by themselves. Corollary 3.8.4. Suppose that the coefficient matrix A = (aij) in (3.8.1) has at least one real negative eigenvalue say f3 which is such that 0<
1f3lle::; 1;
3.8.32
then (3.8.1) has at least one bounded nonoscillatory solution. Proof. The portion of the characteristic equation of (3.8.1) (see (3.8.12)) corresponding to f3 is 3.8.33 which is equivalent to J.1. = I f31 eILT where J.1. = -..\. It is easy to see that there exist positive real numbers J.1. such that J.1. = I f31 eILT when I f31 Ie::; 1 and corresponding to such J.1., we will have a solution of (3.8.1) in the form p/L(t)e- At where PIL(t) is a polynomial in t. A solution of the form PIL(t) exp[-..\t] is not oscillatory since Pli can have only a finite number of zeros and the proof is complete. [] The previous results have been concerned with bounded solutions of delay differential systems. The next result does not suffer from such a restriction.
261
§3.8. 08Cillations in linear systems Theorem 3.8.5. Assume tbat aij E (-00,00), Tjj E [0,00), i,j
Let
=
1,2, ... , n.
n
=
II.
r
min 1a ]1.. I}. <.< {a ".. - ~ ~ 1_I_n j=l
3.8.34
j#
If j1.Tiie
> 1, i = 1,2, ... , n,
3.8.35
tben all solutions of dXi(t)
-d-
t.
( +~ ~aijXj t .
Tjj
)
.
= 0, z = 1,2, ... ,n
3.8.36
)=1
are oscillatory.
Proof. Suppose (3.8.36) has a nonoscillatory solution
such that Ix;(t)1
> 0 for t
~
to, to E IR. Then we have from (3.8.36)
dlxi(t)1 dt- -< -a··lx·(t - r")1 + II
1
II
I: la"llx ·(t - r")1 n
I)])]
3.8.37
j=l j~i
and therefore 3.8.38 We have from (3.8.38) that
t
IXi(tll + J-l ],'
i=1
t
IXi(s -
T i==1
1"iill ds :::;
t
IXi(Tll-
3.8.39
;=1
Thus, E~=l IXi(t)1 is uniformly bounded for t ~ OJ as a consequence, :ri(t) is unifonnly bounded. It is easily seen from (3.8.39) that
[
IXiCs -
1"iill ds ::; '" < 00;
3.8.40
since :ti(t) is bounded, Xi is unifonnly continuous and therefore by Lemma 1.2.2, we can conclude that lim x;(t)
t-oo
= 0,
i = 1,2, .. ,n.
3.8.41
§3.8. Oscillations in linear systems
262
Integrating both sides of (3.8.38) on [t, (0) and using (3.8.41),
or equivalently
To
If we let
=min{Tll, .. Tnn }.
3.8.42
n
Vet)
= L /Xi(t)/, i=l
then (8.8.42) implies
Vet) 2: I'
1:0
3.8.43
V(s) ds.
Define a sequence as follows:
o(t)
= Vet)
A.. () _ 'f'k+l t -
{
Vet) - VeT) + /-L I';'r o k(S) ds; roo /-L Jt-ro k(S) ds; t ~ T.
t 5:: T
One can show that for and therefore the pointwise limit of k as k lim
k-oo
~ 00
exists; if
= *( t),
then
*(t) = {V(tL - VeT) + /-L I';'r o *(s)ds, /-LIt-ro *(s)ds, t > T. It follows, * is an eventually positive bounded solution of
dy(t)
-;It
=
-/-Ly(t -
To),
t > Tj
t
269
§9.8. Oscillations in linear systems
a necessary and sufficient condition for this is, that the characteristic equation
has a real root which is negative; let -}. =
f3 and note that
[]
which contradicts (3.8.35). Thus the result follows.
It is left as an exercise to consider the oscillation of other general cases such as the following:
dXi(t) + ~ -;It ~Pij(t)Xj(t - Tjj(t))
=0
I
)=1
i = 1,2, ... ,n.
dXj(t) -;It
~ %(t)Xj(t)+ ~Pii(t)Xi(t ~ +~ 'l
dXj(t)
Tjj(t)) = 0
I
= 1,2, ... ,n.
~
-;It + L...JPij(t)Xj(fLj(t)) = 0,
i=1,2, ... ,n
j=l
as
t
~ 00,
j
I
= 1,2, ... , n.
3.9. Simple stability criteria Sufficient conditions for the uniform asymptotic stability of the trivial solution of the linear nonautonomous scalar delay-differential equation
.3.9.1 and the vector matrix system
dX(t)
-;It
= -C(t) X (t) + B(t) X
(t - T)
3.9.2
have been obtained by Busenberg and Cooke [1984]. It is found from the results of these authors that a decisive role is played by the coefficients c(t) and C(t) of
§3.9. Simple stability criteria
264
the nondelay terms in (3.9.1) and (3.9.2) in the process of constructing positive definite quadratic Lyapunov functionals whose derivatives are negative definite. If terms such as e(t) and G(t) in (3.9.1) and (3.9.2) are absent, it is usually difficult to construct positive definite functionals; however, it is a routine procedure to construct quadratic semidefinite functionals. In this section we propose a practical method for Gonstructing functionals with which one can obtain sufficient conditions for the trivial solution of the nonautonomous vector-matrix system dx .(t)
dt
_ I_
n
+ '" L a' ·(t) x' (t I}
)
T") I]
= o· ,
i=1,2, ... ,n
3.9.3
j=l
to be asymptotically stable and to apply this result in the next chapter for the derivation of sufficient conditions for the global asymptotic stability of the positive equilibrium of a time-delayed competition system modelled by the Lotka-Volterra equations
dNi(t) [r.1 dt = N·(t) Z
- L~ b· ·N·(t -
.
I)]
j=l
T' .)] I]
3.9.4
i = 1,2, .. ,n.
Our technique is based on the construction of a 'degenerate' Lyapunov functional and an application of a Lemma 1.2.2 due to Barbalat [1959]. Although our construction and the subsequent calculations are routine, they become lengthy and therefore we shall consider only the case n = 2 in (3.9.3). For literature related to the construction and application of Lyapunov functionals for linear systems with nondelayed terms we refer to the works of Datko [1977], Infante and CasteIan [1978], Castelan and Infante [1979], Carvalho et al. [1980], Castelan [1980], Abrahamson and Infante [1983J and the references therein. We consider the coupled system of two ·nonautonomous delay differential equations 3.9.5
where Tij E [0,00) and
aij
(i,j = 1,2) are continuous real valued functions defined
265
§3.9. Simple stability criteria on [0,(0). We shall write (3.9.5) in the equivalent integrodifferential form : [Xl (t) t
t
all (s
it-Tn
.= - [an(t : [X2(t) t
-it = -
[a
2l (t
-it
t- T12
a12(S + T12)X2( s) dS]
+ Tn)Xl(t) + a12(t + T12)X2(t)]
a21
t-Tn
+ Tll)XI (S) ds
(s + T2dXI (s) ds
-it
3.9.6
t-Tn
a22( S + T22)X2( s) dS]
+ T2dXl (t) + a22( t + T22)X2( t)].
We shall assume throughout this section that
all(t) > OJ
a22(t) >
°
t>0
for
3.9.7
and consider a functional VI where
VI (XI,X2)(t) = [Xl(t) -
t
all(s
+ Tll)XI(S)ds -
t
al2(s
+ T12)X2(S)dS]
it- Tl2
it-Tll
+ [X2(t) -
2
t
a21(s
it- T21
-it
+ T2t)XI(S)ds
a22(s + T22)X2(S) dS] 2.
t-Tn
The rate of change of VI along the solutions of (3.9.6) can be computed:
dV1(Xl,X2)(t) dt
= - Q( Xl,X2 )( t ) - [aul t
+ Tll)X;(t) + a22(t + T22)X~(t)]
+ 2an(t + Tll)Xl(t) l~ru all(s + Tll)Xl(S)ds
+ 2an (t + Tll )Xl (t) l~r" a12( s + T12)X2( s) ds + 2al2(t + Tl2)X2(t) l~ru ants + Tn)xl(s) ds
+ 2a12( t + Tl2 )X2 (t) 1~
",
a12( s
+ Tl2 )X2( s) ds
3.9.8
§9.9. Simple stability criteria
266
+ 2a21 (i + T2.)XI (i) J.~T" a21 (s + T21)XI (s) ds + 2a21 (t + T21 )XI (i)
J.~r"
a22( s + T22)X2( s) ds
+ 2a,,(t + T22 )X2 (i)
J.~T"
a21 (s
+ 2a22( i
+ T21 )X2 ( s )ds
+ T22 )X2 (i) J.~T" a22( s + T22 )X2( s) ds
3.9.9
where 3.9.10 and
A(t)
= [
al1(t+7"u) a12(t+7"12)+a21(t+7"2d]. a12(t + 7"12) + a21(t + 7"2I) . a22(t + 7"22)
The right side of (3.9.9) is estimated so that
dVi(Xl,X2)(t) Q( )( ) dt ::; Xl, X2 t - [au(i
+TU)X~(t) + a22(i + T22)X~(t)]
+ au (i + TU )x~(i) J.~TH au (s + TU)
all(s + 7"ll)xi(s)ds
t-Tll
+ all(t + 7"ll)xi(t) it
/a12(S + 7"p)/ ds
t- T 12
+ au(t + TId
J.~", lau(s + T12)lxi(s) ds
+ I a12( i + Tl2 )Ixi( i) J.~TH au (s + TU) ds + Ia12( s + T12) IJ.~TH au (s + TU )x~( s) ds + Ial2(i + T12) I xi(t) J.~", Ial2(s + T12)1
§9.9. Simple stability criteria
26'1
+ Ia2I(t + T2,) Ix~(t) 1~r" Ia2I(s + T2,) Ids t
..
+ I a21(t + 'T2d 11-T21 I a21(s + 'T21) I xi(s) ds + Ia2I(t + T2I) I xi(t) l~T" a22(s + T22) ds + Ia21(t + T2,) '1~T" a22( s + T22)X~( s) ds +a22(t+T22)xi(t)
l~T" la2I(s+T21)lds
+ a22(t + 'T22)
it
I a21 (s
+ 'T21)1 xi(s) ds
t- T 21
+ a22(t + T22)
l~T" a22(S + T22)X~(S) ds.
3.9.11
We choose another functional V2 defined on the solutions of (3.9.6) such that V2(Xl,X2)(t) =
it
all(s
+ 2'Tll) {
t-Tu
+ (
it
all(U
all (s
+ 'Tll + 'T12 ){
(la12( U +
Jt-T12
+ l~TH
+ 'Tll)Xi(U)dU} ds
s
Js
lal2(S + TI2
+ Tn)l{
1.'
an (u +
'T12)lx~( U)dU} ds
Tn)x~(u)du } ds
+ 1~", la12( s + 2T12) I{ [lal2 (u + T12) Ix~( u)du } ds +
l~T" la2I(S + 2T21)1
+
it
la21(S
t-T22 t
+ [
{1.'
+ 'T21 + 'T22 )1{
Jt-Tn
1.t
.
a22(S
+ 'T22 + 'T2d{ + 2'T22) {
+ 'T22)x~(U)dU} ds
t
la21(U s
a22(S
+ T2I)lx~( u)du } ds
a22(U
s
Jt- T 21
+ (
la2I(U
(a2Z(U Js
In terms of VI and V2 , we define a functional
+ 'T21 )lx i (U)dU} ds
+ 'T22)X~(U)dU} ds.
3.9.12
§3.9. Simple 3tability criteria
268
and calculate the rate of change of V along the solutions of (3.9.6). Estimating such ~~, we derive from (3.9.10), (3.9.11) and (3.9.12), 3.9.13 where
1t1 (t) == all (t
+ 1"11) -
[all (t
+ 1"11)
it
+ 1"11) ds
all (.5
t-Tll
+ all(1 + Tll) l~r" la12(s + T12)1 ds
+ la21(t + T2dll~r"
/all(S + T2dl ds
+ la21(1 + T2dll~r"
a22(s + T22)ds
+ all (t + 1"11)
it it
all (s
+ 21"11 ) ds
t-Ttl
+ au(t + 1"11)
/alZ(S
+ 1"12 + 1"11)/ ds
t-Tl1
+ la21(t + T21)ll~r,. lads + 2T2dl ds
+ la21(t + T2')ll~r" la21(s+ 2T2')1 ds t
+ !aZ1(t + 1"Zl)11
a22(S + 1"Z2
+ 1"Zl) dS]
t-T21
1t2(t) == a22(t + 1"22) - [la 1z (t
+ 1"12)1
it
all(S + 1"11)ds
t-Tu
+ la12(t + T12) 11~r" lal. (s + T12) Ids
+ a•• (t + T2.) l~r" la21(s + T.')I ds t
+ azz(t + 1"2z)11
a2z(s
+ 1"Z2) ds
a11(s
+ 1"ll + T12) ds
t- T 22
t
+ lan(t + T12)11 t- T 12
3.9.14
§9.9. Simple stability criteria
+ la12(t + T12)1 J.~T"
+ a22(t + 7'22) + a22(t + T22)
t
Jt- r
t
Jt-
269
+ 2T12) Ids
la12(s
la2I(s
+ 7'21 + 7'22)1 ds
22
a22( s + 27'22) dsj.
3.9.15
r 22
With the above preparation we fonnulate our principal result. Theorem 3.9.1. Assume the following: (i) Tij E [0,00); i,j = 1,2. (ii) aij (i, j = 1,2) are continuous real valued functions defined on [0,00) and bounded on [0,00) satisfying all (t)
> 0, a22(t) > 0 for t
~
0
such that
inf fJ.(t)
t~O
= fJ.l > 0;
inf fJ.2(t) = fJ.2
t~O
> O.
3.9.16
(iii) "A"=
I [::: ::: III <
1
where
3.9.17
(iv) The quadratic form Q satisfies 3.9.18 Then every nontrivial solution of (3.9.5) satisfies lim [xi(t)
t-oo
+ x~(t)] = o.
3.9.19
Proof. From our preparation preceding the fonnulation of the theorem we derive that the functional V = Vi + Vi satisfies
3.9.20 and
§3.9. Simple stability criteria
270
where J-L
= min{J-Ll' J-L2}' From (3.9.18), (3.9.21) and definition of V, we derive that V( Xl, X2)(t)
+p
1.'
[x;( s) + X~( s)] ds
~ V( x" X2)(O) ~ Vo.
3.9.22
From the definition of VI and (3.9.22) ,
I Xl(t) I ~ 1~TU ants + TU) I X, (s)l ds +
IX2(t)l~ 1~T"
1~T" Ia,2(s +
Ul(t)
=
vv.
3.9.23
vv..
3.9.24
I X2(S) Ids +
la21(s+T2lll x l(S)lds
+ 1~,." We let
Tl2)
sup
a22(s + T22) I X2( s) Ids
I Xl(S) Ii
SE[-T,t]
U2(t)= sup Ixz(s)1 SE[-T,t}
r
+
= [l~i,i~z] max ri" )
3.9.25
3.9.26
It is easy to see from (3.9.23), (3.9.26) and (3.9.17), 3.9.27 where I denotes the 2 x 2 identity matrix and the vector inequality in (3.9.27) is in the componentwise sense. From (3.9.25) and (3.9.27), it follows that the solutions of (3.9.6) are bounded on [-r,oo). The boundedness of aij on [0,00) and the equations (3.9.5) show that the derivatives Xl and X2 remain bounded on [0,00) and hence Xl and Xz are uniformly continuous on [0,00). We note from (3.9.22) that xi(t) + x~(t) E L 1 (0, 00). An application of Barbalat's lemma (see Lemma 1.2.2) implies (3.9.19) and this completes the proof. [J The method proposed above for linear delay differential systems is also applicable for a class of linear neutral delay differential systems, details of which can be found in Chapter 5.
§9.9. Simple stability criteria
211
We conclude this section with a brief derivation of sufficient conditions for the trivial solution of
';(1) = AX(I) +
J.'
H(I,s)x(s)ds
3.9.28
to be asymptotically stable where A E Rnxn, H is an n X n matrix valued continuous function defined for 0 ::; s ~ t < 00. Equations of the form (3.9.28) have been extensively investigated by Burton [1983]. A crucial assumption in most of these investigations is that there exists a symmetric B such that ATB+BA=-I
3.9.29
where I is the n x n identity matrix. The next result and several similar results can be found in the book by Burton [1983].
Theorem 3.9.2. Assume (3.9.29) holds and suppose there is a constant M > 0 such that
II B II
[J.' II
H(I,s) lids + ['" II H(u, I) IIdU] $ M < 1
3.9.30
Then the trivial solution of (3.9.28) is stable, if and only if x T Bx > 0 for each x E IR n , x =f 0 E IRn . We note that the verification of (3.9.30) is nontrivial due to the explicit dependence of (3.9.30) on B. In the following result we derive a sufficient condition explicitly in terms of A and H rather than directly through Bas in (3.9.30).
Theorem 3.9.3. If the elements
aij
of A and Hjj of H satisfy
3.9.31
then every solution of (3.9.28) satisfies n
L xHt) i=l
--t
0
as t
--t 00.
3.9.32
§S.9. Simple stability criteria
272
Proof. Consider a Lyapunov functional V = V(x)(t) defined by
V =
t
[X:C t ) +
,=1
t Jot (1
00
I Hij(u,s) IdU)
t
)=1
X~(S)dS] .
3.9.33
Calculating ~~ along the solutions of (3.9.28), dV n [ dt = ~ 2xj(t)
+
{nj;aijxj(t) + j; J.t Hij(t,S)xj(s)ds } n
t, ([I
-t
t {2a
i;
,=1
Hij(U,t) IdU) xJ(t)
([IHi j (t,S)I X 1(S)ds)
)=1
$
0
1
3.9.34
0
+ L (Ia jd + la;j
I)
}=1
jf;i
+
t,[1 Hij(t, s) Ids +t, [0 IHji (u, t) Idu }x;(t)
3.9.35
n
~ -11-
L x;(t).
3.9.36
i=1
One can now see that (3.9.36) leads to
from which the uniform boundedness of both II x(t) II and II x(t) II for t 2:: 0 will follow. An application of Lemma 1.2.2 of Barbalat (see Chapter 1) implies that /I x(t) II -+ 0 as t -+ 00 and this completes the proof. []
273
EXERCISES III 1. Assuming that a, b, I, aj, Ij (j = 1,2, ... ,n) are positive constants, prove that solutions corresponding to positive continuous initial values of the following remain positive and exist for all t ~ 0:
(i)
d~~t) = x(t - I)[a - bx(t)].
(ii)
d~~t) = x(t - I) - bxZ(t).
(iii)
d~~t) =
Ej=l aix(t - Ii) - bx 2 (t).
Prove that the trivial solution of each of the above equations is unstable while the nontrivial steady state is asymptotically stable with respect to positive initial values. Also examine the absolute stability (independent of delay) of the non trivial steady states.
2. Can you generalize the result of problem (1) above, to the following integrodifferential systems: (i)
d~~t) = (fooo k(s)X(t-S)ds)[a-bx(t)]. oo
(ii) d~~t) = a Jo
k(s)x(t - s)ds - bx 2 (t).
(iii) d~~t) = aJ; k(s)x(t - s)ds - bxZ(t). State your assumptions on the delay kernel k(.). 3. Discuss the stability and instability of the trivial and nontrivial steady states of the following scalar systems (assume a, b are positive constants and 11,12 are nonnegative constants).
(i) d~~t) = ax(t - It) - bx(t - 11)X(t - 12)X(t). (ii) d~~t) = x(t - 1I)[a - bx(t - IZ)X(t)].
(iii)
d~~t)
(iv)
d~~t) = x(t) (a - b[ 10
=
10
00
k.(s)x(t - s) ds 00
[a - bx(t)jooo k,(s)x(t - s) dS].
k( s)x(t - S)ds] ').
(State your assumptions on k1 , k2, k in (iii) and (iv) above).
Exercises III
274
4. Let b, c, r be real constants and let P denote the class of all nonnegative
solutions of
dy(t)
d:t
= by(t - r)[l - yet)] - ey(t) ,
for t E [0,(0). Assume b > 0, c ~ 0. Prove that the trivial solution is asymptotically stable within the class P if b < c and the nontrivial constant solution yet) == 1 - (c/b) is asymptotically (locally) stable if e < b. ~ [0, 1], is continuous}. Prove that ife ~ b > 0, then the trivial solution of the equation in problem 4 above, is globally asymptotically stable with respect to S. If c < b, then show that yet) = 1 - (c/b) is globally asymptotically stable for all initial conditions in S with (s) > O,S E [-r,01.
5. Let S = {I: [-r,O]
°:;
6. Generalize your discussion of problems 4 and 5 above for systems of the form
d~~t)
=
(b 1= k(s)y(t - S)dS) [1 -
yet)] - cy(t).
State your assumptions on the delay kernel k(.). 7. Let the nonnegative function y denote a solution of the difference inequality:
yet) ::; ay(t - ret)) + bexp( -,Bt) y(t)::; (t),
° °: ; ret) ::; r*. °
where a ~ 0, b ~ 0, ,B > and exist constants a > and N >
°
t E [-r*,O]
yet) ::; N exp( -at), where a
< min{,B,a o }
If a < 1, then prove that there
such that
t
~
°
and a o is the unique positive root of
and
N=
sup
1(s)l+b[l-aexp(ar*)]-I.
sE[-r· ,0]
What type of generalization to vector - matrix systems can be developed? (for more details see Xu [1989].)
Exercises III
215
8. Let 71, 72 ,73 ,7 be nonnegative constants such that 7 Show that the set G = {
maX{71 ,72 ,73}'
is invariant with respect to
dxd(t) t
= -x(t) + x(t -
7"1) - x(t)x(t - 7"2)
fO
_~
[xCi + s) - x(t)]2 ds.
Show that every solution x(.) corresponding to nonnegative initial conditions converges to a constant as t -+ 00. Generalize your result to a system of the form
f=
dx(t)
-;It = -x(t) + 10
k1(s)x(t - s)ds
-X(t){J.~ k (s)x(t-s) }dS J.~ k (s)[x(t-S)-X(S)r dS . 3
2
9. If k : (-00,0] H- (-00,00) is continuous such that J~= Ik( s)1 ds that the trivial solution of
d~~t) =
[ - x(t) +
f~ k(s)x(t + S)dsf,
< 1, prove
8 = 1,3,5, ..
is asymptotically stable. What can you say about the oscillatory behavior of x(t) for t -+ 00 if k : (-00,0] H- [O,oo)? 10. Assume that all the parameters appearing in the following systems are positive constants. Can you prove that, whenever the nontrivial steady state is locally asymptotically stable for 7 = 0, the same conclusion holds under the same conditions for all 7" > O?
dx(t) -;It
= x(t)[rl - allx(t) - a12y(t - 7)] }
(1)
dy(t)
-;It = y(t)[T2 - a21x(t - 7") - a22y(t)]. dx(t) = x(t) dt
[1 - kl + ay(t x(t) - 7")
dye t)
[
l)
1
y( t) -;It = yet) 1 - k2 + f3x(t - 7") .
(2)
Exercises III
276
dx(t) = X(t){kl a:t
}
x(t) - ay(t - T)}
(3)
dy(t)
dt = y(t){ -k2 + f3x(t -
d~~t) = x(t)[al -
T) - Yet)}.
a2x(t) - f3y(t - T)] )
1
-dy(t) _ ()[ hy(t) - ry t 1 - ( ). dt x t - T dx(t) = x(t a:t dy(t)
dt
TI)[rl
+ allx(t) -
(4)
al2y(t - T2)] } (5)
= yet - T3)[r2 - a2lx(t - T4) - a22y(t)].
11. Let aI, 0:2, {31, (32 be positive constants. Prove that for each positive constant e, the system
(1 - Xl) - e(xI - X2) dX2(t) =a 2x 2(1- X2) -e(x2 -Xl) dt f32 dXI(t) = alxI dt
f3I
has a positive steady state (xr ,xi) with state (xr, xi) of
xr > 0, xi > 0. Prove that the steady
dx~?) = ""X, (1 -;:) - ex, + e [= k'2(t dx;?) = "2 X2 where kl' k2 : [0,00)
1.
!---;.
(1 -;:) - eX2 + e['= k
21 (t
S)X2(S )ds
- S)Xl( s)ds
[0,00) are continuous and integrable on [0,00),
00
kj(s)ds=l;
1.
00
skj(s)ds < 00,
i=1,2
is locally asymptotically stable. Under what additional conditions can you derive a similar result for a system of the form
dx~?) = ""X, (1- ;:) - ex,(t) [= kll(t - S)Xl (s )ds
['= k (t - S)X2( S)ds "2 2 (1 -;:) - eX2(t) [= k 22 (t - S)X2(S )ds +e
dx;?) =
12
X
+E
[=
k2,(t - S)x,( s )ds.
277
Exercises III
Formulate sufficient conditions on kij ( i,j = 1,2) for your result. (For more results of this type see Gopalsamy [1983c]) 12. Assume that f is a continuously differentiable function such that there exist positive constants x* and H satisfying
x* f(x*) - H
= o.
Obtain sufficient conditions for the local asymptotic stability of the steady state x* of the following scalar equations:
(a)
d~~t) = x(t)f(x(t - 1'» - H.
(b)
d~~t) = x(t - 1')f(x(t - 1'» - H.
(c)
d~~t) =x(t- 1'df(x(t- 1'2»-H.
(d)
d~\t)
= x(t)f (
(e)
d~\t)
=
J~= k(t -
[f(J~=
s )x( S)ds) - H.
k.(t - s)x(s)dS)]
J~= k2(t -
s)x(s)ds - H.
where 1',1'1,1'2 are nonnegative constants and k, k1, k2 : [0,00) 1-4 [0,00) are piecewise (locally) continuous on [0, 00) such that for i = 1, 2
13. Assume that the function f in problem 12 is given by the following (a, b, n are positive constants, n ~ 1).
(a)
f (x) =
(b)
f(x) = a - blogx
(c)
a - bx
f(x) = a - bxn. Do the same as in problem 12 for these
f.
Exercises III
278
14. Let
h ,12 below be continuously differentiable functions of their arguments.
Assume that Tij (i, j = 1,2) are nonnegative constants and suppose there exist positive constants xi, hi (i = 1,2) such that X;fi(X~
,x;) = hi
;
i
= 1,2.
Derive sufficient conditions for the local asymptotic stability of (xi, xi) in the following:
Do the same as in (1), (2) above if
hex, y) =
an X - a12Y
Tl -
hex, y) = T2 - a21 x - a22Y where
Ti,aij
(i,j = 1,2) are positive constants.
15. Can you formulate and analyse the local asymptotic stability of (xi,xi) in the systems of problem 14, when the delays are continuously distributed over an infinite interval? 16. Assume that kl , k2 , b1 , b2 are positive constants. Under what conditions the system dx(t) = kl _ b x(t) dt 1 +y(t) 1
dy(t)
d1 = k2X(t) -
b2y(t - T)y(t)
has a steady state (x*, y*), x* > 0, y* > O? Derive sufficient conditions for the local asymptotic stability of (x*, y*). Generalize your result to a system of the form
dx(t)
kl
-dt- = 1 + y( t dy(t)
d1
TI)
-
b1x(t - T2)X(t)
= k2X(t - T3) - b2y(t - T4)y(t).
279
Exercises III
17. Under what conditions the following scalar systems will have locally asymptotically stable positive steady states; examine stability switching also;
dx(t)
---;It dx(t)
=
AX(t - .r) a+xn(t-r) -,x(t).
(i)
Ax(i)
-at =
a
sup xes). + xn(t) - ,xCi), xCi) = sE[t-r,t]
( ii)
( iii)
d~~t)
= -,xCi)
+ ;Jx([iDe-ax([tj).
(iv)
;J
dx(t)
-at = -,x(t) + 1 + x(t -
(v)
r)
(A,a,r,rl,r2,r3 are positive constants). 18. Derive sufficient conditions for the existence of a locally asymptotically stable positive (componentwise) steady state of the system
dx(t)
a
---;It = 1 + ;Jx'Y(t - rd
AX(t) 1 + J-l y6(t - r3)
dy(t) = AX(t - r4) dt 1 + tLy6(t _ r5) - wy(t) where rl, .. ,r5 ,a,;J",8,J-l,A,W are positive constants.
Wheldon [1975]
19. Formulate and examine the asymptotic behavior of integrodifferential analogues of the systems in problems 17 and 18 above. 20. Examine the local asymptotic stability of nonnegative (componentwise) steady states of the following systems:
dx(t) = x(t) [ b -at
1
+ aux(t) + a12 Jo[00 k 12 (s)x(t - s)y(t - s) ds
1) 1
dy(t) = yet) [ b + a21 Jo[00 k (S)X(t - s)y(t - s)ds + a22y(t) . ----;It"" 2 21
Exerci.'3e.'3 III
280
dx(t) = dt
roo 10
kn ( s )x(t +
dy(t) dt
= roo JO
s) ds
a121°° k
[b
I
+ allx(t)
12 (S)X(t
- s)y(t - s)ds ]
k22 (S)y(t-s)ds [b 2 +a 21 [00 k 21 (S)X(t-s)y(t-s)ds
Jo
+ a22y(t) ]. Formulate your conditions on the various delay kernels appearing in the above systems. 21. Consider the scalar system
dx(t)
---;It
= -ax(t)+ Jot
where a is a positive number, k : [0,(0)
/.00 k(s)ds <
00,
H-
- a
k(t-s)x(s)ds
[0,(0) is continuous and satisfies
+ /.00
k(s) ds =I- O.
Can you prove the equivalence of the following statements ? (i) all solutions tend to zero as t --t 00; oo (ii) - a + fo k( s) ds < 0; (iii) each solution satisfies fo I x( s) Ids < 00 ; (iv) the trivial solution of the system is asymptotically stable. Generalize and derive a two variable version of the above result.
oo
22. Consider a prey-predator dynamics governed by
dx - = xg(x) - yp(x) dt dy dt = y[cp(x) - q(x)] where c is a positive constant and g, p, q are of the form given below. 1.
g(x)=,(l-~)
2.
g( x ) -
3.
g(x)
-y(k-z) k+cz
= ,[I -
(x/k)O']
281
Exercises III
4.
p(x) = a~;r;
5.
p(x)
6.
p(x) =m(1 - e- Ox );
7.
p( X )
8.
q(x) =
= mxfJ;
0::; (3 ::; 1
>0
m
__ mxn • - a+;r;n,
(j
Assume that the parameters" k, €, fr, a,m, (3, b, n, positive constants and there exists x* > 0 such that
cp(x*) = q(x*);
(j
are real
* x*g(x*) y = . p(x*)
Assume also that for k > 0, g(k) = 0, (x - k)g(x) < 0 for x -I k. Derive sufficient conditions for (x* , y*) to be (i) locally asymptotically stable and (ii) examine the global attractivity of (x* , y*). The following are additional models of prey-predator systems; examine the local asymptotic stability of the nonnegative steady states; also examine the absolute (delay independent) stability of the various equilibria:
[1- IOgr;(t)]] - aH(t)log[P(t)])
d~y)
= TH(t)
d~;t)
= -b[P(t)]
d~?) = TH(t) dP(t)
-;It
= TH(t - T)
d~;t)
= -bP(t)
d~it) = dP(t)
[1- H%)] _ aH(t)p(t))
= -bP(t)
d~it)
(1)
+ (3P(t)log[H(t - r)].
+ fjH(t -
(2)
r)P(t - r).
[1 - Hi)] - aP(t) [1 - e-'H(t)] )
[1- e-,H(t-T)]. + bexp~~!~(t - r)] - cP(t) )
(3)
+ I1P(t _ r)
+a
-;It = P(t)[ -
(3 + bH(t - r)].
(4)
Exerci3e3 III
282 dH(t) =
dt
r
= bP(t) [1 -
dP(t) dt dH(t) dt
H()[ _ H(t - r)]_ aP(t)H(t) ) t 1 K f3 + H (t)
(5)
pet) ]. f3H(t)
= rH(t) [K - H(t)]_
aH(t)P(t)] f3 + H(t)
1 + cH(t)
) (6)
dP(t) = pet _ r) [_ f3 + ,H(t) - 6P(t)]. f3 + H(t) dt
d~~t)
H(t)[rl - alH([t]) - b1P([t])]
=
} (7)
d~;t) = pet) [ -
+ a2H([t]) -
T2
bzP([t])].
23. Consider the dynamics of a one prey and two predators modelled by dS =
as
dt
[1 _~K _Yl(al + S)
dx 1 dt
= Xl [
dxz dt
= x2 [
ml Xl
m IS
al
+S
-
m2 x Z ] yz(az + S)
- Dl]
m 2S - DZ] az + S
where a, al, az, mI, mz, Yl, Yz, k, D 1 , Dz are positive constants. Examine the above system with respect to the local (or global) asymptotic stability of nonnegative steady states. Let HI, Hz : [0,(0) ~ [0,(0) be piecewise continuous such that i = 1,2.
Discuss the existence of asymptotically (local or global) stable nonnegative steady states of the integrodifferential system dS =
dt dXl = -d t dxz = -dt
as [1 _~ k
[1 Xz [1 mz
Xl
ml
_
mi _X_l-
Yl a 1
00
HI ( s)
0
Hz(s)
-
m2
~l
Yz
a2
+S
al
s(tS(- s) ) ds - DI ] + t- s
az
Set -( s) ) ds - Dz ] . +s t - s
00
o
+S
283
Exercises III
24. Discuss the existence of bounded and nonnegative solutions for all t 2::. the following (see Hsu and Hubbell [1979]).
°of
where Ti, Ri, kij, bij, Di (i,j = 1,2) are positive constants. Investigate the asymptotic behavior of the following modification of the above system;
Assume such that
Hij :
[0,00)
1-+
[0,00), i,j
= 1,2; Hij
are piecewise continuous
i,j
= 1,2.
25. Examine the characteristic return times (or decay rate) associated with the scalar systems:
dx(t)
--;It
= -ax(t) + bx(t (a < 0;
dx(t)
--;It
= -ax(t)
(a < 0;
+ bx(t -
I),
(1)
Ia I > I b I). It) + cx(t - 12),
I a I > I b I + I c I).
(2)
Exercises III
284
26. Consider a vector matrix system
dx(t) --;u= Aox(t) + AIX(t -
t>O
r)j
in which x(t) ERn; T > OJ A o, Al are real matrices. Let Q = II Ao II; f3 = 1/ Al " denote operator norms of the matrices consistent with some norlillR n • Assume that Ao is such that
for some constants a ~ 1; b > O. Let (J" = 2(-. If (J" < 1 then prove that for any r > 0 the system (*) is asymptotically stable and the following estimate is valid:
II x(t) 1/
::;
a{ sup "x(s) sE[ -r,rj
lI}e-
8t
;
t
~0
where 6 is the unique solution of the equation b - J.L = af3ep.r.
27. Let
~
denote the characteristic quasi polynomial defined by m
<.p(z) = P(z)
+L
Qj(z)e- rj %
where
j=l n-l
P(z)
= zn + L akz k;
n
Qj(z) =
k=O
L bkjzk. k=O
Prove the following result due to Zhivotovskii [1969]. If m
L
m
Ibn}1 < 1
j=l
and
L Ibojl < laol, j=1
then a set of necessary and sufficient conditions for all the roots of ~(z) to have negative real parts is the following: (i) The real parts of the roots of P( z) = 0 are negative. (ii) For any real y > 0, m
L IQiCiy) I < IP(iy)l· j=l
=0
285
Exercises III
28. Suppose p, q are real numbers such that p + q < O. Prove that there exists a real number say 8 = 8(p, q) > 0 such that all the roots of
..\ = p + qe-).r have negative real parts if 0 < I given by 8(p, q) < 7r[lp I + I q IJ/8.
< 8. Prove also that an estimate for- 8 is
29. Let all the roots of
D("\) = det[ aij
+ bij -
..\8ij J = 0
have negative real parts. Prove that there exist two positive numbers 8 8( aij, bij ) > 0 and € = €( aij, bij ) > 0 such that all the roots of
satisfy
~e(..\) ~ €
provided 0
~ lij ~
8.
=
Qin Yuan-Xun et al. [1960]
30. Prove that positive constants € and '" can be suitably selected so that the trivial solution of (for details see Qin Yuan-X un et al. [1960])
dx(t) -;It =€x(t) + yet) + ",[y(t) - yet - I)] dy(t) -;It = €y(t) - x(t) - ",[x(t) - x(t - I)] is unstable for
I
= 0 but is asymptotically stable for some positive I.
31. Obtain a set of necessary and sufficient conditions for the trivial solution of
to be asymptotically stable for all
I
~
0.
32. Assume that all the parameters are positive constants in the following population systems. (a) derive a set of sufficient conditions for the systems to have a positive steady state.
Exercises III
286
(b) obtain the variational systems corresponding to a positive steady state. (c) examine whether the trivial solution of the variational system can be asymptotically stable in the absence of delays. (d) whenever the trivial solution of the variational system is asymptotically stable in the absence of delays, examine whether a delay-induce~ switching from stability to instability can take place. (e) if delay induced switching from stability to instability cannot arise, can you prove that the positive steady state of the full nonlinear system is globally asymptotically stable with additional assumptions?
I: aj x(t - rj) - bx (t). n
dx(t) dt
-- =
2
dx(t)
-;It = axm(t- r) - bx(t)j
mE [1,00).
dx(t) = /X{l - X(t)} - bx(t)y(t) dt K
d~~t)
d~~I)
(1)
j=l
1
= ')'x(l) { 1-
x~)} _ bX(I)ym(l) (4)
r)ym(t - r) - 8y(t)
by(t) dx(t) = X(t){l _ X(t)} _ ax(t) dt /. K a + x(t) b + ym(t)
=c
dx(t)
--;It
(3)
= cx(t _ r)y(t - r) - 8y2(t).
dy(t) --;u = cx(t -
dy(t) dt
(2)
ax(t - r) by(t - r) _ 8y(t) a+x(t-r)b+ym(t-r) = x(t)[b1
-
(5)
allx(t) - a12y(t)]
dy(t) --;u = b2x(t -
rl)y(t -
dz(t) --;u = b3y(t -
z rz)z(t - r2) - a33 z (t).
2
rl -
d~~t) = ')'x(t) { 1 - (x~)
a22Y (t) - a23y(t)Z(t)
r} -t,
aj
X(I)yi' (I)
dYj(t) ] ] (t - r·) ] - d·y ] ]·(t) dt = c'a ] ]·x(t - r·)y~i j=1,2,···,nj
0<
mj
< 1.
(6)
(7)
287
Exerci3e3 III
dS(t) dt
= [S
dy(t) dt
=c
- S(t)JD 0
as(t) by(t) a + S(t) b + ym (t)
) (8)
as(t - I) by(t - I) _ 8y(t). a + S(t - I) b + ym (t - I)
dS(t) = rS(t _ dt
1)(1 -:- Set») _ K
ml
Yl
(1-
- ;22(1 _e-S(t)!a, ) dX~t(t) = Xl(t - r) [ffil (1 - e-S(t)!a,) dx~?) = X2(t - I) [m2 (1 - eS
(t)/a 2 )
-
e-S(t)/a1)
(9) D1Xl(t)] D2 X2(t)].
t) al - bIX! () t ) dt 1 + alYl(t - 11) dYl (t) 2 ---;It = (31X 1 (t - 12) - O'lYl(t) .
dx 1( -
dx(t) = x(t) [ (_ X(t») _ yet) ] dt I K a + ym(t) dy(t) = bx(t _ I) [ ay(t - I) dt a + ym(t - I)
c];
o <m
(10)
<1.)
dH(t) = aH(t - I) - €H2(t) - bH(t)P(t) ) dt 1 + aH(t) dP(t) _ P() dH(t - I)P(t - I) - TJ. p 2( t ) . - - - -c t + dt 1 + aH (t -- I) 33. Prove that the trivial solution of the linear system
dxn(t)
~ =
-anxn(t)
+ bnXn-l(t)
is asymptotically stable if ai > 0, bi 2:: 0 (i = 1,2, ... ,n) and
(11 )
(12)
Exercises III
288
Can you prove the same result for the delay differential system
where
Tj ;:::
0 and
E;=l Tj > 0 .
34. Let m be a positive integer j let x(t) E Rnj A E Rnxnj T E (0,00). If (i) det A = [aij] 1= 0 (ii) the eigenvalues f31, f32, ... ,f3n (real or complex) of A satisfy j = 1,2, ... ,n,
then prove that all bounded solutions of dmx(t) dtm
= Ax(t _ T)
are oscillatory. What can you say about the oscillation of unbounded solutions? 35. Discuss the feedback regulation, linear stabilization and oscillation of the feedback control systems given below (the interested reader should formulate models of predation and mutualism subject to feed back controls with time delays similar to the following):
d~?) duet)
dt
d~?) du(t)
dt
= rN(t)
[1- N(t;
r) - cu(t
-17)]) (1)
= -au(t)
= rN(t)
+ bN(t -
17)·
[1- <>,N(t - r) ~<>2N([t - mll - CU(t)]) (2)
= -au(t) + bN(t - 17)·
Exercises III
289
[1- N)P - CU(t)]
d~it) = rN(t)
duet) = -au(t) + bN(t); dt N(t) = sup N(s).
(3)
sE[t-!",t]
dN(t) +K o2N(At) _ cu(t) ) - = rN(t) [OlN(t) 1dt
(4)
duet) dt
= -au(t) + bN(t),
dN(t) dt
= rN(t) [ K K
(0 < A < 1).
- N(t - r) - cu(t)] ) r)
+ r N (t -
(5)
duet) = -au(t) + bit N(s)ds. dt t-r dN(t) = N(t) [_ r + f3Nm(t - r) - CU(t)] ) dt o+Nm(t-r) duet) dt
(6)
= -au(t) + bN(t)e-CN(t).
dN(t) = rN(t) dt
[1 - t
ajN([t - j]) - curt)] )
j=O
(7)
duet) = -au(t) + bN([t]). dt dN(t) = dt On
f3()n
+ Nn(t -
r)
_ N(t)[{ _ cu(t)] ) (8)
duet) = -au(t) + bN(t). dt dN(t) = f3()n N(t - r) _ N(t)[{ _ cu(t)] ) dt ()n + Nn(t - r) duet) dt dNi(t) dt
= -au(t) + bN(t -
= Ni(t) [r, -
t
r).
C<,jNj(t - T'j) -
j=l n
L
t
C,ju;(t)]
J=1
n
dUi(t) = -aiiui(t) + aijUj(t - O"ij) + ~ bijNj(t); dt j#i J=1 i = 1,2, ... , n.
(9)
(10)
Exercises III
290
36. Let a be a real number and J{ : [0,(0) 1-+ [0,00) be of exponential order. Prove that a necessary and sufficient condition for all solutions of
dx(t) - +a dt
1
00
K(s)x(t - s)ds
0
=0
to have at least one zero on (-00,00) is that the equation
has no real roots. Prove that each of 00 1. a 10 K(s)e AS ds > A, AE R 00 2. a 10 K(s)sds > ~ is a sufficient condi tion for
to have no real roots. Prove also that a sufficient condition for all solutions of
dx(t)
roo
d:t + pet) Jo
K(s)x(t - s) ds
where P : [0,00) 1-+ [0,(0) and I{ : (0,00) on (-00,00) is, that
1-+
=0
[0,00) to have at least one zero
has no real roots where
Po
= liminf pet). t-+oo
Can you generalize your results to linear vector matrix systems of integrodifferential equations? 37. Let Pi(t)
~
0 (i = 1,2) be differentiable functions on [-7, (0) which satisfy
FI(t) ~ allPl(t) {
+ a12P2(t) + bllPI(t) + b12 P2(t) o ~ a21Pl(t) + a22P2(t) + b21 Pl(t) + b22 P2(t)
(t
~
0)
where,
(i,j = 1,2).
291
Exercises III
If aii + bii < 0 and the real parts of all eigenvalues of the matrix (aij + bij ) 2 x 2 are negative then prove that there exist constants M ;::: 1, a > 0 such that
Pi(t)
~ M(tp;(Ol)e-
OIt
t E [-T, (0).
,
J=1
(for more details and a generalization, see Li-Ming Li [1988]);-
38. Assume that the trivial solution of the autonomous vector - matrix system d (t)
~t :::: Ax(t) +
?= Aix(t) m
J=1
is asymptotically stable. Prove that the trivial solution of the delay differential system d (t) m
L
~d t :::: Ay(t) + _ Aiy(t - Ti) J=1
is asymptotically stable independent of delay, if the matrix
AT Ho
+ HoA + mHo AfHo AfHo
HoAl -Ho 0
HoA2 0 -Ho
o
o
HoAm 0 0
is negative definite where Ho is the solution of the Lyapunov matrix equation
( A+ fAi)THo+Ho(A+ fAi):::: -I. J=1
}=1
Use your result to discuss the stability of equilibria of the following logarithmic population models with i :::: 1,2, ... ,n,
(1)
(2) (3) dN-(t) : : Ni(t) [ ----it ri n + ?= aij }=1
it 0
1
I{At - s)log[Nj(s)] ds .
(4)
CHAPTER 4
GLOBAL ATTRACTIVITY
4.1. Some preliminaries
In this chapter we study the global behavior of nonlinear systems of autonomous equations occurring in models of population dynamics. We restrict our analysis to those systems which have a unique positive (componentwise) steady state; systems with multiple steady states are not considered here. First we develop a few preliminary observations in this section. We note that the unique solution of the scalar initial value problem
t > 0,
x(O)
= xo > 0
4.1.1
is given by
t
~
o.
4.1.2
We assume that Xo, a, b are positive constants in (4.1.1). It is elementary to see that every solution of (4.1.1) is defined for t ~ 0 and furthermore, (i) ( ii) ( iii)
x(t) > 0 for t ~ 0; } (bja)1 is nonincreasing; x(t) ~ bja as t ~ 00.
Ix(t)
4.1.3
"7
The existence of solutions of (4.1.1) for all t ~ 0 and their behavior described by (4.1.3) is usually referred to as the global asymptotic stability (or global attractivity) of the positive steady state (bja) of (4.1.1). One of the consequences of (4.1.3) is that for arbitrary constants C2 > 0, there exist tt, t 2 , (tl = tl(c!), t2 = t2(C2» such that
x(t) S (bja)
x(t)
~
+ CI
for
t ~
(bja) - cz for t
~
tl
tz .
CI
> 0 and
4.1.4
We have been able to note (4.1.4) directly from the solution (4.1.2). We can also make the same observation, if we show that the positive steady state (ajb) of (4.1.1) is globally asymptotically stable by other methods, which do not need an explicit knowledge of the solution.
§4.1. Some preliminaries
293
As a second example, let us consider the scalar equation dx dt
x) = AX (1·;.... k -
a
ax
+xA
4.1.5
where A, k, a, A are positi~re constants. One can see that, if A > A, then (4.1.5) has a Wlique positive steady state say x· and that x(O) > 0 will imply x(t) > 0 for all t ~ O. We consider the Lyapunov function v where vex)
=x -
x* - x*log(x/x*).
4.1.6
Calculating ~; along the solutions of (4.1.5), we have 4.1.7 As a consequence of (4.1.7) one can derive the following for (4.1.5).
"If the positive constants A, k, a, A in (4.1.5) are such that A > max(A, Ak/a), then every solution x(t) of (4.1.5) with x(O) > 0 has the asymptotic behavior lim x(t) = x*
t-oo
4.1.8
where x* is the unique positive solution of
x* [A (1 - x*) - ~l = 0" . K a+x·
4.1.9
Consider now the scalar equation dx dt
=~-Dx b + xm
where b, f3, D and m are positive constants. The above system can be put in the form d:i; = ~ {(f3 - D)(b/D) _ xm}. 4.1.10 dt b + xm
If f3 > D, (4.1.10) has a unique positive steady state x; also solutions of (4.1.10) corresponding to x(O) > 0 exist for all t ~ 0 and satisfy x(t) > 0 for t ~ O. If we consider a Lyapunov function v defined by 4.1.11
§4.1. Some preliminaries
294
then we note
dv(x(t)) = _ (xm _ xm)2 dt
4.1.12
showing that, every solution of (4.1.10) has the following asymptotic behavior (details are left to the reader)
x(O) > 0 ~ x(t) > 0 and x(t)
-P
X as t
4.1.13
-P 00.
From the foregoing examples one can observe the following: if in an equation of the form
dx dt = K - f(x),
4.1.14
where K is a positive constant and f : [0,00) f--+ [0,00), f is continuously differentiable and monotone increasing such that f(O) = 0, f'(x) 2 c > 0 for x ~ 0, then (4.1.14) has a positive steady state say x such that f(x) = K. The reader should be able to verify that (4.1.14) has the following behavior:
x(O) > 0
~
x(t) > 0 for t
~
0 and
x(t)
-P
X as t
-P
00.
4.1.15
It has been relatively easy to verify the existence of positive steady states in the above systems due to their scalar nature. When we consider the dynamics of multispecies population systems described by nonscalar systems of differential equations, the problem of ascertaining the existence of positive steady states becomes difficult; it is not uncommon to assume that such steady states exist and then proceed to analyse the asymptotic behavior of the relevant systems. However, in a number of multispecies model ecosystems such as the Lotka-Volterra competition equations, it is possible to propose sufficient conditions for the existence of positive steady states. It is found that the same set of sufficient conditions which guarantee the existence of a positive steady state, sometimes can also guarantee the global attractivity of such a steady state. The following result is of the above type and is due to Kaykobad [1985].
Lemma 4.1.1. Suppose 'xi, aij (i,j
that aji
aij
'xi > E7:~ J .,.'
>0 20 aij('xj!ajj)
= 1,2, ... , n) are nonnegative constants such i = 1,2, ... ,n i,j = 1,2, ... ,n i = 1,2, ... ,no
4.1.16
295
§4.1. Some preliminarie3 Tben tbe Lotka-Volterra competition system i = 1,2, ... ,n
bas a componentwise positive steady state x· tions
= (xi, ... , x~)
4.1.17
satisfying "tbe equa-
n
L aii x; = Ai
and
xi > 0;
i
= 1,2, ... , n.
4.1.18
i=l
Proof. Let AD denote the n x n diagonal matrix; AD = diag( au, a22, ... ,ann)' Then (4.1.16) will imply that AD is nonsingular and that (AD)-l > 0 in an elementwise sense. Define an n X n matrix B as follows: B
= A(AD)-l -
I
4.1.19
where I denotes the n x n identity matrix. We note that B is nonnegative (elementwise) and also that A
= (I + B)AD;
4.1.20
The assumptions in (4.1.16) will imply that the components of a column vector c = col. { Cll C2, ••• , cn } defined by
c= (I -
B).,
4.1.21
satisfy the condition Ci > 0, i = 1,2, ... , n. Since Ai > 0, i = 1,2, ... , nand B 2:: 0 (elementwise), it will follow from (4.1.21) and the componentwise positivity of c that p(B) < 1, (p(B) being the spectral radius of Bj see for instance Berman and Plemmons [1979}, Ch. 6). Let p = p(B)j by the Perron-Frobenius theorem there exists a vector J = col.{ d l , d 2 , ••• ,dn } , d j 2:: O,j = 1,2, ... ,n such that
BT being the transpose of B. Since Ai > 0, Ci > 0, i (J)Tc> 0; but we have from (4.1.21) and (4.1.22),
=
1,2, ... , n we have
§4.1. Some preliminaries
296
which implies that 1- P > OJ a consequence of this is that both (I -B) and (I +B) are nonsingular. The nonsingularity of A = (aij) now follows from (4.1.20) and that of AD. We have
A-IX = (AD)-I(I + B)-IX
= (AD)-I(I + B)-I(I = (AD)-I(I2 _ B2)-le
B)-Ie 4.1.24
= (AD)-l
(tB2i) C
(since pCB)
< 1)
)=0
( componentwise )
[J
and the proof is complete. The set of sufficient conditions n
Ai >
L aii(>\i/aji),
i = 1,2, .. . ,n
j=l j ¢:i
of (4.1.16) has an interesting ecological interpretation; in fact, the motivation for the derivation of Lemma 4.1.1 has come from an analysis of Lotka-Volterra competition equations. We remark that the conditions (4.1.16) are only sufficient conditions for the conclusion of Lemma 4.1.1. One can argue, that by means of Cramer's rule, it is possible to give necessary and sufficient conditions for (4.1.18) to have a componentwise positive solution; such conditions are analytic and unintuitive with respect to the system (4.1.17). It will be found below that the conditions (4.1.16) are also sufficient to make the steady state x* of (4.1.17), a global attractor with respect to solutions of (4.1.17) with Xi(O) > 0, i = 1,2, ... ,n. The proof of the following result (see Gopalsamy [1980], Gopalsamy and Ahlip [1983]) is similar to that of a somewhat more general one to be proved in the next section. One of the implications of the following result is that whatever the size of the delays, nonconstant periodic solutions cannot exist for the system considered and this is contrary to the commonly held expectation of the influence of delays in model ecosystems.
Theorem 4.1.2. Assume that the conditions (4.1.16) of lemma 4.1.1 hold. Let Tij 2:: 0, (i,j = 1,2, ... ,n;, i =1= j). Then every solution of the delay differential
297
§4.1. Some preliminaries system
dUi(t) = Ui(t) [ Ti -;uUi(S) =
=
max
l$:~/n
aiiUi(t) -
~ f;:: j
°
ajjUj(t - Tij)
1
"I'i
4.1.25
-
Tij
satisfies the following:
Xi(t) >
°
for
t ~
°
and
Xi(t) ~X:
t ~ 00 i = 1,2, ... ,njas
4.1.26
where (xi, xi, ... ,x;) is a steady state of (4.1.25)
We leave it to the reader as an exercise, to prove that the steady state x'" of (4.1.25) is a global attractor when the conditions in (4.1.16) hold with
Tij = 0, (i,j = 1,2, ... ,nj it- j) and then to show that (4.1.25) under (4.1.16) cannot have a delay induced switch from stability to instability based on the linear analysis. We note that the result of Theorem 4.1.2 remains valid if some or all of the terms with discrete delays are replaced by continuously distributed delays as in the case of
4.1.27 i = 1,2, ... , n
where k ij : [0,00)
~
[0,00) are continuous and normalised as follows:
1
00
kij(S) ds = 1 j i,j = 1,2, ... , n j i
t- jj
the initial conditions for the system (4.1.27) are of the type
Ui( s) =
>0 ;
sup
i = 1,2, ... , n
in which
298 4.2. Competition: exploitation and interference
In this section we formulate a resource based competi tion model and propose a "method of monotone sequences" using differential inequalities to show the global attractivity of the positive steady state of the model system formulated. The competition considered here is a combination of both exploitation and interference; the resource exploited by the interfering competitors is logistically self renewing. For qualitative details of exploitation and interference competition we refer to the articles of Brian [1956] and Miller [1976]. The result of this section is extracted from Gopalsamy [1986bJ. We consider in what follows, a model without time delays. If there are time delays in the interspecific competition, then the proposed method will still be applicable and the details are similar to those in Banks and Mahaffy [1978a, b] and Gopalsamy [1980]. Let Xl (t) denote the biomass at time t of a logistically self-renewing resource which is essential for two other species. We suppose that the dynamics of the resource in the absence of exploitation by other consumers (or predators) is governed by
in which bl and all are positive constants. Let X2(t) and X3(t) denote the biomasses (or population densities) of two species which feed on the resource exclusively. Assume that the consumers of the resource exploit the resource and also indulge in intraspecific and interspecific interferences. The dynamics of the above type of resource based exploitative competition with interference can be modelled by a system of equations of the form
dXI (t) ---;u=
Xl(t) [b 1
dX2(t) ---;u=
X2(t) [b 2xl(t) - a22 x 2(t) - a23 x3(t)]
dX3(t) ---;u=
X3(t) [b 3xl(t) - a32 x2(t) - a33 x3(t)]
-
allxl(t) - a12 x 2(t) - a13 x 3(t)] 4.2.1
where bi , aij( i,j = 1,2,3) are positive constants (for a model with no intraspecific or crowding effects, see Hsu (1981)). It is assumed that the per individual consumption of the resource in (4.2.1) is a linear function of the resource density and the conversion of the resource into competitor (or consumer) biomass is also a linear function of the resource density; that is, both the functional and numerical
§4.2. Competition
299
responses are linear functions. A wide variety of different functional and numerical responses can be considered instead of the linear ones as in (4.2.1); some alternatives will be suggested in the exercises. The presence of the terms a22, a33 denote intraspecific interference while those with a23, a32 denote interspecific interference. The possible steady states of ( 4.~.1) are Eo, E1 , E 2 , E3 and E. where
(0,0,0) all' 00) ,
Eo E1
E* x*I xi x;
= =
=
E2
(.h.. (a ~
E3
(;3 ,
, a22 '
0)
°~) , a33
a= bI/[all + aIz(b2/a22)] ;3
= bI/[all + a13(b 3/a33)]
6 2X r 6 3xr
It is found that if and
xr
°,
4.2.3
°,
then > xi > x; > 0. If E* exists, then one can ask, under what additional conditions (if needed), the steady state E. : (xi,xi,xi) is globally attractive in the sense that
Xi(O) > 0 =} lim Xi(t) t-oo
= xi ,
i
= 1,2,3.
4.2.4
An intuitive examination of (4.2.1) together with (4.2.4) suggests the following: if both the consumers do not overexploit or "overkill" the resource (as measured by the consumption (or predation) parameters al2 and aI3), and if the resource can reproduce itself sufficiently (as measured by the potential regeneration rate parameter bI ) so as to withstand consumption pressure, then the three species community described by (4.2.1) not only can "persist" in the sense
Xi(O) > 0 =} lim t-.oo inf Xi(t) > 0 , i = 1,2,3, but also satisfy (4.2.4). We proceed to establish a set of sufficient conditions under which all solutions of (4.2.1) with positive initial values will converge as t -+ 00 to the positive equilibrium E. of (4.2.1). The sufficient conditions will be of such a type, one can intuitively foresee. Precisely we prove the following:
§4-2. Competition
300
Theorem 4.2.1. Assume that the constants bi, aii (i = 1,2,3) are positive and aij ~ 0 , i =f j , i,j = 1,2, ... , n. Suppose the following hold: ( i)
(ii) ( iii)
b2 hI b3 bi bi > a 1 2 - - - a 1 3 - a22 all a33 all b2 bi b3 bI ] 1 b2 .[ bi - a I 2 - - - a 1 3 - - > a22 all a33 all au b3 [ bI
b2 bi
b3 bI
- aI2-- - a13-a22 all a33 all
]
-
1
au
4.2.5 b3 hI
a23-a33 all
2 bI > a 3b2 --' a22
all
4.2.6 4.2.7
Then the equilibrium point E.: (xr,xi,xi) of (4.2. 1) defined by (4.2.2) exists and all solutions of (4.2.1) have the following behavior:
Xi(O) > O:::} lim Xi(t) = t-oo
xi
j
i
= 1,2,3.
4.2.8
Proof. We first note that (4.2.5) - (4.2.7) are kept in a form in which it is easy to interpret, rather than in a compact and simple form. It is found from (4.2.5) ( 4.2.7) that b3 b2 b2 > a23- and b3 > a32a33
a22
which will imply the existence of E. : (xi, xi, xi), xi > 0, i = 1,2,3. It is easy to establish that all solutions of (4.2.1) are defined on [0,00) and Xi(t) > 0 on [0,00) when Xi(O) > 0, i = 1,2,3. Using such a positivity of the solutions of (4.2.1) and the property of logistic growth one can show that for any C 1 > 0 there exists a tl > 0 satisfying
ui 1) X2(t) < ui 2)
Xl(t)
<
X3(t) <
=
4.2.9
for
=
=
U?)
The detailed arguments leading to (4.2.9) are based on differential inequalities and the solution of the usual logistic equation; for more details of this technique of derivation of (4.2.9), we refer to Gopalsamy [1980]. Our strategy, for the proof of (4.2.8) is to derive sequences of "asymptotic upper" and "asymptotic lower" estimates of the solutions of (4.2.1) and then show that such sequences of upper and lower estimates converge to the positive steady state E. under (4.2.5) - (4.2.7). We begin by choosing CI > 0 and the corresponding tt > 0 such that
[b b3 [b
b2
bi l i -
-
a12U?) - aI3U?)
?) - a I3 U?)] a!t 3 aI2 U ?) - a I3Ui )] a!t a12 U
>0
> a23 U ?) > a32 U ?)
} .
4.2.10
§4.2. Competition
301
The possibility of such a choice of Cl > 0 satisfying (4.2.9) - (4.2.10) is guaranteed by (4.2.5)-(4.2.7). Having selected C1 > 0, tl > 0 we choose C2 > 0 small enough to satisfy
and
4.2.11
The possibility of choosing C2 > 0 satisfying (4.2.11) is a consequence of (4.2.10). It will follow from (4.2.1) and (4.2.9) that
leading to the existence of a t2 > tl for which
Xl(t) > Lil)
=
X2(t) > Li 2 ) X3(t) >
Li
3
t > t2 •
4.2.12
)
It is a consequence of (4.2.1) and (4.2.12) that
dXl(t) < Xl(t) ({ bi
~
-
a12Ll(2) - a13Ll(3)} - allXl(t) ) t
> t2'
4.2.13
We also note
>0
(by (4.2.5».
4.2.14
§4.2. Competition
302
One can now show from (4.2.13) - (4.2.14) that there exists a 0, C3 < min { c2} such that
t3
> t2 and
C3
>
t,
Xl (t)
< UJI)
U
= {bi
(2) Z
=
X3(t) < UJ3)
=
X2(t) <
a12 L
-
i
2
{b {b U?) -
) -
a 13L
i a!l + C31 3
)}
2 u(1) 2 -
a23 L(3)} I
_I
a22
+ 9.2
3
a32Li2)}
a!a
+~
t>
.,
t3'
4.2.15
The positivity of the estimates U?) and U~3) are verified as follows:
b2U2(1)
-
a23Ll(3) = b2 [ ( bl - a23 [
> b2 [ (b i > bz [ >
°
(b
(3»)
(2) - a12 L I - a13 L I
(b 3LP) -
a12 Li
2
a~3
a3z U?») ) -
a 13L
+ C3 ]
-
c; 1
i3») 2-]a2a~LiI) all a33
i
3
a12Ui2) - a I3U »)
l -
1
all
2-]- a23~~ all
aaa all
(by the second of (4.2.11»;
4.2.16
and similarly,
I baU2(1) - a32 L (2)
= ba [ (b I - a32 [
b[(b
1 I aI2Ll(2) - a13 L (a») ~
-
(b 2Lil) -
+ c3 ]
i3») a~I - c; 1 a 13 U?») 2-]- a32~~ all a22 all
a 2aU
> a
l -
>0
(by the third of (4.2.11).
a12Ui2) -
4.2.17
Now using the upper estimates in (4.2.15), we derive a set of lower estimates as before; first we need the following verification: bi
-
(2) (a) a12 Uz - a13 Uz
= bi
-
aiZ -
> bi
> bi >0
[
(1) bU
an [
Z
2
-
(3»)
a23 L l
1
a22
cal + 2"
(b U?) - LF») a~3 + "; 1 3
a 32
-
aI2
(~u?) + ~) a22 2
-
a 12
U?) - alaU~a)
al3
(by the first of (3.16»,
(~U?) + C1 ) aa3 2 4.2.18
303
§4.2. Competition •
(1)
(2)
(3)
in which we have used e3 < e1' Let us define £2 '£2 '£2 1 } [ b1 - a12U2(2) - a13U2(3)] ~ (1) U(3)
b2 £2 b3 £2(1)
-
a23
-
an
as follows:
.
2
t > t2 •
'
U(2)
4.2.19
2
Using (4.2.18) we derive that
4.2.20 Similarly,
4.2.21 It will follow from (4.2.18)-(4.2.21) that there exists a positive e4 satisfying £~1) _ e4 > 0 } { b2 (£~1) - e4) - a23U~a)} a~2 - T > 0 .
{b3(.e~1) - e
4) -
32
a UJ2)}
< min {i, e3} 4.2.22
a;3 - T > 0
Now using the upper estimates in (4.2.15), we have
dXl(t)
~
(2)
(3)
> Xl(t) { [b 1 - a12 U2 - a13U2 ] - allXl(t) } t > ta
with which and (4.2.22) one can show, there exists a t4
X1(t) > L~l)
=
X2(t) > L~2)
=
X3(t) > L~3)
=
[b 1
-
a12 UZ(Z) - a13 U(3)] Z
1 Ci7i'
C4
4.2.23
> ta such that
>0)
- a U(a)]..L - ~ > 0 [ bZ L(l) 2 23 2 a22 2 - a32 U(2)]..L - ~ >0 [ b3 L(l) 2 2 a33 Z
. '
t > t4 • 4.2.24
§4.2. Competition
904
The positivity of the estimates L~i), i We thus have
= 1,2,3 is a consequence of the choice of C4.
i
< Xi(t) < U?); L~i) < Xi( t) < U~i);
i = 1,2,3; i = 1,2,3;
Li )
4.2.25
At this stage let us compare the respective lower-and upper estimates: for instance, U2(1)
-
UI(1)
= { bI < c3 -
(2) (3)} a12 L l - a13Ll -
1
all
CI
a22
<
a22
{ - bI
all
+ c1 }
< 0;
U?) - U?) = {b 2 U?) - a 23Li3 )} ~ + c3 bz
+ c3 -
2
_
{~ui1) + c 1 } 2
a22
(1) (1») +(C3- CI)2'1 U -U 2
I
< 0; similarly, we will have
> 0;
L
(2)
2
-
L(2) _ 1
-
{b
2
L(I) 2
-
a23
U(3)} 2
-
1
a22
- [ {b,L\l) - a23U!')}
-
a~2
c4
-
2
-
e; 1
= bz { L2(I) - L1(I)} - 1 - a23 {(3) U2 - UI(3)} - 1 a22
a22
+ (c3 -
c4)-1 2
> 0;
a further similar analysis will lead to
Thus, we have from the above
U?) }
LP) < L~I) < XI(t) < UJ1) < L~2) < L~2) < X2( t) < U?) < U?) LP) < L~3) < X3(t) < <
U?) U?)
;
4.2.26
305
§4.2. Competition Now repeating the above procedure we can derive L(i) 1
<
L(i) 2
<
L(i) 3
< ... <
L(i) < x·(t)
< ... <
U(i) 2
i=1,2,3, .. j where
{ b1
{ b U(1) _
=
1 + e2n-l a13 L(3)} n-1 ii"i7 L(3)} _1 + ~ a23 n-1 a22 2
a12 L (2) n-l -
-
2
n
{ b3 U(I) n
_
a
32
n
= [b 1 =
a12 Un(2) -
b L (1) [2 n b L (1) [3 n
a 13 Un(3)]
L (3)
n-1
U(i) 1
t > t2n
4.2.27
}
4.2.28
+~ 2
} _1_ aS3
= 2,3,4, ...
1 - e2n ii"i7
-
a 23 U(3)].L _ n an
.!2n..
-
a
U(2)] -L _ n a33
.!l.n.
32
<
2
I
,
for n = 1,2, ..
4.2.29
2
By the choice of em(m = 1,2,3, ... ) we know that (since em < ~), em ~ 0 as m ~ 00. Also the monotone sequences U~i) and L~) converge to positive limits as n ~ 00. We let i and L~i) = lim L~). ) = lim U~i) 4.2.30
ui
n-=
n-=
We have from (4.2.27) - (4.2.30) on using limn-+= en = 0, that
Similarly,
§4.2. Competition
906
Under the hypotheses of our theorem, the linear system of equations all x -
a12y- a 13 z
b2x -a22y- a 23 z b3 x -a32y- a 33 z
: =
~} 0
4.2.33
has a positive solution E* : (xt,xz,xi) which is unique such that xi > 0, i == 1,2,3; one can verify that (xi, xz, xi) satisfies the same relations (4.2.30) - (4.2.32) satisfied by U!i), i = 1,2,3. The uniqueness of solutions of (4.2.33) shows that { U!I), U!2), U!3)} is the positive equilibrium of (4.2.1). Similar to the above, one can show that {L~l), L~2), L~3)} is also the positive equilibrium of (4.2.1) and by the uniqueness of such an equilibrium we have (i) - U(i) - x~ . L* -*-1'
1 2 3 -".
z· -
4.2.34
The result follows from x~ = lim L(i) I
n--oo
n
< tlim Xi(t) < lim U(i) = ..... oo - n--oo n
-
x~ 1
4.2.35
[]
and this completes the proof.
The result of Theorem 4.2.1 has an important ecobiological (ecological and biological) interpretation regarding the principle of competitive exclusion. More details and references to the related literature can be found in Gopalsamy [1986b].
4.3. Delays in competition and cooperation The principal aim of this section is to derive sufficient conditions for all positive solutions of certain classes of autonomous systems of delay differential equations to converge to equilibrium states. We first consider the initial value problem
{I: n
dXi(t) dt- -- x-(t) I
b-I -
a-I}-x ) -(t - r-I-))} '.
t>O
j=l
Xi(S)='Pi(S»O sE[-rO]' r= max r--, "l~i,j~n I)
4.3.1
§4.9. Competition and cooperation
907
where bi, aij, Tij (i, j = 1,2, ... , n) are nonnegative constants and Tii > 0 for one or more i E (1,2,3, ... , n). In theorem 4.3.2 below, we provide a set of sufficient conditions for the local asymptotic stability of the positive (componentwise) steady state of (4.3.1). Subsequently, we derive sufficient conditions for the global attractivity of the positive steady state of (4.3.1) with respect to all ecologically meaningful solutions. It has been shown by May and Leonard [1975] that a LotkaVolterra three species competition system can exhibit aperiodic oscillations of ever increasing cycle time; Smale [1976] has demonstrated that one can expect any type of dynamical behavior from a general n-species competition system. Shibata and Saito [1980] have shown that time delays in two species Lotka-Volterra competition can lead to "chaotic behavior". The structure of limit sets corresponding to solutions of ordinary differential equations modelling competition and co-operation has been examined by Hirsch [1982, 1985] who has shown that bounded solutions of co-operative systems converge to the set of equilibrium solutions. Complex dynamical behaviour induced by time delays in certain physiological systems has been discussed by an der Heiden [1979], an der Heiden and Mackey [1982] and Hale and Sternberg [1988J. Our result below (Theorem 4.3.2) provides sufficient conditions for the nonoccurrence of complex behavior in competitive systems. Let us suppose that bi > 0, 1,2, ... , n; i =f j; furthennore, let
aji
> 0, (i = 1,2, ... ,n) and
aij
2:: O,i,j =
n
bi >
L
i
aij(bj/ajj);
= 1,2, ... ,no
j=l j#i
Then it will follow from the result of Lemma 4.1.1 that (4.3.1) will have a steady state x* = (xi, ... ,x~) with xi> 0, i = 1,2, ... ,n. If we let i
= 1,2, ... , n
4.3.2
in (4.3.1), then dYi(t) ~
* (t = - ~ L...J aijXjYj
Tij
)
- Yi
( t ) L...J ~ aijXjYj * (t -
j=l
Tij
j=l
i
) 4.3.3
= 1,2, ... ,n
together with
Yi(S) = [
xil/xi ; S E [-T,O];
i = 1,2, ... ,no
4.3.4
§4.3. Competition and cooperation
308
A linear variational system corresponding to the steady state x· = (xi, ...
,x~)
in
( 4.3.1) can be obtained from (4.3.3) and is of the form i
= 1,2, ... ,no
4.3.5
The characteristic equation associated with the autonomous linear system of differential-difference equations (4.3.5) can be shown to be
4.3.6 where 5ii = 1, i = 1,2, ... ,n; 5ij = 0, i =lj, i,j = 1,2, ... ,n. If>' is any root of (4.3.6), then it will follow from Gershgorin's theorem (Franklin [1968]) that n
>. + a"x~e-..\rii 1 < ""'" a ··x~e-..\rji U I L....J }I I I
4.3.7
j=1 j 'Fi
for some i E (1,2,3, ... ,n). As a consequence of (4.3.7), complex constants kj = kj(>'), I kj(>') I < 1, j = 1,2,3, ... , n will exist, so that for any root>. of (4.3.6), n
i;x~ >. + a"e-"\r H I
+~ L....J aJI"x~ kJ·(>')e-..\r = 0 j ;
I
j=1
i'Fi
4.3.8
for some i E (1,2, ... , n) . The following result provides a set of sufficient conditions for all the roots of (4.3.6) to have negative real parts. Lemma 4.3.1. Assume the following: (HI) the real constants 7"ij ~ 0 (i,j = 1,2, ... , n) satisfy n
7"ii
~ ffi;in
1<)
7~
(H2)
aij
7"ji j
i = 1,2, ... , n
;
if L....J ~ 7"ji
=I 0
4.3.9
j=l
(i,j = 1,2, ... , n) are nonnegative constants such that
i=1,2, ... ,nj
4.3.10
S09
§4.S. Competition and cooperation
(T;;X i
t.
a j ;)
(1 -
a;;T;;x i ) -1
i#i
t
aii cos ({ TiiXi
<,,/2
i
= 1,2, ... ,n
4.3.11
n
aid / {I - aiiTiixn) > Laji j=1 j'l"
)=1
j'l"
i
4.3.12
= 1,2, ... ,n.
Then all tbe roots of (4.3.6) bave negative real parts.
Proof. It is sufficient to show that when (4.3.9) - (4.3.12) hold, (4.3.8) cannot have roots with nonnegative real parts. Let us suppose that). = a + ij3 is a root of (4.3.8) with a, j3 real and a ~ O. Then considering the real and imaginary parts in (4.3.8),
a = -ammx:ne-OTrnrncosj3Tmm n
- L ajmX:ne-OTjm~e [ki ( a + ij3)e-
i
(3 Tj m]
4.3.13
j=1
j'l'm
j3 = ammx:ne-OTmmsinj3Tmm n
- L
ajmx:ne-OTjm~m [kj(a
+ ij3)e i,B Tj m]
4.3.14
j=l
j'l'm
where (4.3.13) and (4.3.14) hold for some m E (1,2, ... , n). Roots of the type ). = a + ij3 , a ~ 0, j3 = 0 (i.e. real nonnegative roots) are not possible since in such a case we will have from (4.3.13), n
a ::; -aiixie- OTii
+L
ajixilki().)le-OTii
j=l
j'l'i
::; -xie-
OTii
[aii -
t
aji ] for some i E (1,2, ... , n)
4.3.15
)=1
j'l'i
<0
4.3.16
since (4.3.11) - (4.3.12) will imply aii > L:7:~ aji (i = 1,2, ... ,n) which contra) r' diets a ~ O. Thus, (4.3.16) cannot have real nonnegative roots. Suppose now that ). = a + ij3,a ~O,j3 t- 0 is a root of (4.3.16); then we will have from (4.3.14)
IfJRI
[1
-
Tiiaii X*. e t
T -01"'"1 sin(j3 ii) j3 ii T
I] <
_ L j=1
i :pi
a l'"x~e-OTji ,
4.3.17
§4.3. Competition and cooperation
310 and hence
n
1,81 [ 1 -
ajj'TiiXi J :::;
I: ajixi
for some i E (1,2, ... ,n).
4.3.18
i=l
i ,ti
Now (4.3.18), (4.3.10) - (4.3.18) together imply that n
a :::; -aiixie-OTiicos [ ( TiiXi'
I: aji ) / (1 -
aj(T'jjxi) ]
j=l j ,ti
4.3.19 j=l j ,to
~ -xi e
-OT;;
[a"cos ( { Tiixi
t
aj' } / {l - aiiTiixi} )
i ,to
-tai'] i
4.3.20
,to
which contradicts a ~ 0 showing that when (4.3.9) - (4.3.12) hold, (4.3.16) cannot have roots with nonnegative real parts and this completes the proof. [] One of the immediate applications of Lemma 4.3.1 is in the proof of the following: Theorem 4.3.2. Assume tbat the parameters of (4.3.1) satisfy tbe assumptions of Lemma 4.3.1. If in addition, a positive (componentwise) steady state x* = (xi, ... ,x~) of (4.3.1) exists, then x* is locally asymptotically stable. Proof. The result follows from Lemma 4.3.1 by an application of Theorem 11.2 of Bellman and Cooke [1963]; we omit the details of proof. [] We remark that our motivation for the formulation of Theorem 4.3.2 has been to provide a set of somewhat "easily verifiable" sufficient conditions for the conclusion of Theorem 4.3.2 to hold. It is interesting to note that the sufficient conditions of lemma 4.3.1 reduce to a set of diagonal dominance-type conditions if Tii = 0, i = 1,2, ... ,n in which case, global asymptotic stability of x* has been established by Gopalsamy and Ahlip [1983]. It is not unreasonable then to expect
§4.9. Competition and cooperation
911
that if 'Tii (i = 1,2, ... , n) are positive but sufficiently small, then x* can in fact be globally asymptotically stable (or at least is globally attractive). We will show in the following that such an intuitively expected result "is in fact valid. In order to reduce the complexity of notation, we consider the case n = 2; the extension to arbitrary n is otherwise routine. Accordingly, we consider a two species competition system modelled by the delay differential equations,
4.3.21
in which ri, bij E (0,00) and 'Tij E [0,00), i,j = 1,2. We show how the results of Chapter 3 can be used for the derivation of sufficient conditions for the global attractivity of the positive steady state of (4.3.21); that is, we derive sufficient conditions for all positive solutions of ( 4.3.21) to satisfy i
= 1,2
where
4.3.22
2
L
bijN;
=
ri,
i
= 1,2.
4.3.23
j=l
Since the system (4.3.21) is known to be capable of complex behavior ("chaotic") (see for instance Shibata and Saito [1980]) it is of some interest to know under what verifiable conditions on the parameters of (4.3.21), all the positive solutions of (4.3.21) will have the asymptotic behavior in (4.3.22). We assume that the system of equations (4.3.21) is supplemented with initial conditions of the type
i = 1 2' "
'T
= 19,j:52 max 'T".
4.3.24
I)
A few elementary facts about the solutions of (4.3.21) and (4.3.24) are obtained in the following lemmas: Lemma 4.3.3. Assume
bij, ri
E [0,(0) ; i,j = 1,2 and satisfy 4.3.25
312
§4.3. Competition and cooperation
Then there exists a unique equilibrium (Ni, N;) of (4.3.21) satisfying bllN:
+ b12 N; = rl
Ni > 0,
N;
> O.
4.3.26
The proof follows by solving the linear equations in (4.3.26) in the unknowns and verifying the inequalities in (4.3.26) using (4.3.25). We omit the details. [] It is easy to see that solutions of (4.3.21) and (4.3.24) are defined for all t and remain positive; i.e.
Nl(t) > 0,
N2(t) > 0
for all t
~
~
0
0;
4.3.27
t~O
4.3.28
t
4.3.29
therefore, NI and N2 satisfy
~
O.
Lemma 4.3.4. Let ri , bi i E [0,00), i = 1,2. There exist positive numbers Tl and T2 such that every solution of (4.3.21) and (4.3.24) satisfies the estimates
NI(t) 5: Ml
for t
~
Tl
4.3.30
N2(t) 5: M2
for t
~
T2
4.3.31
in which
~ ) b
e r1 Tll
4.3.32
(~ ) b22
er2T22 •
4.3.33
MI = (
ll
M2 =
Details of proof are similar to the case of the scalar logistic equation carried out in the Chapter 1 and therefore we omit the details. [] We shan henceforth assume that all the real numbers satisfy
ri,
bij,
Tii,
i,j = 1,2
(~22) exp[r27"22]
4.3.34
r2 > b'1 (;'1.) exp[r.TllJ.
4.3.35
TI
> b12
919
§4.S. Competition and cooperation
It is seen from Lemma 4.3.4 that there exists a number say T such that solutions of (4.3.21) satisfy 4.3.36 4.3.37
for all t
~
T where
hI
= 1'1 -
h12 (1'2/ b22) exp[1'2 T22]
4.3.38 4.3.39
b2 = 1'2 - b21 (rI/b ll )exp[rITll]'
Lemma 4.3.5. Assume (4.3.34) and (4.3.35) hold. Then there exist positive numbers TI and T2 such that every solution of (4.3.21) satisfies the estimates
N, (t) 2': (b,fb ll ) exp [(b N,(t) 2': (b,/b,,)exp
l -
bl1 M,)Tll] ,
[(b, - b"M')T"] '
4.3.40 t
> T2
4.3.41
where 4.3.42
Proof. We first note
and remark that the details of proof are similar to those of a similar lemma in Chapter 1. Suppose a solution of (4.3.36) is nonoscillatory about (b 1 /b ll ). Then there exists a number s*, such that either Nl(t) > bdbll or N1(t) < b1/bl l for t > s* ~ T + T11' In the former case there is nothing to prove for N I , while in the second case Nl(t) > 0 so that NI (t) ~ mast -+ 00 where m ~ bdbll and hence. there exists a number si satisfying
This proves (4.3.40) for solutions of (4.3.36) which are nonoscillatory about (bi /b ll ).
§4.S. Competition and cooperation
914
If Nl is oscillatory about bl/b 11 , then (4.3.40) is established by considering
a local minimum of NI and arguments are similar to the corresponding details carried out in Chapter 1 for the delay logistic equation. We omit these details. [] We shall proceed to discuss the global attractivity of the positive equilibrium (Ni,Ni) of (4.3.21). We let
NI(t) == N;[l
+ YI(t)],
N2(t) == N;[l
+ Y2(t)]
4.3.43
and derive from (4.3.21) that YI and Y2 are governed by
dYl (t) -;[t
= -[all(t)Yl(t -
1'11) + aI2(t)Y2(t - 1'12)]
dY2(t) -;[t = -[a21(t)Yl(t - T2d
4.3.44
+ a22(t)Y2(t - 1'22)]
where for t > 0,
a11(t) = bllNI(t);
a22(t) = b22 N 2(t)
al"2(t) = bI2 (N; /N;)NI(t) ;
a2I(t)
= b21 (N; /N;)N2(t).
4.3.45
We can conclude from the results of the above lemmas that there exists a number (J > 0 such that for all t > (J, mll ::; all
(t) ::;
Cll;
m22 ::; a22 (t) ::; C22 j
a12(t) ::;
CI2
a2I(t)::; C21
for
where
t;:::
(J"
I
b11 Md Tll] } m22 = b2 exp[(bz - b22 M 2)T22] mll
= bI exp[(b i
-
r1
Cll
=
TI e
Tll
C22
= T2er2T22
CI2
= b12(N;/N;)(Tl/bll)erlTll
4.3.46
4.3.47
4.3.48
C21 = b2I (N; /N;)(T2/b22)er2T22. For convenience we define two numbers J.Li and J.Li as follows:
J.Lr
= bI exp[(bI -
-
bllMt)Tll]
+ C12 T12) + CZI (C21 1'21 + CZ2 T22) + CllTll(Cll + CI2) + C2I T21(C21 + C22)] [Cll (Cll 1'11
4.3.49
J-l; = b2 exp[(b2 - b22 M 2)T22] -
+ C12 T12) + C21 (C21 1'21 + CZZT22) + CIZ T12( Cll + C12) + C22 TZ2( C21 + cn)]. [C12( Cll 1'11
4.3.50
315
§4.9. Competition and cooperation
The following result provides a set of sufficient conditions for the global attractivity of the positive equilibrium (Ni, NZ) of (4.3.21). Theorem 4.3.6. Assume the following conditions hold:
(i)
ri, bij, Tij E
i,j
[O,oo)j
= 1,2.
(ii) bl =
rl -
b12(r2/b22)eT2T22 > 0
b2 = r2 - b21(rI/bn)eT1Tll
4.3.51
>0
(iii) JL~
4.3.52
> 0,
(iv) The quadratic form
= [Yl
Q(y}' Y2)
Y2] [ mn C12 + C21
C12
+ C21 1[Yl 1
m22
Y2
is nonnegative on the set
Then all positive solutions of (4.3.21) satisfy 4.3.53
Proof. We define afunctional V = V(Yl,Y2)(t) = VI + V2 where VI and V2 are as in the case of (3.9.8) and (3.9.12) of Chapter 3. We estimate the rate of change of V similar to that in Theorem 3.9.1 of Chapter 3. On using (4.3.45)-(4.3.50) and assumptions (i)-(iv), we will be led to an inequality of the type
V(Yt,Y2)(t)
+ J.1.
itot[Yi(s) + y~(s)] ds ~ V(Yl,Y2)(tO)
where JL = min{J.L~,JL;}. The remaining details of proof are similar to those of Theorem 3.9.1 of Chapter 3 and we omit these details. Thus, we conclude that lim YI(t)
t--oo
= 0;
lim Y2(t) =
t-oo
o.
[]
§4.3. Competition and cooperation
316
We proceed to derive an alternative set of sufficient conditions for the validity of (4.3.53). Let us rewrite (4.3.44) as follows;
dVl(t) --;It =
-all (t)Vl(t)
- a12(t)V2(t)
+ al1(t)J.~", dV2(t) --;It
rileS) + a,2(t)
J.~r" il2(s)ds 4.3.54
= -a21(t)VI(t) -
a22(t)V2(t)
+ a21 (t) J.~,." il, (s) ds + a2'( t) J.~r" !i,( s) ds , For any fixed t ~ to ~ (J' (see (4.3.46), we can without loss of generality assume that VI(t) 2:: 0, since otherwise, for that t we can consider -Vl(t). Thus, for fixed t the sign pattern of (Vl(t),V2(t» can be
(+,+) ,
(+,-)
and we can write [to, 00) such that
[to, 00) = J 1 U J2 J 1 = {t ~ to IVI(t) ~ 0, Y2(t) ~ O} J 2 = {t ~ to !V1(t) ~ 0, V2(t)
< O}.
We recall that
aij(t) > 0 for
t
~
to
and m"I)
< a' ·(t) < c··'
-
I)
_
i,j = 1,2
')'
from (4.3.46) .
Now for t E J 1 , the system (4.3.54) simplifies to
~ dt
[!IY2VI I(t)I(t)] -< P [!IY21(t)I(t)] + c [I11f21(t) 1ft-l(t)] YI
1
where PI =
c_[
cil III
[-;11
+ CI2 1 12 C21 + C22122C21
C21 121 Cll
Cll III C12
+ C12 1 12 C22]
C21 121 C12
+ C~2T22
4.3.55
917
§4.9. Competition and cooperation
Uhl(t)=
sup
IY21(t) =
IYII(s),
sE[t-r,t]
sup sE[t-r,t]
I Y21(s)
If we assume that the matrix -( PI + C) is an M- matrix then by the result of Tokumaru et al. [1975J (see section §3.6 of Chapter 3) it follows that there exist positive numbers kP), k~2), 61 such that 4.3.56 For t E J 2 one can similarly show that (4.3.54) leads to
.:i [I YI let)]. < P. dt
IY21(t) -
2
[IIY21(t)I(t)] + [IIY21(t)I(t)] C
YI
YI
4.3.57
where C12
]
-m22 Again if the matrix -(P2 + C) is an M-matrix, as before there exist positive numbers k~I), k~2), 62 such that 4.3.58 This discussion leads to a sufficient condition for the global attractivity of the positive steady state of the competition system (4.3.21). We summarise the result as follows:
Theorem 4.3.7. If the matrices -(PI
+ C)
and
- (P2
+ C)
4.3.59
are both M -matrices, then solutions of (4.3.54) satisfy lim YI(t) = 0;
t-oo
lim Y2(t) = O.
t-oo
Proof. Proof is an easy consequence of (4.3.56) and (4.3.58) .
4.3.60
[J
We recall that there are simple criteria for verifying whether or not a given matrix is an M -matrix (section §3.6 of Chapter 3). We ask the reader to carry
318
§4.3. Competition and cooperation
further and simplify the conditions of Theorem 4.3.7 in order to obtain these conditions in terms of the parameters of the competition system (4.3.21). We shall now consider the asymptotic behavior of models of cooperation and in particular "facultative mutualism"; as an example, we study the following model of "hypercooperation";
4.3.61
where at, a2, K l , K2 E (0,00); ; al > K 1 , a2 > K2 j 81 and 82 are odd positive integers. When 81 > 1 or 82 > 1, (4.3.61 ) is a model of hypercooperation; models of hypergrowth have been discussed by Turner et. al. [1976], Turner and Pruitt [1978] and Peschel and Mende [1986] where some evidence of the relevance of hypergrowth models to reality is illustrated. Basically hypergrowth models correspond to situations where in the early phases, a population system flourishes with exponential growth and near saturation, the rate of saturation slows down in nonlinear way. It is this nonlinear slowing down near saturation, that makes hypergrowth different from the other well known growth models considered in mathematical ecology and biology. For instance if 81 ~ 3 or 82 ~ 3, the positive equilibrium of (4.3.61) (for details of this see Chapter 3) is not linearly asymptotically stable and therefore the local asymptotic stability of the positive equilibrium of (4.3.61) cannot be studied by linearization (or variational) methods. Dynamical behavior of cooperative systems without time delays has been discussed by Krasnoselskii [1968], Selgrade [1980], Hirsch [1982-85, 88a, b] and Smith [1986a, b, c]. Cooperative systems with time delays have been considered by Martin [1976, 1978, 1981]' Banks and Mahaffy [1978a,b], Ohta [1981]. One of the crucial assumptions used by Martin [1981] is that the growth rates are dominated at 00 by an affine function (assumption F4) and that all the eigenvalues of the matrix of such an affine function have negative real parts. It is the opinion of the author, that the existing results on the global convergence of time delayed cooperative systems are implicitly based on the following assumption: "the corresponding linear variational system has a negative stability modulus". This is equivalent to the assumption of linear asymptotic stability of positive equilibrium of (4.3.61).
§4.9. Competition and cooperation
919
We show below that even if the linear variational system associated with a unique positive equilibrium of a cooperation model is not asymptotically stable, such an equilibrium can be a global attractor with respect to all other positive solutions. For instance, if 81 2 3 or f)z 2 3, the unique positive equilibrium of (4.3.61) is not linearly asymptotically stable; we show, however, it is a global at tractor. The following lemma is due to :NIartin [1981J and establishes the property of preservation of upper bounds.
Lemma 4.3.8. Let
~
= ('PI, 'Pz) and P = (PI, pz) satisfy tbe following:
= 1,2
Pj E C([-Tj, O] ,R+);
j
'Pj E C([-Tj,OJ, R+);
j = 1,2
'Pj(O)
> 0;
j
s E [-Tj,O];
= 1,2
j = 1,2.
corresponding to j = 1,2, then
Nj(t, if?) s;. pj(t) Proof. Let satisfies
€
for
t 2 0, j = 1,2.
4.3.63
be an arbitrary fixed positive number. First we show that p}E)(t)
j = 1,2
where
4.3.64
§4.3. Competition and cooperation
320
dp~: (I) = F, ((P~') (I), p~,) (I -
dP~(t) = F2 ((p;'\t p)E) (t) = pj(t)
T,),v\') (I) )
+ €,
Suppose there exists s >
T2))
[p,( I) -
F, (p, (I), P2(1 - T2))]
+€
+ [P2(t) -
F2(p, (t - T,),P2(tll]
+€
+
t E [-7"j, 0],
j = 1,2.
4.3.65
°such that
p(E)(t) 2:: N(t, if!) for all t E [O,s]
4.3.66
and either (i) piE\s) = NI(s, 4» or (ii) p~e)(s) = N 2 (s, 4»; if (i) holds, then
pid(s) - NI(s, 4» = FI(pie)(s), p~E)(S - 7"2)) + {PieS) - F1(PI(S), P2(S - 72)) + €} - FI(Nt(s, if!), N 2 (s - 7"2,4>))
> FI(piE)(s),p~E>Cs - 7"2)) - Ft(NI(s, 4», N 2 (s - 72,4»
>
°
4.3.67
°
by the quasimonotone property of FI since %f; 2:: (verify this). From (4.3.67), it follows that p~E)(t) > NI(t, 4» for t E (8, S + 8) for some small positive 8. Now letting € ~ 0, we have PI(t) 2:: NI(t,4» for t E (0, S +8); if necessary, one can repeat this argument to conclude PI(t) 2:: NI (t, 4» for t 2:: 0. Similarly, one proves p2(t) 2:: N 2(t, 4» for t 2:: 0. [] The next result deals with the preservation of lower bounds. Lemma 4.3.9. Let 4> = ('PI, 'P2) be as in the case of Lemma 4.3.8. and let q( t) = (qi (t), q2 (t)) satisfy the following:
qjEC([-7"j,O],R+)
;
j=1,2
qj(s) S 'Pj(s) , s E [-7"j, 0], j = 1,2 dqI(t) < (t) [KI +atq2(t-7"2) _ (t)] 8 dt - ql 1 + q2 (t _ 7"2) qt
1
4.3.68 2
dq2(t) < (t) [K2 + a2qI(t - 7I) _ (t)] 8 dt - qz 1 + q1 (t - 7"1 ) q2
921
§4.9. Competition and cooperation
If N(t) = {N1(t),Nz(t)} = {N1(t,cp),Nz(t,cp)} denotes the solution of (4.3.61) corresponding to
Nj(s,1!) == 'Pj(s), S E [-rj,O], j
= 1,2,
then
Nj{t,1!)
~
qj(t)
for
t
~
0, j
= 1,2.
4.3.69
Proof. Details of proof are entirely similar to those of Lemma 4.3.8 and therefore [J are omitted.
The next result is an analogue of Kamke's Theorem (see Coppel [1965]) for delay differential equations and has been established by several authors in many different forms (Mikhailova and Podgornov [1965J, Sandberg [1978], Ohta [1981J, Martin [1981] and Smith [1987]). Theorem 4.3.10. Let
in the following sense:
'PI(S) > 'ljJ1(S); S E [-rt,0Jj 'ljJl(O) >
°
'P2(S) > 'ljJ2(S); S E [-r2,0]; 'ljJ2(0) > O. Assume also 'PI, 'ljJ1 E C([-rl,O],R+) and 'PZ, 'l/Jz E C([-rz,O],R+). Then the solutions N(t,1!) = {N1(t, 1!), Nz(t,
Proof. Let
€
be any positive number and let N(f)(t, 1!) denote the solution of
4.3.70
322
§4.3. Competition and cooperation
vVe shall first show that
By hypothesis, N(£)(O, 4» > N(O, '11); that is
and therefore there exists a positive t2 such that
Suppose N(E)(t, 4» ~ N(t, '11) is not valid for t ~ 0; then there exists a t3 > t2 such that at least one of the following holds:
If the first of (4.3.71) holds, we can define t4 as follows:
We have
However,
and therefore 4.3.72 Furthermore, 4.3.73
929
§4.9. Competition and cooperation
From (4.3.72) and (4.3.73),
for sufficiently small positive 7J and this contradicts the definition of t 4 • Thus, Ni€)(t, q;) 2:: Nl (t, 'It) holds for all t 2:: and for arbitrary positive €. It will follow from Nl(t, q;) = lim Ni€)(t, q;), t 2::
°
°
€ ...... o
that Nl(t,q;) 2:: Nl(t,'I!) for t 2:: 0. IT the second of (4.3.71) holds, the proof is similar. Thus, the result follows. [] Corollary 4.3.11. If in Theorem 4.3.10 ,
then
Similary, if
then
Proof is immediate from that of Theorem 4.3.10.
[]
Lemma 4.3.12. Let q; = (/>I, <1>2) satisfy the assumptions of Lemma 4.3.8. Suppose there exist positive numbers Pi', pi such that Fl(p;,p;)
< 0,
pj > cPj(s),
F2 (p; ,pi) < S
°
E [-lj,O] , j = 1,2.
4.3.74
Then the following limits exist: lim Nj(t,p*) , j = 1,2,
t-+oo
where
p*
= (* Pl,P2*) ,
4.3.75
324
§4-3. Competition and cooperation
is the solution of (4.3.61) satisfying S
E
[-Tj,O].
Proof. By choice, Fl (pi, pi) < 0, F2(pi, pi) _< 0; if we choose pj( t) == pj for all t ~ -Tj , j = 1,2 then
'PI (t) = 0 > Fl (pr ,p;) = Fl (PI (t),P2(t - T2» P2(t) = 0 > F2(pr ,pn = F2(Pl (t - Tl) ,P2(t».
4.3.76
By Lemma 4.3.8, it follows 4.3.77 By the semigroup and order preserving properties of solutions of (4.3.61), 4.3.78
for all t, h ~ O. Thus Nl (t,p*) is non-increasing and bounded below; hence limt-+oo Nl(t,P*) exists. The existence of limt_oo N2(t,P*) follows by similar arguments. O. Lemma 4.3.13. Suppose there exist numbers qi, qi such that
qi > 0, qi > 0 and 4.3.79
Then the following limits exist: lim Nl (t, q*), t-+oo
j
= 1,2
where {N1(t,q*) , N2(t,q*)} = N(t,q*) is the solution of (4.3.61) satisfying S E [-Tj,
Proof is similar to that of Lemma 4.3.12.
0], j
= 1,2. []
The next result shows that the unique positive equilibrium of the hypercooperation model is a global attractor with respect to all positive solutions of (4.3.61).
925
§4.9. Competition and cooperation
Theorem 4.3.14. Let N(t,p) = {NI(t,p), N 2(t,p)} be tbe solution of (4.3.61) corresponding to tbe initial condition
wbere
PI(S) ~ 0, PI(O)
P2(S)
~
> 0, PI E C([-rI,O], R+) 0, P2(0) > 0, P2 E C([-r2,0] , R+).
Tben 4.3.80 P roof. Choose posi ti ve numbers PI , P2, ql , q2 such that
(Pl,P2) > (N;,N;); Fj(PI,P2) < 0; qj < pj(s) < Ph
< (N;,N;) Fj(qI,q2) > 0, j = 1,2 (ql,qZ)
4.3.81
s E [-rj,O], j = 1,2.
4.3.82
It is not difficult to verify that such a choice of qI, q2, PI, P2 is always is possible. We have from the above Lemmas and Corollary, that
{N1 (t,q),N2(t,q)}
~
{N I (t,p),N2 (t,p)} for
~
t~
By Lerruna 4.3.12, there exist positive numbers
{N1 (t,p),N2 (t,p)}
°
O"j ~
j
Nj, j
= 1,2
4.3.83
= 1,2 such that
4.3.84
and therefore by Lemma 1.2.3 of Barbalat (see Chapter 1)
4.3.85
showing (0"I,0"2) = (N;,N;) since (N;,N;) is the unique positive solution of (4.3.85). Similarly we conclude that lim Nj(t, q) = NJ;
t--oo
j = 1,2.
4.3.86
§4.9. Competition and cooperation
926
The conclusion (4.3.80) now follows from (4.3.83), (4.3.84), (4.3.85) and (4.3.86). The proof is complete. [] Dynamical systems modelling cooperation have been considered by Matano (1984] who has assumed that the flow generated by such systems is "eventually monotone" . A sufficient condition for the generation of such a flow has been obtained by Hirsch [1982, 1984} in terms of irreducibility of the Jacobian matrix of the vector field modelling the cooperative dynamics. A consequence of the result of Theorem 4.3.14 is that if the cooperative system (4.3.61) is stable without time delays, then a delay induced instability leading to a Hopf-type bifurcation to periodic solutions is not possible; in short, delay induced stability switching in cooperative systems is not possible if the time delays appear only in cooperative interactions. We wish to emphasize that our result on global attractivity of the equilibrium in the hypercooperation model is obtained with minimal hypotheses on the system compared with other relevant results in the literature. One of the reasons for the specific choice of the model has been to make the results more transparent for applications. The reader can examine the global attractivity of the positive equilibrium of each of the following models of cooperation:
dx(t) -;It
[Kl
= rlx(t) 1 + e-y(t)
dy(t) --;tt =
1
x(t) ;
i(t) =
[K2 1 r2y(t) 1 + e-x(t) -yet) ;
ii(t) =
-
J.' J.'
x(s)ds y(s)ds.
I
§4.9. Competition and cooperation
where ()
= 1,3,5, .. etc.
and
]{l
>
al
> 0,
I{2
d~~t) = TIX(t)[Kl(l _ d~~t) = T2y(t) [K2(1
>
a2
> 0,
927
T
E [0,00) , 8 E [1,00);
e-y(t-T») _ x(t)] }
I
_ e-X(t-T») - y(t)].
dYl(t) = -a Y (t) + b y~(t - Tm). dt I I I 1 + y~ (t - Tm) ,
yeO) > 0,
dYj(t) = -ajYj () n ( ). -;ut + bjYj-l t - Tj-l; J = 2,3"", m Yj(S) =
4.4. Method of Lyapunov functionals In this section we are concerned with Volterra integrodifferential equations of the form
dx.(t) (n n jt -;it = Xi(t) bi + f,; aijXj(t) + f,;b ii
-00
i
fij(t - s)xj(s)ds
4.4.1
= 1,2, ... , n
where bi, aij, bij (i,j = 1,2, ... , n) are real constants and iij : [0,00) continuous scalar functions normalised such that
1.
)
1--+
[0,00) are
00
/ij(s)ds = 1 ; i,j = 1,2, ... , n.
4.4.2
Worz-Busekros [1978] has derived a set of sufficient conditions for the global asymptotic stability of systems of the form (4.4.1) - (4.4.2) assuming that the delay kernels iij are convex combinations of the functions
im(t)
= (ma~ 1)!t m - l e- at
;
m
= 1,2, ...
; a E (0,00).
4.4.3
Systems like (4.4.1) - (4.4.3) can be reduced to a higher order system of autonomous ordinary differential equations due to the special nature of iij (see (4.4.42) below). In an elaborate discussion, Cushing [1977] has considered various aspects of local stability and bifurcation to oscillations in integrodifferential equations. In the
§4.4.
928
Lyapunov functionals
following we consider a more general class of integro-delay differential equations of the form
dx.(t) -it=
Xi(t)
+
(n n + ~aijXj(t) + ~bijXj(t bi
t
- Tij)
4.4.4
c;; l=k;;(t - s)x;(s)ds )
t>0; i
= 1,2, ... , nj
together with the following assumptions:
(AI) the delay kernels k ij (i, j = 1,2, ... , n) , kij : [0,00) on [0,00) and normalised such that
/.00 k
ij ( s)
ds = 1 ;
/.00 Ik
ij ( s)Ids
< 00 ;
~
( -00,00) are integrable
/.00 slk
ij ( s)Ids
< 00
4.4.5
i,j=1,2, ... ,n.
(Az) the real constants b i , aij, bij, Cij (i, j = 1,2, ... , n) are such that there exists a solution x* = (xi, xi, ... , x~) with xi > 0 (i = 1,2, ... , n) of the linear system n
L(aij+bij+Cij)xj+bi=O; i= 1,2, ... ,nj
4.4.6
j=1
the discrete delays Tij ~ 0 (i, j = 1,2, ... , n) are constants such that 0; i,j = 1,2, ... ,n. (A3)
the real parameters
bi, aij, bij , Cij
satisfy
laid +
[b;;[
a;; < 0; [a;;[ >
t t t +
[a;;[
/.=
[k;;(s)[ds
bijTij
1=
4.4.7
i = 1,2, ... ,no
We note that x* is unique by virtue of (4.4.5) and (4.4.7). Along with (4.4.5) (4.4.7), we consider initial conditions of the form Xi(S)='Pi(S)~O; SE(-oo,O); 'Pi(O»Oi SUpl'Pi(S)I
4.4.8
§4.4.
329
Lyapunov functiona13
where epi (i = 1,2, ... ,n) is continuous on (-00,0]. As a consequence of (4.4.8), it will follow that solutions of (4.4.4) can never become negative and hence we can rewrite (4.4.4) in the form
! {IOg[x;(t)fxil}
= a;;[x;(t) :-- xii +
t,
a;;[x;(t) - xjl
j'l'i
n
+ L bii[xi(t - Tii) - xi]
4.4.9
i=1
+
t
i=1
Cij
/.00 kij(s)(Xj(t -
s) - xi]ds
0
t > 0 ; i = 1,2, ... , n. The following result provides a set of sufficient conditions for the asymptotic stability (stability in the large or global attractivity) of X* (Gopalsamy [1984aJ). Theorem 4.4.1. Assume that the hypotheses (AI) - (A3) hold for (4.4.4). Then all solutions of (4.4.4) corresponding to the initial conditions in (4.4.8) satisfy
lim Xi(t) = xi; i = 1,2, ... ,n.
4.4.10
t-+oo
Proof. Consider the Lyapunov functional vet) = V(t,X1(.), ... ,x n (.» defined by
v(t) =
t.
[JIOg{X;(t)fxi}J + t.Jb;;J tr;; Jx;(s) - xjJds
+ t.Jc;;J {' Jk;;(s)J (t,Jx;(U) for
xjJdU) dS]
4.4.11
t ~ O.
It is easy to see from (4.4.11) and the type of initial conditions that
+
t
;=1
!cijl (sup lepj(s) - xii)
~ Vo
8=:;0
[00 Ikij(s)ls dS] io
< 00 for some positive number
Vo.
4.4.12
§4.4.
330
Lyapunov functionals
and n
vet) 2:
2:: Ilog{xj(t)jx:Jl·
4.4.13
i=l
Calculating the upper right derivative D+ v of v along the solutions of (4.4.4) and simplifying,
D+v(t) ::; -
t, [I +
a;; 1-
{ t.1aj;1 + t.1b;;1
t leji/ Jof=
j
,;:i
Ikji(s)1
dS}] I Xi(t) - xi I
J=l n
~
-62:: IXi(t) -
xii
4.4.14
i=l
where
o < fJ =
l~ifn [ 1a;; 1- t.1aj;l- t. (Ibjd + lejd [0 Ikj;(s )ldS) ]. j
?!i
It can be shown that (4.4.14) will imply (4.4.10); we leave the rest of the details of proof to the reader as an exercise (see Gopalsamy [1984a] for details). []
The next result provides a "mean-diagonal dominance" type sufficient condition for the convergence of all positive solutions of (4.4.4). Theorem 4.4.2. Suppose the hypotheses (AI) and (A2) hold for the system (4.4.4) and assume that in addition the following holds: (A4) aii < 0; i=1,2, .. ,n
(A5)
4.4.15
for i
= 1,2, ... , n.
Then all solutions of (4.4.4) and (4.4.8) satisfy (4.4.10).
§4.4.
331
Lyapunov functionals
Proof. Consider a Lyapunov functional v( t, x(.), ... , X n (.» defined by
v(t) =
t,[
(x;(t) - xi - xilog(x;(t)/Xi))
+ "2 ?= Ibijl 1
n
it
}=1
+~
t, f le;;1
for
(Xj(U) -:- xj)2du
t-r"
4.4.16
I)
Ik;;(s)1
(E.
(x;(u) - Xj)'dU) dS]
t > O.
Calculating the rate of change of v in (4.4.16) along solutions of (4.4.4) we have
d~~t)
=
~ n
[
a;;(x;(t) - xi)'
+ ~ a;; [x;(t) - xi] [x;(t) - xi] n
i't"i
n
+L
bij [Xj(t - Tij) - xj] [Xi(t) - xi]
j=l
t 1
00
+
Cij
j=l
+~
t
kiiCs) [Xj(t - s) - xj] [Xi(t) - xi] ds
0
Ibijl { (Xj(t) - xj)2 - (Xj(t - Tij) - xj)2 }
j=l
+~
t,
Ie;; I['" Ik;;( s)1 { (x;(t) - xj)' - (x;(t - s) - xj)' } dS]
: ; t, [
-la;;I(x;(t) - xi)'
t
laijl {(Xj(t) - xi)2
+ (Xj(t) - xi)2}
2
+~
t
Ibijl {(Xi(t) - xi)2
+ (Xj(t) - xj)2}
+~
j=l j't"i
j=l
+~ ~
t,
le;;I!.oo Ik;;(s)1 {(x;(t) - xi)'
+ (x;(t) - xj)'} dS]
n n[ laiil- 2I(n .I ~(Iaijl + lajd) + 2 ~(Ibijl + Ibjd)
-t;
j't"i
§4 ·4.
332
Lyapunov functional.3
n
2:)Xi(t) - xif
.:; -p,
where
4.4.18
j=1
In
o < p, = 1
~;~
-
1 (ICijllkij(S)1 +
1
n
Ibijl + Ibjd
]=1
00
+
{
-2 '~ "
ICjilkji(S))
4.4.19
dS}] .
Since p, > 0 by (4.4.15), op.e can show that (4.4.18) will imply (the reader should supply the additional arguments) the convergence in (4.4.10) and the proof is complete.
o
It is found from Theorems 4.4.1 and 4.4.2 that if the instantaneous (nondelayed) responses in (4.4.4) dominate (see for instance (4.4.7) or (4.4.15)) all delayed responses, then the positive steady state x* of (4.4.4) is a global attractor. One is now entitled to ask the following: if the system (4.4.4) is such that there are no instantaneous responses in the average growth rates, then one has a system of the form
dY~;t) = Yi(t) ( bi -
t.bij
i'=
t>0; i
kij(t -
S
)Yj(s) dS)
4.4.20
= 1,2, ... , n
(b i , Cij being real constants) where the linear system
4.4.21 for
i = 1, 2, ... , n
has a solution y* = (y;, ... ,y~) with yi > 0, i = 1,2, ... , nj under what conditions the steady state y* of (4.4.20) will be a global attractor with respect to nonnegative initial conditions? It should be noted that the system (4.4.20) does not contain delay independent stabilizing negative feedbacks as in the case of (4.4.4). We will derive an answer to the above question using the concept of positive definite kernels defined as follows:
§4.4-
999
Lyapunov functionals
Definition. Let K denote the n x n matrix of elements (K)ij = bij k ij where bij (i,j = 1,2, ... , n) are real constants and k ij : [0,(0) 1-+ (-00,00) are such tbat
1.= kij(s)ds = 1 ;
1.=
Ikij(s)lds < i
00;
1.T(kij(S))2dS <
00
4.4.22
= 1,2, ... ,n
for each positive T. Tbe matrix kernel K is said to be positive definite if and only if for every i = (h, h,·.·, in), Ii : [0,00) 1-+ (-00,00), Ii E e(o, T] , T> 0, tbere exists a positive constant J.l such that
r;,1. n
T
fi(t)
(t 1. f,;.bijkij(s)!;(t-S)ds n
21' {
{tfM}
)
dt 4.4.23
dt
for each finite positive number T. The following result is not new to the existing literature.
Theorem 4.4.3. Assume tbat the delay kernel K = (k ij ] in (4.4.20) is positive definite satisfying (4.4.23) and
1
00
slkij(s)lds < 00; i,j
= 1,2, ... ,n.
4.4.24
Let y* = (y;, yz, . .. , y~) , yj > O,j = 1,2, ... , n be a positive steady state of (4.4.20). Let (4.4.20) be supplemented witb bounded continuous initial conditions of the form
Yj(s)
=
1-+
[0,(0) ;
> 0;
sup
< 00.
4.4.25
s$O
Then every solution of (4.4.20) and (4.4.25) exists on [0,(0) and satisfies the attractivity condition lim Yi(t) = Y;; i = 1,2, ... ,n.
t-oo
4.4.26
Proof. One can show that solutions of (4.4.20) and (4.4.25) exist locally on an interval of the form [0, t*) for some (possibly) small positive t* and the global
§4.4.
334
Lyapunov functionals
existence (for all t 2:: 0) of solutions will follow from our arguments below. Since solutions of (4.4.20) and (4.4.25) remain positive so long as they exist, we can let 4.4.27 in (4.4.20) and derive that for i'= 1,2, ... ,n
where
hi(t) = -
t /.00 bij
j=l
k ij ( s)[Yj(t - s) - yi] ds.
4.4.29
{t,lh;(t)l} dt < 00
4.4.30
t
We have from (4.4.29) that
implying
t
t
II h(t) II dt =
10
00
slkij(s)lds < 00, i,j = 1,2, ... ,no We consider a Lyapunov function v(t) == v(t, Yl(t), ... , Yn(t)) defined by
due to
v(t) =
n
Lyi
l
i=l
Y ;(t)
[e U
-
1] du ; t 2:: O.
4.4.31
0
Calculating the rate of change of v along the solutions of (4.4.28),
d~~t)
= _
t,
yi {ey,lt)
-t. s: - t
-I}
I}) 1.'
1.'
k;j(s) (e''j(t-·)
-1) ds
-t
4.4.32
b;jyj [e1';('-') -
1] k;j(s)ds
0 j=l
i=l
+
b;jyj
-I} h;(t)]
yi {ey,(t)
(Yi {ey,(t) -
[t.
{t. Yil -II} eY '(')
{t,lh;(t)l} .
4.4.33
§4.4.
335
Lyapunov functionaZ3
Integrating both sides of (4.4.33) with respect to t and using (4.4.33),
vet) - v(O) ::;
-p
1.' (t. [yi( + 1.' II /I [t. Yil
eY;(') -1 ) r)dS
h(s)
4.4.34 eY;(') -
11] ds.
We note that
for some positive constant M > 0 and derive from (4.4.34),
vet) ::; -p
1.' {t. (yi[ + 1.' /I /I + 1.' /I Y
e ;(') -1
M
::; v(O)
h(s)
M
J
y}
ds
v(s)ds + v(O)
4.4.36
/I v(s)ds
4.4.37
h(s)
which by Gronwall's inequality and (4.4.30) implies vet) ~ v* for some constant v*
>0
4.4.38
showing that vet) is bounded uniformly in t for t~ O. By continuation, it will follow that solutions of (4.4.28) exist for all t ~ 0 since we have from (4.4.31), n
I::Yi
[eY;(t) -
Yi(t)
-1] ~ v*
4.4.39
i=l
and v* is independent of t. We have from (4.4.36),
t.
yi[ eY;(t)
-
Y;(t) - 1 J + JL
1.' {t,(y;(S) - vi)' }
ds
4.4.40
~ v(O) + Mv* ].00 II h(s) II ds ~ N < 00. From the boundedness of E~=l IYi(t)1 on [0,00) and the hypotheses, the boundedness of for t > 0 and i = 1,2,3, ... , n (see (4.4.28) and note that
%
§4.4.
996
2:7:11hi(t)!
~ 0 as
t
Lyapunov functionals
~ 00). It will then follow that
2::=1 lY;(t)1
is uniformly
continuous on [0,00). Thus, we have from (4.4.40) that (i)
(ii)
yn 2:7=1 {Yi(t) - yn
2:7=1 {Yi(t) -
2
is uniformly continuous on [0,00).
2
E LI[O, 00). which together imply 2:~1 !Yi(t) - yil ~ 0 as t ~ 00 and this completes the
0
~~
An immediate question now is the following: are there verifiable sufficient conditions for a matrix kernel K = [kij] to be positive definite? The answer is yes and a result for this purpose is formulated below whose proof is similar to that of the scalar case treated in Chapter 1 and is left to the reader as an exercise. Proposition 4.4.4. Let K = [kijl ,i,j = 1,2, ... , n be such that (4.4.22) holds. Then K is positive definite satisfying' (4.4.23) if the matrix k where
K = K(i7J) = ?Re(Kij(i7J» =?Re
1.
00
kij(t)e il1t dt ; T/ E (-00,00),
4.4.41
is a positive definite matrix whose eigenvalues are bounded below by a positive constant J.l. Let us consider briefly a class of integrodifferential equations with a special type of delay kernels:
dx.(t) (n n jt T = Xi(t) b + ~ aijXj(t) + ~ ,Bija i
-00
e-a(t-s)xj(s)ds
i = 1,2, ... ,n ; t
) 4.4.42
>0
where bi , aij, ,Bij (i = 1,2, ... ,n) are real constants and a is a positive constant. The linear "chain trick" introduced by Fargue [1973] and used by Worz-Busekros [1978], MacDonald [1978], Post and Travis [1982] for analysing (4.4.42) is as follows: define a new set of variables xn+i,j = 1,2, ... , n so that xn+j(t)
="
1'=
>0
4.4.43
j=1,2, ... ,n.
4.4.44
e-a('-')Xj(s)ds ; t
and immediately derive
§4 ·4·
337
Lyapunov /unctiona13
Thus, the system (4.4.42) of n-integrodifferential equations becomes a system of 2n autonomous ordinary differential equations i = 1,2,"', nj
4.4.45 j = 1,2, .. . ,n.
If x* = (xi,xi, ... ,x:),xt > 0, i = 1,2, ... ,n is a solution of n
L(aij
+ f3ij)xj + b
j
= OJ i = 1,2, ... ,n;
4.4.46
j=l
* * ... ,xn,xn+l, * * *) ,xn+i * *. = 1, 2 then, ( xl,x2, ... ,x2n = Xj') , ... ,n'IS a componen-t wise positive steady state of (4.4.45). Asymptotic stability of (xi, ... , xin) for the system (4.4.45) is equivalent to that of (xi, ... , x:) for (4.4.42). We formulate our next result in terms of M-matrices; for properties of M-matrices we refer to Chapter 3 (or Araki and Kondo [1972], Plemmons [1977]). The following result concerned with the stability of the system (4.4.42) is due to Post and Travis [1982].
Theorem 4.4.5. Corresponding to the system (4.4.45) define a 2n x 2n matrix B as follows:
B12] B 22
;
Bij (.. Z,)
( B ll ) ..
')
(B 12 )ij=-If3ijlj
=,1 2) are n x
= {la ii1i
-Iaiil;
. n rna t rIces.
i =j i=fj
i=1,2, ... ,nj j=n+1,n+2, ... ,2n
(B21 )ij = diag( -a) (B22)ij
= diag(a).
The positive steady state x* = (xi, ... , x:) of (4.4.42) is globally asymptotically stable if B is an M-matrix and aji < OJ i = 1,2,3, ... , n. Proof. Consider a Lyapunov function
§4.4.
338
Lyapunov junctiona13
defined by 4.4.47 where d 1, d2, . .. ,d2n are positive constants to be chosen suitably. Calculating the derivative of V along the solutions of (4.4.45) and simplifying one can verify that (see Post and Travis [1982]), 4.4.48 where
x-
X* = col.{(Xl - xi), (X2 - x;), ... , (X2n - x;n)}
D = diag.{dI, d 2 , ••• , d2n }.
Since by assumption B is an M-matrix, there exists a positive diagonal matrix D such that DB + BT D is positive definite and hence we have from (4.4.48) that, dd~ calculated along the solutions of (4.4.45), is negative definite from which the [J result will follow. If we let (3ij = 0, i,j = 1,2, ... , nand Q = 0 in (4.4.45), then (4.4.45) will simplify to a system of n ordinary differential equations
i=1,2, ... ,n
4.4.49
yilog(yi/yi)}
4.4.50
for which a Lyapunov function of the form n
V(Y1, Y2,···, Yn) =
I: di{Yi i=l
where Y; > 0 and 2:'}=1 aijYj = .Ai, i = 1,2, ... , n has been used by numerous authors (see Harrison [1979] for a narration). A calculation of ~~ in (4.4.50) along the solutions of (4.4.49) leads to 4.4.51 4.4.52
§4.4. in which
339
Lyapunov functionals
= diag( d 1 , d 2 , ••• , d n ) ; A = {aij} y* = col{(y - y;), (Y2 - y;), ... , (Yn D
y-
y~)}.
It will follow from (4.4.52) that a sufficient condition for the global asymptotic stability of y* = {Yi, ... , Y:} for (4.4.49) is that there exists a diagonal matrix D = diag(dll ... ,dn ) such "that DA + ATD is positive definite. It has been noted by Krikorian [1979] that the algebraic problem of finding necessary and sufficient conditions, for the existence of a positive diagonal matrix D such that DA + ATD is negative definite for a given square matrix A, remains unsolved (see also Barker, Berman and Plemmons [1978]). Furthermore, the negative definiteness of (D A + AT D) demands that all the diagonal elements aii (i = 1,2, ... , n) of A be negative (a condition which we have extensively used); if possible such a requirement is worth relaxing. In many cases, it is not difficult to find a positive diagonal matrix D so that (DA + ATD) is positive semi-definite; in such a case although Lyapunov's stability theorem is not applicable, the following extension (see LaSalle and Lefschetz [1961], Barbashin [1970]) of Lyapunov's stability theorem can be used: "if ~~ in (4.4.52) is negative semi-definite, then every solution of (4.4.49) approaches as t ~ 00, the largest invariant subset of the set of points in Rn for which ~~ = 0". For instance, consider the example of Krikorian [1979]; dx}
dt
= Xl('x1 - allxl - a12 x 2)
dX2
dt = X2( -'x2 + a2l x l dX3
dt
= X3( -'x3
a23 x 3)
4.4.53
+ a32 x 2).
Consider a Lyapunov function v = vex!, X2, X3) for (4.4.53) defined by 3
V(Xl,X2,X3) =
?= ai [Xi - xi - xil09(Xi/x n]
4.4.54
.=1
where al, a2, a3 are positive constants to be selected suitably. Computing ~~ for (4.4.54) along the solutions of (4.4.53) we have
~~
= -alall(xl - x;)2
+ (a3 a32 -
+ (a2a21
- ala12)(x2 - X;)(X3 - x;)
a2 a 23)(X2 - X;)(X3 - xi)·
Suppose we choose aI, a2, a3 such that
§4.4.
Lyapunov functionals
4.4.55 Now ~~ = 0 only when Xl = xi. Let us look for invariant (with respect to (4.4.53» sets of the form
E = {(x}, X2, x3)lxI = x~ , xz'> 0 , X3 > o}. If E is invariant with respect to (4.4.53), then we have the following implications: Xl
= x~
= 0 =::} ).1 -
=::}
Xl
=::}
Al -
all xi
=::}
-A2
+ a21 x I -
=::}
X3 = x;.
anXI - al2x2
=0 = x; =::} X2 = 0
= 0 =::} Xz a23 x 3 = 0 =::} -A2 + a2lxi -
- alZ x 2
a23x3
=0
Thus, the only invariant subset of (4.4.53) is the point (xi, xi, xi) which is a positive steady state of (4.4.53) whose existence is assumed. We can conclude by LaSalle's extension of Lyapunov's stability theorem that (xi, xi, xi) of (4.4.53) is globally asymptotically stable. Other examples solvable by this technique are listed in the exercises. 4.5. Oscillations in Lotka - Volterra systems In competitive and cooperative model systems with no time delays, solutions can converge to equilibria monotonically with time; our discussion in Sec. 4.2 illustrates this phenomenon. The introduction of time delays in model equations, has been to produce certain observed fluctuations in the population densities both in controlled and field environments; furthermore, time delays are natural in many population systems due to maturation processes among many others. It is in this spirit one is interested to examine whether or not delay induced oscillations exist in model systems. Also a knowledge of fluctuations in population densities can prove useful in devising appropriate feedback control strategies. The results of this section are from Gopalsamy [1991]. We discuss the oscillation of solutions about the equilibria of delay differential equations of the type
dx~~t) = Xi(t)[b i -
taijXj(t-rjj)],i = 1,2, .. ,n; J=l
bi,aij E (0,00), i,j = 1,2, ... ,n.
4.5.1
§4.5. Oscillations in Lotka . Volterra systems
341
We have seen in Chapters 1 and 2 that time delays have a tendency to produce oscillations in otherwise nonoscillatory systems. A familiar example of this aspect is provided by the scalar equation with a single delay
duet) dt
= u(t)[b - au(t - r)]
4.5.2
°
which is nonoscillatory if r = where a, b are positive constants and is oscillatory about its positive steady state if (ber) > 1. Usually together with (4.5.1) we consider initial conditions of the form Xi(S)=
sE[-r,O]
with
r=max{rii; i = 1,2, ... ,n}
i = 1,2, ... ,nj
4.5.3
where
°
°
Definition. An IRn-valued function u(.) = {UI(')"'" u n(.)} defined on a balfline [to, (0) is said to be oscillatory if at least one component of U is oscillatory; a vector U : [to, (0) ~ IRn is said to be nonoscillatory if every component of u is nonoscillatory. Definition. The system (4.5.1) is said to be oscillatory about its steady state x* = {xi, xi, ... , x~}, xj > 0, j = 1,2,3, ... , n if every solution x = {Xl, X2, ••• , x n } of (4.5.1) corresponding to (4.5.3) has at least one component, sucb tbat [Xj(.) - xj] is oscillatory on [0,(0) for some j E {1,2,3, ... ,n}. Tbe system (4.5.1) is said to
342
§4.5. Oscillations in Lotka - Volterra systems
= {xi, ... , x~}
be nonoscillatozy about its steady state x*
(4.5.1) has at least
if
one solution corresponding to (4.5.3) such that the vector x(.) - x* = {Xl(.) -
xi , X2(') -
x;, ... , x n (.)
x~}
-
is nonoscillatory on [0, 00 ). We remark that the above definitions constitute one of several possible directions of generalizing the concept of oscillatory and non-oscillatory scalar systems to the case of finite dimensional vector systems.
=
Let us now consider (4.5.1) by relaxing the requirement aij ~ 0, bi > 0, (i,j 1,2, ... ,n) and examine under what conditions all positive solutions of (4.5.1) will be oscillatory about a positive equilibrium.
Theorem 4.5.1. Suppose the parameters of (4.5.1) satisfy the following: bi, aij (i,j = 1,2, ... , n) are real constants such that aii > 0, i = 1,2, ... , nand the system (4.5.1) has a componentwise positive steady state . and 1 pro> -
where
e
p
4.5.4 4.5.5
= l~~n (xi [a ii -
-
tI
aji
j=l j 1'0
I]).
Then every nontrivial nonconstant solution of (4.5.1) and (4.5.3) existing on = {xi,x 2, ... ,x~}.
[-r,oo) is oscillatory about the steady state x*
Proof. First we show that every nontrivial and nonoscillatory solution of (4.5.1) and (4.5.3) converges as t -+ 00 to the positive steady state x*. For instance, suppose x(t) = {Xl (t), X2(t), . .. ,xn(t)} is a nonoscillatory (about x*) solution of (4.5.1) and (4.5.3) on [-r,oo). As a consequence there exists a t1 > 0 such that
Xi(t) -xi
=f. 0
for
t ~ tl ; i
= 1,2,; .. ,no
4.5.6
We can rewrite (4.5.1) in the form
d dt Ui(t)
n
=-
Laid xj(t-rjj) - xj] j=l
t
> 0 ; i = 1,2, ... , n
4.5.7
§4.5. Oscillations in Lotka - Volterra systems in which
Ui(t)
= log[xi(t)/x:J
; t > O.
We have from (4.5.6) and (4.5.7) that :t I Ui(t) I $ -a .. I Xi(t - Tii) -
xi I +
t,1
aij
IIXj(t -
Tjj) - xj
I
i~i
4.5.8
> tl + T
t
and hence
~ {t, IUi(t)l} $ - t, [aidxi(t - TiO) - xil-
t, j
$ -
t>
t [(a t .
T .. ) -
XiI}.5.9
+T
tl
lajd) IXi(t -
ii -
1=1
lajdlxi(t -
7f:i
TiO) -
xi I].
4.5.10
J=1
i#i
It follows from (4.5.5) that P=
.':2ifn [a .. - t,1 aji I] > 0
4.5.11
i#i and therefore d (
dt
t; IUj(t) I n
)
::; -p
t; IXi(t - Tii) - xi I· n
An integration of both sides of (4.5.12) over [t2
+ T, t]
4.5.12
leads to 4.5.13
One can conclude from (4.5.13) that E?=llui(t)1 is bounded on [0,(0) and hence the derivative of this sum is also uniformly bounded. From these it will follow as
t
-+
00;
i=1,2, ... ,n.
4.5.14
To proceed further we now rewrite the system (4.5.7) in the form d
-d Ui(t) t
=-
n
I: aiixj[exp{Uj(t-Tjj)} -1] . )=1
i = 1,2,3, ... , n ; t > t3
4.5.15
§4.5. Oscillations in Lotka - Volterra systems
344
and show the existence of
~jj = ~jj(t)
on [t3
exp{Uj(t - 1'jj)} - 1 =Uj(t -
+ 1', 00), i,j
= 1,2, ... , n such that
1'jj)exp{uj(~jj(t))}
4.5.16
i,j = 1,2, ... ,n ; t > t3 +1'.
Let t, t 1 be such that
We note 4.5.17
i,j=1,2, ... ,n
where uj(B jj ) lies between Uj(t - 1'jj) and Uj(tI). Considering the limiting case of (4.5.17) as tl -1- 00, we derive
exp{Uj(t - 1'jj} -1 = Uj(t i,j
1'jj)exp{uj(~jj(t))}
for some ~jj(.)on [ta +1',00) such that ~jj(t) -1- 00 monotonically as t 1,2, ... ,no Using (4.5.18) we rewrite the system (4.5.15) in the form
d
dt Ui(t)
4.5.18
= 1,2, ... ,n ; t > t3 +1' -1-
00 ;
i,j =
n
=-
I.: aijxjUj(t - 1'jj)exp{uj(~jj(t))} j=l
= -aiixi(t -
1'ii)exp{ui(~ii(t))}
n
- L aijxjUj(t - 1'jj )exp{ Uj(~jj(t))}
4.5.19
j=l j¢i
for
i = 1,2,3, ... ,n ; t
> ta + r.
As a consequence of the facts Ui(t) -1- 0 and ejj(t) -1- 00 as t -+ 00 (i,j == 1,2, ... ,n), it follows that there exists a t4 2: ta + r such that (4.5.19) leads to
n
+
L laijlxjluj(t -
rjj)lexp{luj(~jj(t4))1}
j=l j¢i
t > t4 ; i == 1, 2, ... , n
4.5.20
§4.5. Oscillations in Latka - Volterra systems with the implication
+
t j
lajilxilui(t - Tii)leIU;«;;(t. m]
4.5 ..21
'Fi
We can simplify (4.5.21) to obtain
-t
lai ;le/ ui ({ii(t 4 »/]
xii Ui(t -
'Tii) I
4.5.22
}=l
For convenience let us set 4.5.23
From (4.5.22), 4.5.24 Using the facts Ui(t) -7 0 (since Xi(t) sides of (4.5.24) on (t, 00),
xi) as t
-7
-7
00,
and integratiing of both
4.5.25 which will lead to Wet)
~
0:
where
1.:. n
Wet) =
W(s) ds
I: IUi(t)l· i=l
4.5.26
§4.5. Oscillations in Lotka - Volterra systems We let
F(t) =
fr
1:.
4.5.27
W( s) d.s
and derive
F(t)
= -aW(t -
To)
< -aF(t - To).
4.5.28
It follows that the scalar delay differential inequality (4.5.28) has an eventually positive solution. Since pTo > ~, it is possible to choose t4 large enough so that aTo
1 e
>-
(from)
pTo
1
> -.
4.5.29
e
It is well known that when (4.5.29) holds, (4.5.28) cannot have an eventually positive solution. This contradiction proves the assertion. [] We remark that (4.5.5) provides a sufficient condition for the oscillatory (not necessarily periodic) coexistence of the n-species Lotka-Volterra system (4.5.1). The nonoscillation of competition systems of the type
(L: n
dXi(t) r-I dt- -- x-(t) Z
) • a-Z]-x ] ·(t - 1"-) I]'
i = 1,2, ... ,n
i=l
has not been considered except when
Tij
=
l'
(see Gopalsamy et al. [1990a]).
4.6. Why positive steady states? An n-species population system modelled by coupled integrodifferential equations of the form dx -(t) (n n jt -it = Xi(t) Ai + [; aijXj(t) + [; bij
-00
kij(t - s)xj(s)ds
) 4.6.1
t> 0 ; i = 1,2, ... , n is said to be capable of equilibrium coexistence if and only if each solution x(t) = {Xl(t)"",xn(t)} of (4.6.1) with
Xi(O) > 0 ; Xi(S)
=
sUPs::;o
= 1,2, ... ,
n}
4.6.
2
§4.6. Why p03itive 3teady states?
is defined for all t 2: 0 and is such that
where x* = {xr, ... , x~} is a componentwise positive vector of IRn; the system (4.6.1) - (4.6.2) is said to be capable of none~uilibrium (limit cycle type) coexistence if (4.6.1) has a strictly positive (componentwise) periodic solution and such a periodic solution attracts all other solutions of (4.6.1) - (4.6.2). This concept of coexistence is borrowed from the dynamics of two species systems described by a pair of autonomous ordinary differential equations; in the case of such two species dynamics, the Poincare-Bendixson theorem implies that a solution path which does not leave a bounded set in the nonnegative quadrant of R2 must approach as t ~ 00, either a steady state or a limit cycle which contains a steady state in its interior. Thus, the equilibrium or nonequilibrium coexistence of a two species system requires, by necessity, the existence of a steady state in the interior of the positive quadrant. The concept of equilibrium coexistence is basically a convergence criterion and requires a somewhat severe mathematical requirement of ecological model systems. Another concept of practical significance is that of "persistence" (also known as permanence) which is more general and weaker than that of coexistence. It is known (see for instance Hofbauer and Sigmund [1988]) that for systems governed by ordinary differential equations, the existence of a positive steady state is necessary for persistence; the author believes that the same is true for systems governed by difference, delay and integrodifferential equations. The following is a definition of "persistence" of an n-speCies system described by (4.6.1). Let C+ and L(C+) respectively denote the sets
C+
=
{
x( t) is a solution x(t) = (Xl(t), X2(t), ... , xn(t» E Rn
}
of (4.6.1) and (4.6.2) p = lim m _ oo x(t m ) for some sequence} L(C+) = p ERn; {t m } ~ 00 as m ~ 00 and x(t) is a { solution of (4.6.1) - (4.6.2). The system (4.6.1) is said to be persistent if L(C+) is contained in a bounded open set in the positive orthant of Rn. One can show that when solutions of (4.6.1) (4.6.2) are bounded on [0,00), L( C+) is not empty and
dist{x(t),L(C+)} ~ 0 as t ~ 00.
§4.6. Why positive steady states? Thus, it will follow that if ( 4.6.1) is a persistent system, then no species can become extinct eventually as t -+ 00 and the population densities are eventually bounded below by positive numbers. Nonpersistence will then mean that at least some points of L(C+) C Rn wi1llie on the co-ordinate planes bounding the positive cone 9f Rn (note that solutions of (4.6.1) - (4.6.2) cannot become negative). There are various different but related definitions and concepts of persistence in the literature. For convenience, let us assume that each of the delay kernels kij appearing in (4.6.1) does not change sign on [0,00). Since the signs of the constants bij(i,j = 1,2, ... , n) will not be restricted, we can in fact assume that k ij in (4.6.1) are nonnegative on [0, 00). The following result (see also Noonberg [1971 J) provides a necessary condition for the persistence of the integrodifferential system (4.6.1). Theorem 4.6.1. Assume the following for the system (4.6.1); (a) kij are nonnegative on [0,00) and integrable such that
0:::;
].= kij(s)ds <
00; i,j
= 1,2, ... ,n.
(b) Ai, aij, bij (i,j = 1,2, ... , n) are real constants such that the system of linear equations i = 1,2, ... , n
in the unknowns
Qi,(3)i) Q'1
4.6.3
(i,j = 1,2, .. . ,n) bas no solution oftbe type
>0
,
>0
f.l\i) _ JJ)
(i,j = 1,2, ... ,n).
( c) The set C+ is contained in a bounded open cube of tbe form
S = {(Xl,X2, ... ,Xn) E IR n
10 < Xi < Zj
, i = 1,2, ... ,n}.
Then tbere exists no constant r > 0 sucb that Xi(t + r) = Xi(t) for any t > 0, i == 1,2, ... , n and the set L( C+) bas at least one point on the boundazy of S with at least one of whose components equal to zero. Proof. Suppose there exist positive constants tt, r such that
Xi(tl +r) = xi(td; i = 1,2, ... ,n.
4.6.4
§4.6. Why positive steady states P Since solutions of (4.6.1) satisfy
Xi(t 1
+ r) = Xi(t 1 )exp(1"+T {Ai + tl
+ tb: j ;=1
1""
t
aijXj(s)
j=l
kij(u)x;(s - U)dU}
dS)
4.6.5
0
i
= 1,2, ... ,n,
it follows from (4.6.4) that
i
= 1,2,3, ... , n . 4.6.6
It is found from (4.6.6) that (4.6.3) has a solution
i,j=1,2, ... ,n
4.6.7
and this contradicts our hypothesis of nonexistence of solutions of (4.6.3). Thus, (4.6.4) is not possible for any tl > 0 and r > O. To prove the nonpersistence of (4.6.1), let us suppose otherwise; that is let
L(C+) be contained in a positive open cube of Rn. Since L(C+) is not empty for C+ C S (how?), there exists a sequence say {sm} -7 00 as m -7 00 such that
Let min (Pl,P2, ... ,Pn) = p* > 0 and choose an increasing subsequence of iSm} say {Zm} -7 00 as m -7 00 such that
Zm - Zm-l > 1 ; m
= 1,2,3, ...
4.6.9
n
{Xl(Zm), ... , Xn(Zm)} E I)Xi - Pi)2 = (P* /2)2. j=l
4.6.10
§4.6. Why positive steady states ?
350 It follows from (4.6.10),
4.6.11 Now using the relation
+
I:, b J.= k ij
U
)dU] dS)
4.6.12
0
J=l
i
ij ( u)x j( S -
= 1,2, ... , n
; m = 1,2,3, ...
we compute
t
1Xi(Zm) -
Xi(Zm-,)
+
I' =
t[
X;(zm_,) J' { exp (
L~, [Ai +
t.
a;jxj(s)
I:, b;j J.= k;j(u)xj(s - u)du] dS) - 1}2 J=l
2:: (p* /2)2
4.6.13
0
I:n [exp{n Ai + I:aij i=l
+I:,bij j=l
j=l
{Zm
_1 Zm
Zm-1
jzm
1 Z
_ Z m
_ m
1
xj(s)ds
4.6.14
zm-l
({OOk jj (U)Xj(S_U)dU)dS}_l]2
JZ 1 Jo m _
where the last inequality is obtained using the fact for any real y and (3 > 1. We note that since C+ is contained in S, we have in (4.6.14),
1 Zm -
Zm-1
l
zm
Zm-l
Xj(s)
= Xj(Zm-1 + Bj(zm -
zm-I))
m = 1,2,3, ...
4.6.15
951
§4.6. Why p08itive 8teady 8tate8 ?
for some Bj, 0 < Bj < 1 and since Xj(zm) ~ Pj > 0 (j = 1,2, ... , n) as m - t 00, it follows that the left side of (4.6.15) is bounded away from zero for all j = 1,2, ... ,n and is independent of mj from our hypotheses (a )-( c) we have
0::;
1 Zm - Zm-1
l
zm
((OOkij(U)Xj(S-U)dU)dS
J~
Zm-l
Thus, we have from the assumed persistence hypothesis of our proof that
~
lzm zm
n 1 exp Ai + ~ aij Zm _ Zm-l
n [{
+
t
i=1
bij
Zm
_1Zm-l l
Zm-l
Zm-l
xj(s)ds
(fOO kij(U)Xj(s _ U)dU) dS} _
Jo
4.6.16
1] > 0 2
whose left side has a positive minimum, since (4.6.3) has no solutions with (ti > 0, /3;i) ~ 0, i,j = 1,2, ... , n. The positive minimum of the left side of (4.6.16) is also independent of m. But this implies that (4.6.8) is not possible. This contradiction shows that all points of L( C+) cannot lie in the interior of the positive cone of R n • The proof is complete. [] We shall consider an example to demonstrate the relevance of the existence of a positive steady state to the persistence of a prey-predator system with time delays modelled by Lotka - Volterra equations; (for the analysis of a number of models without time delays, we refer to the monograph by Hofbauer and Sigmund [1988] and Kirlinger[1986] as well as the literature cited therein). Consider the prey predator system,
4.6.17
in which ri, aij (i,j = 1,2) are positive numbers and Tij (i,j = 1,2) are nonnegative numbers. Positive solutions of (4.6.17) remain bounded since
We have from (4.6.17),
§4.6. 'Why positive steady states ?
352
where Ml is an upper bound of the solution Xl (t) on [-0"1,00), 0"1 = max( T11, T2t). We leave these details to the reader. It is not difficult to verify that if 4.6.18
then the system (4.6.17) has a positive steady state (xi, x;) where
If (4.6.18) is satisfied, then one can show that for all the Tij = 0, the prey predator system (4.6.17) has equilibrium coexistence and this can be established by considering the Lyapunov function V where
V(Xl,X') =
Cl [Xl - x; - X; log
Gi)] +
c,
[x, - x; - x; log Gi)]
for suitably chosen positive numbers Cl and C2. If at least one of the delays in (4.6.17) is not zero, then it is of interest to ask whether the prey predator system can persist. Proposition 4.6.2. If (4.6.18) holds, then the prey - predator system (4.6.17) is persistent; if (4.6.18) does not hold, then the system (4.6.17) is not persistent. Proof. Details of proof are based on persistence functionals and can be found in the articles of Wendi and Zhien [1991] and Jiong [1991]. We shall be brief here. Define a functional PI as follows:
PI (Xl, X,)( i) =
[Xl (i)l"" [x,( i)l"U
-a 21 a 12
exp [ -
jt jt
all a21
.
X2(s)ds+a 21 a ll
t- r 12
-a 11 a22
J.~r
Xl (s)
jt
ds xl(s)ds
t- T 21
X2(S)dS];
i>O.
4.6.19
t - r 22
By direct verification it is found from (4.6.17) that
dPl(i) ~
= PI (t)[(rl a 21 -
r2 a ll) - (a12 a21
+ alla22)x2(i)].
4.6.20
If x2(i n ) -+ 0 as in -+ T ~ 00, then PI (in) -+ 0 as tn -+ T ~ 00 : but dPd~tn) > 0 for all large n. But this is not possible and hence X2 cannot approach zero along any
353
§4.6. Why positive steady states f
sequence tn - t 00. The proof of the persistence of the Xl is similar by considering the persistence of X2 and functional P2 where
P2(X}, X2)(t)
= [Xl (t)Ja
22
[X2(t)]-a 12 exp [- a 22 a ll
jt
Xl (s) ds
t-rll
It can be found that
and as before using that X2 persists, it is possible to conclude that Xl persists. Now if we suppose that (4.6.18) fails to hold, then it will follow from (4.6.20) that
and this will imply at least one of the two species will eventually become extinct in the sense of failing to persist. (] It is left to the reader to investigate, whether or not the existence of a positive steady state is necessary, for the persistence of the following population systems most of which, in the absence of time delays have been discussed by Kirlinger [1986J (assume that all the parameters are positive numbers):
§4.6. Why positive stea.dy states ?
354
dXl (t)
~
= xl(t)[rl-- allxl(t -
r) - alzxz(t - r) - QIYl(t - r)
]
dxz(t)
~ = x2(t)[rz - a21xl(t - r) - aZ2xZ(t - r) - bZY2(t - r)]
~ = Yl(t)[ - Cl
dYl (t)
+ d1Xl(t -
r)]
dY2(t)
+ dZ X2(t -
r)].
~ = Y2(t) [ - Cz
dXl (t)
~ = Xl(t) [rl - anXl(t - r) - a12x2(t - r) - b1Yl(t - r)]
dX2(t).
= X2(t) [r2 -
a21xl(t - r) - a2Zx2(t - r) - a23x3(t - r) - bZY2(t - r)
dX3(t)
= x3(t)[r3 -
a32xZ(t - r) - a33x3(t - r) - b3Y3(t - r)]
~
~
dYl(t)
~ = Yl(t)[ - Cl
+ d1Xl(t -
r)]
~
dyz(t)
= yz(t)[ -
Cz
+ dZX2(t -
r)]
dY3(t)
= Y3(t)[ -
C3
+ d3X3(t -
r)].
~
dXl (t)
.
~ = xl(t)[rl - allxl(t - r) - a12X2(t - r) - b1Yl(t - r)
]
]
dX2(t)
~ = xz(t)[r2 - a21xl(t - r) - aZ2x2(t - r) - b2Y2(t - r)] ~
dYl (t)
= Yl(t)[ -
Cl
+ d1Xl(t -
r) - elY2(t - r)]
dY2(t)
= Y2(t) [ -
Cz
+ d2X2(t -
r) - eZYl(t - r)].
~
Investigate also the above systems, with terms such as x(t - r) replaced by terms of the type
N([tJ),
sup sE[t-r,t]
N(s),
and
tT
N(s)ds.
355 4.7. Dynamics in compartments
Compartmental systems are widely used in modelling the dynamic behavior found in chemical reactions and ecological dynamics wherever exchange of material or biomass among compartments takes place. A "compartment" is characterised by the type and amount of material it contains and it is a hypothetical container or pool. In ecology a compartment can be associated with the biomass of certain species, say for instance, the zooplankton population in a lake. Excellent introductions to the subject of compartmental dynamics can be found in the extensive literature (Atkins [1969), Jacquez [1972J, Maeda et al. [1978], Sandberg [1978], Lewis and Anderson [1980a, b) and the literature cited therein). For the relevance of compartmental systems to ecological models, we refer to Mulholland and Keener [1974] and Mazanov [1976]. The dynamics of a simple n-compartment system is modelled by the autonomous system of ordinary differential equations i
= 1,2,···,n
4.7.1
where Xi(t) denotes the amount (or density) of a drug, chemical concentration or biomass of a species in the i-th compartment at time t. The nonhomogeneous term U = COl,{Ul""'U n } in (4.7.1) denotes a constant rate of input of the drug into the system. The constraints on the parameters of (4.7.1) are as follows: aij ( i, j = 1,2, ... ,n) are real constants such that n
i=/=j;
aii
+L
a ji = -aOi;
i, j = 1, 2, ... , n
4.7.2
j=l i'l'i
where aOi is defined by (4.7.2). For convenience we rewrite (4.7.1) in the following form
dx .(t) ( T = -
aOi
n)
+L
j=l j:f;i
aji
Xi(t)
n
+L
aijXj(t)
+ Ui;
i = 1,2, .. ·,n
4.7.3
j=l
i'l'i
and interpret that aijXj(t) (j =/= i) denotes the rate at which material reaches the i-th compartment from the j-th compartment. aOixi(t), i = 1,2, ... n denotes the rate at which material of the i-th compartment leaks to the environment (outside of the compartmental system). If Ui = 0, aOi = 0, i = 1,2, ... , n then the system (4.7.3) is said to be a closed compartmental system and otherwise, the system is
§4.7. Dynamics in compartments
956
said to be an open system. If in an open system we have aOi > 0, 1,2, ... , n, then the system is said to be a leaky system.
Ui?: 0,
z=
It is a simple exercise to show that corresponding to an initial condition of the form
i=1,2,:···,n
4.7.4
where c = col. {Cl , .•• , cn } is a (componentwise) nonnegative constant vector, solutions of (4.7.3) - (4.7.4) are defined for all t > 0 and are such that Xi(t) ;::: o for t?: 0 , i = 1,2, ... ,n. We first consider the following problem which is of interest in drug administration. Assume that (4.7.3) is a leaky system with no input (i.e. Ui = 0, i = 1,2, ... , n) and the state of the system is impulsively altered at a specified sequence of time points so that the modified system is described by
dXi(t)
~
--;It = LtaijXj(t)j
4.7.5
j=l n
Xi(tm
+ 0) -
Xi(tm - 0)
= L Cij(tm)Xj(tm -
0)
4.7.6
j=l
i = 1,2," ',n;
m = 1,2,3""
o = to < tl < t2 < ... < tm
-+ 00
as
m -+
00
where Cij(t m ), i,j = 1,2,"" nj m = 1,2",' are real nonnegative constants. Intuitively one expects that in a leaky system, if the impulsive perturbations are not "too frequent" and if the perturbations Cij(t m ) are not "too large" then the impulsive system (4.7.5) - (4.7.6) should eventually lose all the substance from the system as t -+ 00. The following result provides a set of sufficient conditions under which the above intuitively expected result holds.
Theorem 4.7.1. Let A denote an n x n matrix wi th elements aij, i, j = 1,2, ... , n. Suppose there exist positive constants a, f3 and Co such that (i) tm - t m - 1 ?: f3 > OJ m = 1,2,3, ... , 4.7.7 (ii) 0 ~ ciiCtm) < Co for i,j = 1,2"", n m = 1,2,3",' , 4.7.8 (iii) J.l(A) + log(l + nco) = -a < 0, 4.7.9 where J.l( A) denotes the matrix measure induced by the matrix norm
!
\I A \I
t
= l:$J:$n m?-x laijl i=1
where J.l(A)
t
= l:$J:$n m?-X {an + I aij I}. i=1 i:f;j
957
§4- 7. Dynamics in compartments Tben all solutions of (4.7.5) - (4.7.6) satisfy n
II x(t) II =
L
IXj(t) I:::; II x(to) Ile-o(t-to)
t ~t6·
for
4.7.10
i=l
Proof. Let X(t) denot-e the flUldamental matrix eAt. ~y direct calculation one can derive that for t in the open interval (tk, tk+I), k = 0,1,2"",
x(i) = X(t - i.) { [I + C(i.) I X(i. -
i.-I)} {[I + C(i.-I)][ X(i'_1 - i.-
... { [I+C(tl)][X(tl-tO)] }x(t o)
2 )]}
4.7.11
where x(t) = C01.{Xl(t), ... , xn(t)},C(tk) denotes the n x n matrix with entries Cij(tk) i,j = 1,2, ... ,nj k = 1,2,3, ... and I denotes the n x n identity matrix. Maintaining the order of the terms on the right side of (4.7.11), we can rewrite (4.7.11) compactly in the form k
x(i)
= X(i -
i.) { }] ([ I
+ C(ij) ][X(tj -
ij_I)
I) }X(io);
i
E
(i •• ik+J). 4.7.12
It will follow from (4.7.12), on using the fact
II X(t) II = II eAt II :::; eJl(A)t j
t~O
that
4.7.13
k
II x(i) II :<:; II X(i -
i.) II}]
:::; eJl(A)(t-tk) (1 :<:; exp
[ (II III + II C(ij) II) (II X(tj -
ij_I) 11)]11 x(to) II
+ nco)keJl(A) (tk-tO) II x(to) II
{ptA) [i - io + k log(1+ nco) ]} II x(to) II
:::; II x(to)1I
exp [ (fl(A)
+ ~ log(l + nco»)(t - to) ] []
from which the result follows and the proof is complete.
Consider now a nonlinear compartmental system described by the equations dx ·(t) = T
1
[ /o;(x;(t)) + ~ n n fj;(x;(t)) . + ~ f;j(xj(i))
+ U;
4.7.14
§4.7. Dynamics in compartments
358
1,2 ... , n) are nonnegative valued scalar functions and u = C01.(Ul,U2, ••• ,U n ) denotes a nonnegative (componentwise) constant input vector. One of the fundamental questions for (4.7.14) is the following: if the system (4.7.14) has bounded solutions defined on [0,00) corresponding to nonnegative initial conditions, under what conditions such solutions will asymptotically converge (as t ~ 00) to a nonnegative steady state ~nd whether (4.7.14) has a nonnegative steady state. If all bounded solutions of (4.7.14) converge to a nonnegative steady state, it will mean that (4.7.14) cannot have a nonconstant periodic solution. A knowledge of the existence and nonexistence of periodic solutions of compartmental systems should be of interest in applications. The following result provides a set of sufficient conditions for the nonexistence of periodic solutions of the autonomous nonlinear system (4.7.14).
where
fOi, fij
(i, j
=
Theorem 4.7.2. Assume tbat fOi and J;j (i,j = 1,2, ... , n) are Lipschitz continuous in their respective arguments and monotonic non decreasing such that fOi(O)
= 0;
= 1,2 . .. ,n. = col.{ Ul, U2, ••• , un}
i,j
4.7.15
Suppose furthermore, that the input vector u has nonnegative constant components. Then either all solutions of (4.7.14) corresponding to nonnegative initial conditions x·(O) = c·z > z _ 0,
i
= 1,2, ... ,n
4.7.16
are defined on [0,00) and remain bounded on [0,00) or no solution of (4.7.14) - (4.7.16) will remain bounded on [0,00). If (4.7.14) - (4.7.16) has at least one solution such that
limsup Xi(t) ~ Mi
< 00,
t-co
then (4.7.14) has a nonnegative steady state x*
- [fOi(xn
+
t
Iii(xn]
+
J=l
= col.{xi, x~, ... , x~}
t
/ij(xj)
+ U; =
0
such tbat
4.7.17
J=l
j#i
i#i
and xi ~ 0, i = 1,2, ... , n; also this steady state is globally attractive in the sense that all solutions of (4.7.14) - (4.7.16) satisfy
i
= 1,2, ... ,n.
4.7.18
359
§4.7. Dynamics in compartments
Proof. Since the right side of (4.7.14) is Lipschitzian, solutions of (4.7.14) (4.7.16) exist locally on an interval of the form [0, T) for some possibly small T> 0. On [O,T) no component of the solutions of (4.7.14) - (4.7.16) can become negative; for instance, suppose xp for some p E {I, 2, ... , n} is the component which becomes negative not later than any other component of a solution; that is there exists a t* E (0, T) satisfying
Xp(S) < 0 for
s E [0, i*];
for some
€
s E (i*,i*
+ €)
> 0. This will mean that dXp(t) dt
and
I
< 0·
t=t.
'
but we have from (4.7.14) that
which contradicts the definition of t*. Thus nonnegativity of solutions of ( 4. 7.14)(4.7.16) will follow. Now let x(t) = C01.{Xl(t), ... ,xn(t)} and yet) = COl.{Yl(t), ... ,Yn(t)} be any two solutions of (4.7.14) on a common interval of existence corresponding to nonnegative initial conditions. We have then d
dt[Xi(t) - Yi(t)] = - [fOi(Xi(t» - fOi(Yi(t»] n
- I:: [!ii(Xi(i»
-!ii(Yi(i»]
j=1 j-¢i
4.7.19
n
+ I:: [fij(Xj(t»
- J;j(Yj(t»]
j=l
j-¢i
for
i
= 1,2, ... , n .
Consider a Lyapunov function vet) = v(t,x(t),y(t» defined by n
vet) =
I:: I Xi(t) - Yi(t) I; i=1
t E [0, T).
4.7.20
§4.7. Dynamics in compartments
960
Calculating the upper right derivative n+v of v and using the monotonicity of fOi, we can derive that n
D+v(t) ~ -
L
4.7.21
Ifoi(xi(t» - fOi(Yi(t»1
i=l
which shows that if one of the solutions x(t) or yet) is defined on [0,00) and remains bounded on {O, 00), then the other is also defined on [0,00) and remains bounded on [0,00). Thus, either all solutions of (4.7.1) remain bounded or no solution remains bounded on [0,00). Suppose (4.7.14) has a solution x(t) = col. {Xl (t), ... , xn(t)} such that (4.7.17) holds. Then every solution of (4.7.14) is bounded on [0,00) showing that the system (4.7.14) has a compact convex invariant set in Rn. As a consequence of Brouwer's fixed point theorem, it will follow that such an invariant set must contain at least one (fixed point) steady state x* = (xr, ... ,x~) of (4. T.1). Since x* lies in a bounded closed set of nonnegative octant of Rn, x* is nonnegative componentwise.
IT we choose now Yi( t) == xi (i = 1,2, ... , n) in (4.7.20), we then have
D+
(t,
Ix;(t) - Xii)
~-
t,
4.7.22
Ifo;(Xi(t» - fOi(xill .
We can show that (4.7.22) and the monotonicity of fOi (i = 1,2 ... ,n) will imply (4.7.18). Suppose (4.7.18) does not hold. We note that Xi(t) is bounded since Xi is bounded and hence the right side of (4.7.14) is bounded on [0,00) implying that Xi is uniformly continuous on [0,00). IT t:~ Xi(t) =:f xi or if Xi(t) does not converge to xi as t -+ 00 for one or more i E (1,2 ... , n), then we can find a sequence {tk; k = 1,2, ... }, to < t1 < ... , tk -+ 00 as k -+ 00 such that
k
= 1,2, ...
4.7.23
for some positive number e. As a consequence of (4.7.23) and the uniform continuity of X for t ~ 0, it will follow that there exists a constant Tf > such that t 1 - Tf > to , and
°
4.7.24
961
§4.7. Dynamics in compartments We have from (4.7.22) - (4.7.24) that n
LI
n
Xi(tj)':"
xi I::; -(€/2)j 1] + L I Xi(tO) - xi I
;=1
i=1
which shows that L:~=1 I Xi(t) - xi I can become ~egative for large t and this is impossible. Thus our assertion (4.7.18) holds and the proof is complete. [] Let us consider next compartmental systems which incorporate "transport delays" (for numerous examples related to this, see Gopalsamy [1983c]). For instance, in the place of (4.7.3) we consider a delay-differential system of the form 4.7.25
i = 1,2,3, ... , n
for
t>0
;
t=
j) are nonnegative constants. It has been an where iij(i,j = 1,2, ... ,n; i implicit assumption in (4.7.3) that the transit time for material flux between any two compartments is negligible. In several physiological systems involving the transport of tracers of blood from one compartment to another, there is usually a finite time iij required for the transport of material from compartment j to the i-th compartment (from right ventricle to left ventricle etc.). Thus, it is worthwhile and perhaps necessary to consider (4.7.25) to be a generalisation of (4.7.3). Detailed mathematical analysis of compartmental systems with transport delays has been done by Lewis and Anderson [1980a,b], Gyori and Eller [1981]' Krisztin [1984] and Gyori [1986]. In the following, we first consider the effects of delays in (4.7.25) on the asymptotic behavior of solutions of (4.7.25) as t ~ 00. Theorem 4.7.3. Suppose the constant parameters of (4.7.25) satisfy the following: iij ~
aOi
> 0;
i,j'= 1,2,3, ... ,nj
0;
aii
=
aOi
+L
a ji
>
i=1,2, ...
0;
,n.
j=1 j#i
Then all solutions of (4.7.25) corresponding to initial conditions of the type
t E [-i, 0];
i
=
max
l$i,j$n i#j
i'"
')'
4.7.26
§4.7. Dynamics in compartments
962
satisfy
Yi(i) where x*
~
0
=
for
i
~
(xr,xi, ...
lim Yi(t)
O· and ,x~)
t-co
= xi;
i=1,2,,,.,n
is a steady state of (4.7.25) with xi
4.7.27 ~
O,i ==
1,2,3, ... ,no
Proof. We note that the existence of x* is not a part of the assumptions. Define a sequence y(k)(t) = {yi k)(t), y~k)(t), . .. , y~k\t)}, k = 0,1,2,3, ... as follows: (0)
_
Yi (t) -
{c/>i(t) for t E [-T; 0] c/>i(O) for t > 0
i = 1,2,,,.,n
4.7.28
c/>i(t) for t E [-T,O]
+ .I: aij
1
j=l
0
n
e-a"t c/>i(O)
+ Ui
1.t
t
e-a,,(t-s)y;k\ s -.:. Tii )ds
4.7.29
ii:-i
t > 0;
e-a,,(t-')ds;
i = 1,2, ... ,n.
It can be shown that the sequence {y(k)(t); t ~ -T} converges as k ~ 00 to a limit function y*(t) and the convergence is uniform on bounded closed subsets of [-T, 00). It will then follow from (4.7.29) that
c/>i(t) for t E [-T,O]
yi(t) =
4.7.30
A consequence of (4.7.30) is that ~
= 0,l,2,,,.,n:::} yi(i) ~ 0,
i=l,2, ... ,n
t
~ -T.
Consider now the linear system (in the unknowns ml, m2, ... , m n ) of the algebraic equations n
Laijmj +Ui j=1
= 0
i
= 1,2, ... ,no
4.7.31
363
§4.7. Dynamics in compartments
It follows from our assumptions on the coefficients aij in (4.7.31), that the matrix A = (aij) is diagonal (column) dominant with ajj < 0, i = 1,2, ... , nand aij ~ 0, i,j = 1,2, ... ,n; i t'j. It is known (see Araki and Kondo [1972]) that (-A) is a stable M-matrix such that (-A) is nonsingular and the elements of (_A)-l are nonnegative. Thus, the linear system (4.7.31) in the unknowns ml, m2, ... , mn has a nonnegative .solution i
= 1,2, ...
,no
We conclude that leaky compartmental systems have unique nonnegative equilibrium states. Let us for convenience, suppose x· = {xI, xi, ... , x~} is the nonnegative steady state of (4.7.25). To prove the convergence in (4.7.27), we let
i=1,2, ... ,n and derive that
i=1,2, ... ,n.4.7.32
Consider the Lyapunov functional v(t,w(.) for (4.7.32) defined by
v(t, w(.»
n ( n =~ I w;(t) I +j;, a;j f.-TO) I w;(s) Ids t
)
;
t > 0.
4.7.33
j#i
One proceeds to show that the upper right derivative D+ v of v along the solutions of (4.7.32) satisfies n
D+v :::; -
L aOil Wj(t) I,
t>O
4.7.34
i=l
from which it can be shown (the reader should' try this) that Wi(t) 00, i = 1,2, ... , n and this completes the proof.
-t
0 as t
-t
[]
Since (4.7.32) is a linear autonomous system of delay differential equations, one is entitled to ask the following; does it follow from the assumptions of a leaky compartmental system that all the roots of the associated characteristic equation have negative real parts? The following result contains an affirmative answer to this question.
§4.7. Dynamics in compartments
364
Theorem 4.7.4. Assume that
aij
2: 0,
Tij
2: 0, i,j
= 1,2, ... , n; i t= j
and
n
aii
= aOi + 2::: aji > 0;
i = 1,2, ... , n.
j=l j#i
Then all the roots of 4.7.35 where parts.
Oij
= 1 if i
Proof. Let
Z
=j
and
Oij
= 0 for i
=f j , i,j = 1,2, ... , n have negative real
be any root of (4.7.35). By Gershgorin's theorem of matrix theory
we know that n
I Z + aii I:::;
2:::
aije-
ZTij
for some
i E (1, 2, ... , n)
4.7.36
j=l
j#i
or equivalently, there exists Mi = Mi(Z) , I Mi(Z)
I :::; 1 such that
Z is a root of
n
Z + aji
+ Mi(Z) 2::: aije-
ZTij
=0
for some
i E (1,2, ... , n).
4.7.37
j=l j#i
It is enough to show that (4.7.37) has no roots with nonnegative real parts. Define Ii and gi as follows:
for
Since aii > 0, fi(Z) has no zeros Z with region ~e(z) ~ 0 , we have
~e
as in
~
Z
(4.7.37) .
4.7.38
0 and on the boundary of the
n
I fi(Z) I = IZ + ail I ~ aii > I:: aji ~ Igi(Z) I; j=l j#i
4.7.39
965
§4.7. Dynamics in compartments
hence IIi(z) I > I gi(Z) I on the boundary of ~e (z) ~ O. Since fi(Z) t- 0 for ~e (z) ~ 0, it follows from Rouche's theorem that fi(Z) + gj(z) i 0 for ~e (z) ~ 0 and this completes the proof. [] We find that although the result of Theorem 4.7.4 provides a set of sufficient conditions for the asymptotic stability of the trivial solution of (4.7.32), this result has not exhibited in any way, the effects of the transport delays on the mode or rate of convergence of solutions of (4.7.25) to its steady state. One expects from the result of Theorem 4.7.4 that the above convergence should be exponential. The following result is concerned with an examination of the effects of transport delays on the convergence of solutions of (4. 7~25) to its steady state. Theorem 4.7.5. Let r be any fixed positive number; let A and matrices such that aj;
<0
,
A be real n
x n
i = 1,2, ... , n
i,j=1,2, ... ,n.
Suppose p = minl
Iaj; I is such that 4.7.40
Then every solution of the linear delay differential system
dWi(t) + I: dt- = a··w·(t) n n
I
a··w·(t - r) IJ J
4.7.41
j=l j¢i
corresponding to continuous initial conditions on [-r,O] satisfies an estimate of the form
II wet) II ~ { sE[-r,r] max
/I w(s) II}
e-p*t j
t>O
4.7.42
where p* is the unique root of the equation 4.7.43
Proof. Proof is similar to that of Theorem 3.6.2 of Chapter 3 and hence is omitted.
§4.7. Dynamics in compartments
366
We consider briefly a nonlinear compartmental system where the compartments are intercolUlected by pipes. It is known (see Gyori [1986]) that a pipe compartmental system can be modelled by an integrodifferential system of equations of the form
dx·(t) -Tt= - L n
Pjj(Xi(t)) +
L n
iT
j=l
j=O
dFij(S)Pij(Xj(t - s)) ds
+ Ui.
0
4.7.44
i = 1,2, ... ,n where (i) Pij: IR ~ IR is continuous, monotone and nondecreasing and Pij(o) = 0; i = 0,1,2, ... , n ; j = 1,2, ... ,n, (ii) T > is a fixed positive constant. (iii) Fij : [0, T] ~ [0,1] is continuous from left, monotone nondecreasing and Fij(O) = 0; Fij(T) = 1; i,j = 1,2" ... , n. (iv) Ui, i = 1,2, .... , n are known nonnegative constants. One of the immediate questions is to ask whether (4.7.44) has a nonnegative equilibrium state and whether such an equilibrium is globally attractive of all other nonnegative solutions. First we derive the following result which is a generalization of Theorem 4.7.2.
°
Theorem 4.7.6. Assume that Pij, Fij, T are as in (i) - (iv) above. Suppose a set of nonnegative initial conditions
t E [-T,O];
4.7.45
are provided for (4.7.44). Then either eve.ry solution of (4.7.44) - (4.7.45) is bounded or no solution of (4.7.44) - (4.7.45) is bounded on [0,(0) . Proof. Let xCi) = {Xl(t), ... ,xn(i)}, y(i) = {Yl(t), ... ,Yn(t)} be any two solutions of (4.7.44) with a common interval of definition. The result will follow easily from an analysis of the upper right derivative of the Lyapunov functional
v(t) = v (t, x(.), Y(.)) where for t > 0 v(t) =
t. [I
Xi(t) - Yi(t)
I 4.7.46
967
§4.7. Dynamic3 in compartments
The next result (due to Krisztin [1984]) asserts the existence of a globally attractive constant steady state of (4.7.44)j a consequence of the following result is the nonexistence of nbnconstant periodic solutions for (4.7.44) Theorem 4.7.7. Suppose the assumptions of Theorem 4.7.6 hold. Assume further that P~i (i = 1,2, ... , n) are strictly monotonic. Then all bounded solutions of (4.7.44) (if such solutions exist on [-T,oo» are such that they converge to a nonnegative equilibrium state of (4.7.44). Proof. Let xCi) = {xl(i), ... ,xn(t)} be a solution of (4.7.44) which is bounded on [- T, 00) satisfying
o ::; lim sup Xi(t) = Mi < 00 t-oo
i
o ::; lim t-oo inf Xi(t) = mi < 00 Define a function G : IRn
1-4
= 1,2, ...
,no
Rn as follows:
n
n
G i (ZI, Z2, .• ' ,zn) = - LPij(Zi) + LPij(Zj)+Uii j=O
i
= 1,2, ...
,no
4.7.47
j=l
First we show that i
= 1,2, ...
4.7.48
,no
If (4.7.48) is not true, then there exists an io E {1, 2, ... ,n} such that
Let a that
= Gio(Ml, M 2 , •••
,Mn). Since Pij are continuous there exists
T
> 0 such
n
n
- L Pjio(Mi o - €) + L Pioj(Mj j=O j=l Now choose
€
such that if t ~
T,
sup x }·(t)
f~r-T
+ e) + Uio < aj2.
4.7.49
then
-< M·} + eo,
j=1,2, ... ,n.
We have by the monotonicity of Pij,
dXi o n - d (t) ~ - LPjio(Mj - e)
t .}=o
n
a
+ LPioi(Mj + e) +Uio < 2" < 0 . }=l
4.7.50
§4.7. Dynamics in compartments
368
t 2: rand Xio(t) E [Mio - €, Mio + €].
for
This contradicts the definition of G io proving i = 1,2, ... ,no
By similar arguments we have Gi(ml, m2,'" , m n ) above and the equality
0, i
= 1,2, ... , n. From the
n
n
L Gi(Zl,."
~
, zn) =
L[Ui - POi(Zi)),
i=1
;=1
it follows that n
n
o ~ LGi(M1,M2, ... ,Mn)- LGi(ml,m2, ... ,m n) i=1
;=1
4.7.51
N
= - 2:.)Poi (Mi )
-
POi(mi)] ~ 0
i=1
showing i = 1,2, ... ,no
By the strict monotonicity of POi, we have that
Mi = mi = tlim Xi(t); --00
i
= 1,2, ... ,n.
It is now easy to see that x* = (xr,x;, ... ,x~), xi = Mi = mi, i = 1,2, ... ,n is a steady state of (4.7.44). The nonnegativity of xi, i = 1,2, ... , n is a consequence of that of Xi(t) on [-T, 00), i = 1,2, ... ,no To prove that x* = (xi, ... ,x~) is attractive of all other nonnegative solutions, one can proceed as in the case of no delays, using the Lyapunov functional proposed in the proof of Theorem 4.7.3 with Yi == xi, i = 1,2, ... ,n. 0 Some examples of compartmental systems are listed below; it is left to the interested reader to investigate the convergence, persistence and oscillatory characteristics of the following:
dx~it) = -.\;Xi(t) + ~ aij l~TJ Xj(s) ds + Ui; i = 1,2,"" n.
l
969
§4.7. Dynamic3 in compartment3
dXi(t) d- = t
\
-/\iXi
() ~ ajjxj -()d t + L...J S S + Ui; i=l i~i
Xj(t) =
sup
i = 1,2,000,
Xj(s),
no)
sE[t-rj ,t]
dXi(t) = -AiXi(i) + ~ aijXj([iJ) ds + Ui; ) -----;[t
f;r
z = 1,2, .. ·,n.
dx~?) =
-AiXi(i) +
~ aijX j(jJjt) + Hi;
0< Jli < 1,
i
)
= 1,2,,,,, n.
dx i ( t) = -AiXi(i) \. -----;[t + ~ aij log{Xj(t )}.+ Hi )
f;r
i = 1,2, .. ·,n.
We remark that the techniques developed for the analysis of compartmental systems can also be used in studying the dynamics of neural networks modelled by (for details of neural networks see Marcus et al. [1991] and the references cited therein) systems of the type
in which f3i, ajj , a i E R, i = 1, 2, ... , n and the responses.
Tji
correspond to delays in neuronal
970
EXERCISES IV 1. In the following equations, prove that solutions corresponding to positive initial conditions remain positive and are defined for all t > O. Derive sufficient conditions for the existence of a positive equilibrium and for its global attractivity. Investigate the P?ssibility of delay induced bifurcation to periodic solutions and obtain sufficient conditions for the oscillation of all positive solutions about the positive equilibria (assume that all the parameters are positive constants).
dN (t) dt
= r N (t) [J{ - N (t -
J{ + N (t - r)]. r)
dN(t) = -rN(t) + ae-PN(t-r). dt dN(t) = -rN(t) + aN(t _ r)e-PN(t-r). dt dN(t)
(3
---;It = -rN(t) + 1 + N(t - rf dN(t) = -rN(t) (3N(t - r) dt + 1 + N (t - r) dN(t) dt dN(t) dt
d~~t)
= -rN(t)
+
(3N(t)Nn(t - r). (}n+Nn(t-r)
= rN(t)[l- N(t -
. J{ r)]_ 1 +N2(t) N2(t)
= - r N(t)+ aN(t - r)
[1+ 11(1 - Nn~;: r)) ].
Investigate the convergence and oscillatory characteristics of the above systems when N(t - r) is replaced by N(t) , N([t]) and Nt respectively where
N(t) =
N(s),
sup sE[t-r,t]
and
Nt
=
it
N(s) ds
t-r
[t] = greatest integer contained in t.
Exercises IV
971
2. In the previous exercise replace N(t - r) by the integral tenn
1
00
K(s)N(t-s)ds
with a suitable nonnegative delay kernel Kj examine the stability (local and global) of equilibria; examine the oscillations and persistence of the resulting integrodifferential systems. 3. Discuss the stability of equilibria, persistence and oscillations about the equilibria in the following systems:
dx(t) = X(t)[7'l - anlog(x(t - r)] --;It dy(t) -;It
= yet) [7'2 + a21 1og[x(t dx(t) = 7'l X(t) dt
[1 -
+ a12 log[y(t - r)J]
lj 1 lj
r)] - a22 log[y(t - r)]]. x(t - r)
al
+ Ilty(t -
r)
dy(t) _ 7' (t)[lyet - r) dt - 2Y a2 + f32X(t - r) . dx(t) dt dy(t) --;It
~X(t)[k-X(t)-
=y
()[ x(t - r) ( )1 t 1 + x(i _ r) - ay t .
dx (t) _ (.) [ _ x (t - r) dt - 7'X t 1 K - ax(t)y(t) [1 dy(t) --;It
y(t-r)
1 + x(t - r)
1 1
-
exp { - /3x(t - r)}
= -8y(t) + y(t)[l -
exp { - /3x(t - r)}.
dx(t) = a - (3x(t)
+x
dt
d~~t) = ax2(t _ r) -
2
(t - r) yet - r)
by(t).
j
}
Exercises IV
372
4. Consider a Lotka - Volterra system
[2: a··x ·(t)1 n
dXi(t) r·, - . dt- = x·(t) '
'1
1
,
i
= 1,2,··· ,no
1=1
Assume that solutions of this system are uniformly bounded in the positive orthant of Rn and there exist positive constants Pl,P2,··· ,Pn such that
holds for all equilibria on the boundary of the positive quadrant of Rn. Then prove (for details see Jansen [1987] and Hofbauer and Sigmund [1987]) that the above Lotka Volterra system is persistent where A denotes the matrix with entries aij. Can you extend this result to delay and integrodifferential system of equations? Derive that a necessary condition for the persistence of the above system is the existence of a positive equilibrium point. 5. Assume that solutions of the Lotka - Volterra system of the previous problem are uniformly bounded in the positive orthant and there exists a differentiable function P : Ri- Ho R with the following properties:
(i) P(x) = 0 for x on the boundary of the positive orthant of Rn; P(x) > 0 for x in the interior of (ii) The function (~) extension to R+.
Ri-.
= 1/1
defined on the interior of
Ri-
has a continuous
(iii) For all x on the boundary of Hi- there exists aT> 0 for which 1 {T T Jo tjJ(x(t))dt > O. Prove that the Lotka - Volterra system of the previous problem is persistent (for more details see Kirlinger [1986], Hofbauer and Sigmund [1988]
373
Exercises IV
and the literature cited therein). Can you obtain a similar result applicable to systems with time delays and integrodifferential equations. 6. Consider the functional differential equations for i = 1, 2, ... , n
oo
where"\ > 0, aij < 0, J.lij : [0,00) 1-+ R is of bounded variation, Jo /dJ.lij( s)/ = 1 and 7]ij( s) = bij( s) - 8jj e->'s obeying 7]ij( s) = constant for s > T. Assume that the nonlinear system has a positive steady state X* = (xi, ... , x~), xi > 0, i = 1,2, ... , n. Suppose there exist constants d j > 0, with n
ddaid >
n
L dj/aij/ + L dj(/bij/ + 18ij !), 1;~
i = 1,2, ...
,n.
j=l
Prove that the steady state x* is locally asymptotically stable. Prove or disprove the statement: x* is globally asymptotically stable with respect to positive solutions (see Busenberg and Travis [1982]). 7. Consider the retarded functional differential equations for i = 1,2, ... ,n
with
aii
/.00 d7]ii( s) + 8ii < 0; fIX> Id7]ij( s)/ = 1 o ~ .
aij7]ij( s) - 8ije ->.s = constant for s > T and ,.\ > Suppose the system has a positive steady state x* 1,2, ... , n and there exist positive constants dj >
°
i, j = 1,2, ... ,n .
= (xi, ... ,x~), xi
> 0, i
=
°, j = 1,2, ... , n such that
Exercises IV
374
d;la;;
1~ cos(vs )dry;;(s) + 8;; J.~ cos(vs),\e -,. d·1 >
t.
dj (la;;1
+ 18;;1)
j~i
for all real v satisfying
Prove that x* is locally asymptotically stable (Busenberg and Travis [1982]). 8. Consider the linear system dXi(t) --;It =
?=k aijXj(t) + Lk [fO_ Xj(t + s) ( L dT/ijm(S) 1 .= 1,2, .. , k f.
J=l
J=l
h
)
j Z
m=l
where aij ~ 0 for i =j:. j and dT/ijm(S) are nonnegative measures on [-h,O]. Prove that a necessary and sufficient condition for the asymptotic· stability of the trivial solution is that the following hold (for more details see Obolenskii [1983]) f.
an
+~
1. 0
<0
drynm
all + 2:~=1 J~h dT/l1m ...... a1n + 2:~=1 J~h d1Jlnm > 0;
(-It anI
+ 2:~=I J~h d1Jn1 m
ann n = 1,2, ... , k. ••••••
+ 2:~=l J~h d1Jnnm
9. Consider a biochemical system modelled by the autonomous ordinary differential equations dXl(t) _ al _ b x (t) dt - 1 + kl Yn ( t ) 1 1
dYl (t)
--;It
= Q:'lXl(t) - (31Yl(t)
dXi(t) dt dYi(t)
=
aj
_
1 + k i Yi-l(t)
bjXi(t)
--;It = Q:'iXi(t) - (3iYi(t)
for i = 2,3, ... ,n. Assume that all parameters appearing above are positive constants. Obtain sufficient conditions for the existence of a positive steady state and for its global asymptotic stability with respect to positive solutions.
375
Exercises IV
If there are time delays in the above model so that
for i = 2,3, ... , n, examine whether there exists a delay induced instability leading to persistent oscillations (Banks and Mahaffy [1978a, bJ). 10. Derive a set of sufficient conditions for the existence of a componentwise positive steady state and its global attractivity with respect to positive solutions in the following (assume all parameters are positive constants).
(1)
(2)
i = 1,2, ... , n.
(3)
i=1,2, ... ,n. (4)
(5)
i = 2,3, ... ,no
Exercises IV
376
11. Consider a competition system modelled by
- i = 1,2, ... , n; where bil aij, Tij are nonnegative constants. Assume n
bi
> Laij(bj/ajj),
i = 1,2, ... ,n.
j=l j~i
Show that the system has a globally asymptotically stable positive steady state. Develop a similar result for a system of the form
= 1,2, ... ,n.
i Do the same for the systems
dXi(t) = x;(tl { b; - a;;Iog[x;(tl] --;It
fu~
a;j log[xj(t - T;jl] } ;
i
= 1,2, ... ,n;
i
= 1,2, ... , n.
12. Suppose there exists a positive steady state x* = (xi, ... , x:), xi 1, 2, ... , n for the Lotka-Volterra system
dx- =x- { b-+"a--xn } _, dt t 1 L....t I] ] j=l
i = 1,2, ... ,no
>
0, i =
377
Exercises IV
Suppose further, there exists a constant positive diagonal matrix C such that CA+ATC is negative definite where A denotes the n x n matrix (aij). Prove that the steady state x* is globally asymptotically stable. Under the same assumptions as above, examine whether x* is (i) locally; (ii) globally asymp.totically stable for the time delayed systems i = 1,2, ... ,n
i
= 1,2, ... ,n.
(1)
(2)
13. Investigate the asymptotic behavior of the dynamics of a prey-predator system modelled by
Y- - c:x -dx = x [ ab - dt 1 + ax dy -d t
dx- = Y [ -c + 1 + ax
fLY
/.00 kl (s )y( t 00
s )ds
1
0
+
1 0
1
k 2 (s)x(t - s)ds .
14. Derive sufficient conditions for the existence of a globally asymptotically stable positive steady state of
i = 1,2, .. . ,n;
where r, fJi, ki' aij ; (i, j = 1,2, ... , n) are nonnegative constants. Consider the cases 0 ~ fJi < 1 and fJi > 1, i = 1,2, ... , n. If there are time delays in the above system so "that Xi () t xi (t ) e· ' . ---;u= .\iXi(t) 1- ( T ) - ~aij [
d
n
(
x·(t 1 k
j
Ii') ) J
e.] J
•
;'
= 1,2, ... ,n,
then under what additional conditions, the global asymptotic stability of the positive steady state holds?
Exercises IV
978
15, Derive a set of sufficient conditions for the existence of a globally asymp-
totically stable positive steady state of an integrodifferential system of the form
d:~t) = x(tl(a -
bx(t) - ey(t)
_/,T k,(s)x(t - s)ds
_/,T k2(s)y(t _ sjdS] d~~t) = yet) [ _ d +px(t) - qy(t) + /,T k,(s )x(t - s) ds
_/,T k (s)y(t-S)dS]. 4
16. Consider the autonomous ordinary differential system 1 dy· dt
= Yi [<(3; +
?= n
O:ijYj
J;
i
= 1,2,."
,Pi
]=1
dYe = dt
~ ai'y'
~]]
.
,
e = P + 1, P + 2, ... , n .
j=l
Assume that this system has a steady state y* E R~ x Rn-p. Prove (see WorzBusekros [1978]) that a sufficient condition for y* to be globally asymptotically stable with respect to the initial values yeO) E IR~ x IRn-p is the existence of positive numbers d 1 , •• • , d n and a positive definite (n - p) x (n - p) matrix D4 such that DA + ATD is negative definite where A
= (O:ij),
D= [Dlo D1
i,j=I,2, ... ,n
0 D4
1
=diag(d1 ,d21 • •• ,dp ).
17. Let e, 0:,8, a be positive constants in the scalar integrodifferential equation
Prove that all nontrivial solutions corresponding to bounded continuous initial conditions on (-00,0] converge as t --+ 00 to a positive steady state.
379
Exercises IV
Investigate the asymptotic stability of the positive steady state of
'jt
dx dt = x ( c - ax - 8a
_(Xl
)
x(s)(t - s)e-a(t-s)ds .
18. Consider the integrodifferential system
with the assumption that it has a steady state x* = (xi, ... , x~, 0, ... ,0) where
x; (bi
+
t
i=1,2, ... ,m
aijx;) = 0;
)=1
x; > OJ x; = 0;
i = 1,2, ... ,mj i = m
+ 1, ... , n
; m::; n.
Derive a set of sufficient conditions for the global asymptotic stability of x* with respect to positive solutions.
19. Assume that all the parameters bi,aij (i,j
dXl at = xdb dX2 at = x2[b2X dX3 at = x3[b3X 1 -
allXl -
= 1,2, ... , n) in the system a12 x 2 - a13 x 3]
l -
a22 X2 - a23 x 3]
l -
a32 x 2 - a33 x 3]
are positive constants. Obtain sufficient conditions for the existence of a globally asymptotically stable positive steady state. If there are· time delays such that the above system is modified to the form
dXl (t)
~ = b1 X1(t - TI) - xl(t)[allxl(t)
+ a12X2(t) + a13 x 3(t)]
~ = b2X1(t - T2)X2(t - T2) - X2(t) [a22x2(t)
dX2(t)
+ a23x3(t)]
dX3(t)
+ a33 x 3(t)],
~ = b3xl(t - T3)X3(t - T3) - X3(t) [a32x2(t)
Exercises IV
980
does the global asymptotic stability of the positive steady state continue to hold (at least for small delays)? 20. Assuming that all the parameters appearing are positive constants, investigate the asymptotic behavior of the following plausible competition models .
.J
(1)
(2)
(3)
21. Consider a resource based two species competition system modelled by dXl
dt dX2
dt dX3
dt
= Xlg(XI) - X2P2(xI) - X3P3(xd = X2[,82P2(XI) - a22 x 2 - a23 x 3]
= X3[,83P3(XI) - a32 x 2 - a33 x 3]
where ,82, ,83, a22, a33 are positive constants while a32, a23 are nonnegative constants; the functions g, P2, P3 are of the type
r, B ,k ,c are positive constants.
mixCi
Pi(X)
=
mjXni
{
mj+x nj
mj, ni, Ci are positive constants; i = 1, 2.
mi(l - e CiX )
Investigate the existence of a positive steady state and the persistence of all the species.
Exerci8e8 IV
381
22. Examine whether equilibrium coexistence is possible in a resource based competition system modelled by
ds = .AS(l-~) _ as x _ ~ __Y_ dt k a + s a + xm b + s f3 + yn dx as x _ - = a l - - - - - - D 1x dt a +s a + xm dy bs y - = a 2 - - - - - - D 2y. dt b + s f3 + yn
Assume that all the parameters are positive constants with 0 m<1.
<
n < 1,0 <
23. Derive sufficient conditions for the equilibrium coexistence of the competition systems modelled by
dx dt =.Ax (1x - k)
-
~
x ~ aiYiI'"•
)
and
1=1
dYi
dt
=
/JiXYi • -
~~ = (XO dYi dt
D
I'"
(.l
Q
iYi;
x)D - x
I'"
=/JiXYj' -
.
Z
t.
D iYi,
(1)
= 1,2, ... ,n;
a,.Yfi
) (2)
1,=1,2, ... ,n.
in which .A, k, ai, f3i, Di, X O are positive constants and !-li are constants such that 0 < !-li ~ 1, i = 1,2, ... , n. 24. Investigate the effects of time lags in the asymptotic behavior of the following model systems (assume 0 < J-l < 1).
dx(t) = .Ax(t) dt
{I - X(t)} - mxn(t)y(t) I + xn(t) k
a
dy(t) = c mxn(t - r) yet _ r) _ Dy(t). dt a + xn( t - T) dx(t) = .Ax(t) dt
{I - x(t)} k
a
ax(t) yll(t) + x(t)
dy(t) = c ax(t - T) yll(t _ r) _ Dy(t). dt a+x(t-r)
(1)
I (2)
982
Exercis es IV
25. Derive sufficient conditions for the convergence, oscillations and persistence of the following multiplicative and additive logistic equations:
d~~t) d:~t)
= rx(t)
[1 - t, ajx(t - Tj)] ,
=rX(t)[l- t,ajIOg[X(t-Tj)J],
[1 - [0 K(s) log[x(t - s)J dS]. d~~t) = rx(t) [1 - J.' K(s) log[x(t - s)J dS].
d~~t)
= rx(t)
26. Let A(t) be an n by n matrix of continuous functions satisfying A(t) ~ 0, to :::; t < 00. Prove that the linear system of equations
dx(t)
-;It
= -A(t)x(t) ; x E Rn
has a nontrivial solution xo( t) satisfying
xo(t)
~
0 , xo(t) :::; 0
on
[to
~
t < (0).
(Assume that the inequalities hold in the componentwise sense). Let T( t) be a nonnegative valued bounded continuous function on [to, 00). Derive a set of sufficient conditions for the existence of a nonoscillatory solution of the linear system
dx(t)
-;It = -A(t)x(t - ret));
t > to.
27. Let aij,bij,rij,CTij (i,j = 1,2, ... ,n) be real constants such that rij ~ O,CTij;::: o (i,j = 1,2, ... , n). Derive a set of sufficient conditions for all nontrivial solutions of the following linear system to be oscillatory:
dx .(t) ---!it+L n
j=l
L n
aijXj(t - Tij)+
bijXj(t
+ CTij) =
j=l
i = 1,2, ... ,no
0; t
>0
383
Exercises IV
28. Derive sufficient conditions for the convergence of all positive solutions of the following competition model systems (see Gopalsamy [1980]);
duet) = uti) [rl ----;[f dv(t) -;[t
-
= vet) [T2 -
1° 1°' -
r al uti) -b 1 -T Kl(S )v(i + s) ds ]
a2
R 2(s)u(t
-T
+ s) ds -
)
]
(a)
b2v(t) .
(b) i = 1,2, ... , n.
dUi ( t)
-;Jt = Uj(t) [ Tj - aji log[Uj(t)]-
Lnaijo 1 liij(, s) log[Uj(t + s)] ds], (c) -T
j=l i -:F;
i = 1,2, ... ,no
29. Discuss the persistence of the competition systems modelled by (see Schuster, Sigmund and Wolff (1979))
dXl
dt
=
dX2 dt
= x2(1 -
Xl (1
-
aX2 - {3x3)
Xl -
{3XI -
(;) •
X2 - aX3)
dX3 = x3(1 - aXl - {3X2 - X3) dt ' dXl
dt
= Xl(XI
dX2 dt
= X2({3X1
+ aX2 + {3X3
- M)
+ X2 + aX3 -
M)
h were
(ii)
dX3
M =
XI(XI
dt = X3(a- XI + {3X2 + X3 - M) + aX2 + {3X3) + X2({3XI + X2 + aX3) + (aXI + f3 X2 + X3)'
30. Discuss the persistence of population systems (for details see Gard [1980]) modelled by the following;
t
dx dt =x{a-bX- .
CiYi }
1=1
~i
= aiYi(X - {3i); i
= 1,2, .. , n.
)
(1)
Exerci3e3 IV
384
(2)
(3)
31. Derive sufficient conditions for the persistence of a simple Lotka-Volterra food chain modelled by
dXn-l
~ = Xn-I( -an-I,O dXn
dt
=
xn( -anO
+ a m -I,n-2 X n-2 -
an-l,nXn
)
+ an,n-lxn-l)'
Assume that all the coefficients are positive constants with the exception of which can be nonnegative (see Gard and Hallam [1979]).
all
32. Discuss the effects of the presence of the predator population, on the dynamics of two competing species modelled by (see Hsu(1981])
385
Exercises IV
33. Obtain sufficient conditions for the equilibrium coexistence of competition systems modelled by the equations i=1,2, ... ,n.
Xi dXi dt =AOxo{l_ k I
I
[l+~bOOXO]}' L..J I]]
,
i
= 1,2, ... , n.
(1)
(2)
)=1
j'l'i
34. Investigate the global asymptotic stability of the positive steady states of the following models discussed by Krikorian [1979J;
dXl
dt
dX2
dt
=
xl(rl -
= x2(rz
a12 x 2 - a13 x 3)
allXl -
+ aZlXl -
aZ2 x Z)
(i)
dX3
dt = X3( -r3 + a31 x t). dXI
dt
dX2
dt
dX3
dt
=
xl(rl -
= x2(rZ
+ a21 x l
= X3( -r3
dXI
dt = xl(rl
allXl - a12 x 2 - a13 x 3) aZ3 x 3)
-
+ a31 x l + a32 x 2).
- allXl - a12 x 2 + a13 X3)
dX2
dt = X2( -r2 + a21 x l
dX3
dt
( ii)
= X3( -r3 -
a31 x l
35. Discuss the asymptotic behavior as t
- a23 x 3)
(iii)
+ a32 x 2)'
- t 00
of solutions of each of the following:
dx(t)
---;It = x(t)[a - bx( At)];
(1)
d~~t)
(2)
= x(t)[a _ bx(tA)J;
(3)
Exercis es IV
386
dXi(t) _ ~ b··x .(>.. ·t)] dt = x·(t)[a· Z Z L.,; ZJ J J
(4)
j=l .
i
= 1,2, ... ,n;O < Aj < 1.
= -ax3(t) + bx 3(>..t)., "
(5)
d:~t) = -ax3(t) + bx 3(t>');
(6)
dx(t) dt
(7)
dx(t)
d:t = x(t) [a - bx(logt)]; t > l. dx(t)
-;It = x(t) [a -
(8)
blog{ x(t)}].
(9)
d:t = x(t)[a - blog{x(At)}].
(10)
dx(t)
36. Discuss the asymptotic behavior of the following prey-predator system due to Anvarinov and Larinov [1978] :
d:~t)
= X(t)[,,->.y(t)->'
1.= K,(s)y(t-s)ds
-1.= 1.= R, (s, u)y(t - s )y(t - u) du dS] d~~t) = yet) [ -
j3 + J1.x(t)
+ 37. Let aij
~
0, i
=f j
+ J1.1°O I(2( s )x(t -
s) ds
1.= 1.= R2(S,U)X(t-u)x(t-S)duds].
, i,j = 1,2, ... , n. Prove that the trivial solution of i = 1,2, ... ,n
is asymptotically stable if and only if
> O.
387
Exerci3e3 IV Under the same conditions prove that the trivial solution of
Yi(t) = ('taijyrj+I)2mi+l;
i = 1,2, .. ,n
}=l
where kj and mj are nonnegative integers, is also asymptotically stable. Generalize (see Martynyuk and Obolenskii [1980]) the above result to systems of the form
38. Investigate the convergence and oscillatory characteristics of the following models of cooperation:
()]3]
= Xl3()[I<1+0'IX2(t)_ t ( ) Xl t 1 + X2 t dX2(t) _ 3() [K2 + 0'2 Xl(t) ( )]3 - d - - X2 t () - X2 t . t 1 + Xl t dXI(t) dt
dXI(t) ( )]3 ) - = x3( t - r ) [KI + 0'1X2(t - r) -Xlt dt I 1 + X2 (t - r) dX2(t) _ 3( _ ) [K2 + 0'2XI(t - r) _ ()]3 1 X2 t . dt - X2 t r + Xl ( t - r ) dx I ( t) =
n( ) Xl t
[K
+ I X 2 (t - r) 1 + X2(t - r)
I
Q
-
Xl
(t -
r
)] m
dt m Q Xl(t-r) dX2(t) n( )[I<2+ 2 ( )]. -- = X t - X2 t - r dt 2 1 + Xl (t - r)
(i)
(ii)
) ;
m,n E [1,00).
( iii)
39. Derive sufficient conditions for the convergence of all positive solutions of the following to positive equilibria:
(i)
(ii)
Exercis es IV
388 40. Suppose that
r is a compact, convex invariant set in
IRn for the autonomous
system
dx(t) = f(x). dt Assume further, that the elements of the Jacobian matrix
are continuous in r and have the following properties; (i) the off-diagonal elements of G are negative; (ii) G is (strictly) diagonally dominant with respect to rows. Prove that there exists one and only equilibrium point b in r and moreover every solution x(t) in r satisfies
where 11.1100 denotes the f= norm; i.e. IIxlloo = max{lxd, 1 ~ i ~ n} where ..\ is a positive constant. Can you prove the same result when there are time delays in the "off-diagonal entries" in the governing equation above? (Eisenfeld [1981». 41. Derive sufficient conditions for the asymptotic stability of the positive equilibrium of the controlled systems:
d~; t) = d~~t) d~;t) d~~t)
r N ( t)
[1 - N (t; r) - cutt)]
= -au(t)
I
+ bN(t - a); a E [0,(0).
=rN(t)[l_log[N;;-r)l_clog[u(t)l] = -au(t)
+ blog[N(t -
a)];
I
a E [0,(0).
42. Investigate the possibility of linear stabilizability of the following control systern;
d~;t) du(t)
-dt
= rN(t)
[1 _(N(t)~~ - r)) - cu(t)]
= -au(t) + bN(t);
N(t)
=
sup sE[t-u,t]
N(s).
989
Exercises IV
43. In the following systems Ul, U2 denote feedback indirect controls; if the positive equilibrium of the uncontrolled system is not asymptotically stable, determine whether it is possible to stabilize the controlled system at a new positive equilibrium:
dN (t)
1 -;u= dN (t)
2 -;u=
N1(t)[rl - allNl(t) - a12N2(t) - QI Ul(t)] N 2(t)[rz - a21Nl(t) - a22N2(t) - Q2 U2(t)] (i)
dUl (t) -;u= -!31Ul(t) + ~lNl(t) dU2(t) -;u= -!32 U2(t) + ~2Nz(t). dN1(t) -;u=
N1(t - r)[rl - allN1(t - r) - a12 N 2(t - r) - QIUl(t)]
dN2(t) -;u= N 2(t dUl (t) -;u=
r)(rz - a21 N l(t - r) - a22 N 2(t - r) - Q2 U2(t)] (ii)
-!31Ul(t) + ~lNl(t - r)
dU2(t) -;u= -!32 U2(t) + ~2N2(t -
r).
44. Derive sufficient conditions for the existence and asymptotic stability of a positive equilibrium of the following models of coupled systems:
(i) i
= 1,2, .. ,n ..
tJ ej(ta~Tj))] [1 - tJ t,,~
dx~;t) =riXi(t)[ldy~; t)
= riYi( t)
i = 1,2, .. , n.
(Yj(
H[Yi(t)-Xi(t)]
Tj)) ] Hi [Xi( t) - Yi( t)]
( ii)
Exercises IV
990
45. Prove or disprove the following: a necessary and sufficient condition for all solutions of
d~~t) = rN(t)[l- ~ K : [0,(0)
1-+
100
K(s)N(t - S)dS]
in which
].00 K(s)ds = 1
[0,(0),
to have equilibrium level crossings (in the sense, that there exists at least one
t* E (-00,00) for which N(t*) - C = 0) is, that I()..) = ).. + r
100
K(s)e- AS ds =
°
has no real roots or equivalently that for all
).. E [0, (0).
Do the same for the equation,
d~~t)
=rN(t)[l-
~
100
K(S)log[N(t-S)]dsj.
46. Derive a sufficient condition for all positive solutions of the integrodifferential systems
i=1,2, ... ,nj (1)
i
= 1,2, ... ,nj (2)
(where for i,j = 1,2, ... , n
ri,aij E (0,00),
K ij : [0,(0)
1-+
[0,(0),
].00 Kij(s)ds =
1)
to have equilibrium level crossings in the sense that there exists a t* E (-00,00) such that n
i = 1,2, ... ,nj
LaijNJ=ril j=l
i=1,2, ... ,n.
391
Exercises IV
(Assume the existence of positive Nt, i = 1,2, ... , n). Derive also a set of sufficient conditions for all positive solutions to satisfy as
t
---+
00;
i=1,2, ... ,n.
47. Consider the following integrodifferential model of mutualism;
Assume
Ki: [0,00)
1-7
J.oo Ki(s)ds = l;Ki >
[0,00),
Qij
i = 1,2.
Derive a sufficient condition for all positive solutions to have equilibrium level crossings. 48. Derive a set of sufficient conditions for all solutions of the system (4.7.44) to have equilibrium level crossings; discuss the cases T < 00 and T = 00. 49. In the following integrodifferential equations, formulate your own hypotheses and derive sufficient conditions for the existence of a positive equilibrium and its global attractivitYi also derive sufficient conditions for the equilibrium level crossings of all positive solutions (assume that all the parameters are positive numbers and the delay kernel has nonnegative values):
dx(t) - = -,x(t) dt
+
Q
100
d~~t) = -f'x(t) + '" exp [ dx(t) - = -,x(t) dt
k(s)e-/h(t-s) ds.
0
+a
1
f3
J.= K( s)x(t -
s) ds
00
k(s)xn(t - s)e-.Bx(t-s) ds.
0
dx(t) = _ x(t)
dt
,+
j.
13J.00 k(s) 0
xn(t - s)
1 + x n (t - s)
ds.
Exerci3e3 IV
392
50. In the three species competition system (see Hofbauer and Sigmund [1988])
assume 0
< f3i < 1 < ai, i = 1,2,3 and r 2:: 3
o. Prove that when r = 0,
3
.Il(a; -1) < Il(1- !3i) ===> i=1
i-I
3
3
> Il(1-!3i) ===>
Il(ai -1) i=1
persistence
and
nonpersistence.
i=1
Can you prove the same result when r > O? Generalize your result to n-species competition and integrodifferential equations? Discuss the persistence of species governed by the following model systems:
dXl(t) = rlx1(t) [1-1og[xl(t ---a:t dxz(t) ---a:t
= r2xZ(t) [1 -
dX3(t) = T3 X3(t) [ 1 ---a:t
r)] - azlog[xz(t - r)] - f33log[x3(t - r)] ]
f3 1 1og[Xl (t - r)] -log[x2(t - r)] - a3log[x3(t - r)] ] Q
1
log[X1(t - r)] - !321og[xz(t - r)] -log[x3(t - r)] ]
dx;?) = TIXl(t) [l-log {Xl([t])} - "2 log{x2 ([t])} - P3 10g{X3([t])}]
dx~;t) = T2 X2(t) dX3(t) ---a:t
[1-
Pllog{Xl([t])} -log{X2([t])} - "3 Iog{ X3([t])}]
= T3 X3(t) [1- allog{Xl([t])} -
!3z log{X2([t])} -log{x3([t])}]
[1 - log[Xl ( >.t)] - o2Iog[x2( >.i)] - P3log[x3 (>'t)]] dx~;t) = r2 x2(t) [1 - PI log [x 1(>.i)] -log[x2(>.t)] - 03 Iog[X3(>'t)]] dx~?) = r3 x3(t) [1 - 01 log [x 1(."t)] - P2Iog[x,(>.t)] -log[X3(>.t)]]
dX;?) = rl Xl (t)
in which [t] denotes the integer part of t and 0
< A < 1.
CHAPTER 5
MODELS OF NEUTRAL DIFFERENTIAL SYSTEMS 5.1. Linear scalar equations
Consider a linear neutral integrodelay differential equation of the form
x(t) + t,b;X(t-U;)+fJ
J."" K (s):i:(t-s)ds 2
+aox(t) + ~a;X(t-T;)+a
J."" K.(s)x(t -s)ds =0
5.1.1
in which x(t) denotes the right derivative of x at t. (Throughout this chapter we use an upper dot to denote right derivative and this is convenient in writing neutral differential equations systematically). Asymptotic stability of the trivial solution of (5.1.1) and several of its variants have been considered by many authors. There exists a well developed fundamental theory for neutral delay differential equations (e.g. existence, uniqueness, continuous dependence of solutions on various data; see, for instance, the survey article by Akhmerov et al. [1984]); however, there exist no "easily verifiable" sufficient conditions for the asymptotic stability of the trivial solution of (5.1.1). By the phrase "easily verifiable" we mean a verification which is as easy as in the case of Routh-Hurwitz criteria, the diagonal dominance condition or the positivity of principal minors of a matrix etc. Certain results which are valid for linear autonomous ordinary and delay-differential equations cannot be generalized (or extended) to neutral equations. It has been shown by Gromova and Zverkin [1986] that a linear neutral differential equation can have unbounded solutions even though the associated characteristic equation has only purely imaginary roots. (see also Snow [1965], Gromova [1967], Zverkin [1968], Brumley [1970], and Datko [1983]); such a behavior is not possible in the case of ordinary or (non-neutral) delay differential equations. It is known (Theorem 6.1 of Henry [1974]) that if the characteristic equation associated with a linear neutral equation has roots only with negative real parts and if the roots are uniformly bounded away from the imaginary axis, then the asymptotic stability of the trivial solution of the corresponding linear autonomous equation can be asserted. However, verification of the uniform boundedness away from the imaginary axis of all the roots of the characteristic equation is usually difficult. An alternative method for stability investigations is to resort to· the technique of Lyapunov-type functionals and functions; this will be amply illustrated in this chapter.
§5.1. Linear scalar equations
394
Let us consider (5.1.1) with the following assumptions:
(Hd
ao,
0,
a, {3, ajTj
are real numbers such that ao > 0, Tj ~ 0, j = 1,2,3, ... n (or m as the case may be).
aj, bj, Tj, (Jj
=I 0,
=I 0,
bj(Jj
(H2) ,KI, K2 : [0,(0)
J-lo
(Jj ~
(-00,00) are piecewise continuous on [0,(0) such that
(H3) A set of initial conditions x(t) =
t t
~ 0
~
0
are provided satisfying the following:
x(l) = x(O) +
J.'
i(s)ds
where the functions x(t) and x(t) satisfy (5.1.1) for t
> O.
Definition. (Kolmanovskii and Nosov [1986]) The trivial solution of (5.1.1) is said to be stable if for any € > 0 there exists a 8( €) > 0 such that Ix(t,
t~O
whenever (H3) holds and sups~o 1
t-oo
For convenience, we shall write x(t) instead of x(t,
395
§5.1. Linear scalar equations Theorem 5.1.1. In addition to (HI) - (H3) assume tbe following:
5.1.2
(t, Ib l+I.8I;'= j
IK2(S)ldS) .
) 5.1.3
+ &,lailTi + la l ;'= IK,(s)lsds < 1. Tben all solutions of (5.1.1) satisfy
lim x(t)
t--oo
= O.
5.1.4
Proof. We rewrite (5.1.1) as follows:
d [m dt x(t) + ~ bjx(t - O"j) +,8 J=l
1 f 00
K 2(s)x(t - s)ds
0
K'(S)U~8 x(u)du )dS]
- &,a i Lr, x(s)ds -a =
-(ao+ tai + 1=1
Q
foo K 1 (S)ds)x(t). 10
We let 1" = ao
5.1.5
+ &, ai + a
;.=
K,( s)ds
5.1.6
and define a Lyapunov-like functional Vet) = V(t,x(.)) where
V(t,x(.)) = [x(t) +
t,
bjx(t -l7j) +.8 ;.= K 2(s)x(t - s)ds
- &, ai L" x(s)ds + 1"
[t, Ibjl + t laill~ . (It
a
;.=
K,(S)(L. x(u)du )ds]'
L.; x2(s)ds + 1.81;'=
1=1
+ la l
t
;'=
T,
IK2(S)IU~. x 2(u)du )ds
x2(U)dU)dS
S
2
IK,(S)i(L. ( [ x (V)dV)dU)dS].
5.1.7
§5.1. Linear scalar equations
396
Calculating the upper right derivative D~r of V along the solutions of (5.1.1) and estimating (using 2ab ::; a2 + b2 ),
D+V
.
IJt ::; -2pIP2 X2 (t)
5.1.8
where 1'2 = 1- {t.1bjl
+ 1,811.= IK2(s)lds + 1;.laih + lal 1.= sIK, (s)1 dS}.
5.1.9
It follows from (5.1.2), (5.1.3), (5.1.8) and (5.1.9) that
V(t,x(.)::; V(O,x(.) and therefore
\x(t) +
t.
-1;.
5.1.10
bjx(t - <7j) +,8 1.= K 2( s)x(t - s )ds
ai
lTi x(s)ds - a
f
5.1.11
K1(S)(l. x(u)du )dS\
::; JV(O,x(.». One can derive from (5.1.11) that
(1 - P2)P(t) ::; [V(O,x(.)]!
5.1.12
pet) = sup Ix(s)l.
5.1.13
in which s$t
A consequence of (5.1.12) and (5.1.13) is that x is bounded for all t ~ 0 from which, the boundedness of x will follow; thus x is uniformly continuous on [0,00). Integrating both sides of (5.1.8),
V(t,x(.))
+ 21'11'2
1.'
X2(s) ds :'O V(O,x(.))
5.1.14
which shows that x E L2(0, (0). By Barbalat's lemma (Lemma 1.2.2) the result
0
~~
We next consider the asymptotic stability of the non autonomous neutral equation
x(t)
+ a(t)x(t -
r) + b(t)i:(t - 0')
where a, b are continuous real valued functions.
=0
5.1.15
397
§5.1. Linear scalar equations Theorem 5.1.2. Assume the following:
liminf aCt) t-+oo
limsup[i' laCs t-+oo
t-r
> 0;
5.1.16
+ r) + a(s + 2r)jds + t(t)l) a t +T
+ I,~T Ib(s + r)lds + 4lb(t +" + r)la(t + r)] < 2;
5.1.17
5.1.18
Then every solution of (5.1.15) satisfies lim x(t)
t-+oo
= O.
5.1.19
Proof. We can assume in view of the hypotheses (5.1.16)-(5.1.18) that there exist positive numbers Cl, C2, T and € such that
aCt) ~
€, PI *(t)
~ CI
< 2,
P2 *(t) ~ C2
< 1 for t
~ T
5.1.20
in which PI *(t)
=
i'
t-T
laCs + r)+ a(s + 2r)Jds + tt)l) a t+r
+ I,~T Ib(s + r)lds + 4lb(t + 11 + r)la(t + r) 1'2(t) = 4Ib(t)/Ib(t + (1)1 + IT a(s + r) ds. We first rewrite (5.1.15) in the form
~ [X(t) -
I,>(S + r)x(S)dS] = -aCt
+ r)x(t) -
5.1.21
b(t)x(t - 0-),
§5.1. Linear 3calar equation3
398
and consider a Lyapunov-type functional vet) = v(t,x(.)) defined by
vet)
= vet, x(.)) = [x(t) -
t r a(s
r
+ T)x(s)ds
+ t r a(s + 2T)([ a(u + T)X 2 (U)du )dS + t r jb(s + T)j ( [ a(u + T)X 2 (U)du )dS
5.1.22
+ 2 t.jb{s + uljx 2 (slds + 41~T Ib(s + r + a)!a 2(s + r)x2(s) ds. Restricting our attention only for t 2: T, we calculate (5.1.15) and simplify it so as to obtain
D+v lit
~
-
( 2 x (t)a(t
D;t
V
along the solutions of .
+ r)[2 - J.Ll(t)] 5.1.23
+ :i;2(t -
a)lb(t)![l - J.L2(t)]).
As in the case of the proof of Theorem 5.1.1, it can be shown that x is uniformly continuous on [T,oo) and furthermore aCt + r)x 2 (t) E Ll(T,oo). By Barbalat's lemma (Lemma 1.2.2) lim aCt + r)x 2 (t)
t-+oo
Since
= 0.
5.1.24
I!:: a( t + r) > 0, the result follows from (5.1.24) and the proof is complete.
Corollary 5.1.3. Let a, b, rl, r2 be real numbers such that a > 0,71 > 0, r2 2 If
arl
o.
+ Ibl < 1,
then the trivial solution of :t [X(t l - bx(t is asymptotically stable.
T2l] + ax(t -
T.) =
0
5.1.25
399
§5.1. Linear scalar equations Proof. Proof is based on the Lyapunov functional
t
[x(t)-bX(t-r2)-a Jt-Tl X(S)dsj2
+ albl
Lr,
+ a'
x'(s) ds
Lr, ([
x'(u) dU) ds ..
[]
We omit the details. If r = 0 and aCt) is a constant functional
v(t,X(.),x(.))
==
a in (5.1.15), then one can consider the
= x2(t) +
.!.It a
x2(s)ds
5.1.26
t-cr
and, derive that
vet)
= -ax2(t) - (.!.a )[1 ~ -ax2(t) if !b(t)!
Once again it is possible to show that if Ib(t)! of (5.1.15) satisfy (5.1.19).
b2(t)Jx 2(t - a)
< 1.
< 1 and aCt) == a > 0, then solutions
The result of Theorem 5.1.2 can be used to obtain sufficient conditions for the global asymptotic stability of the positive equilibrium of
N(t)
= rN(t) [1 - N(t -
r)
K
+
aN.(t - a)
1 + N2(t - a)
j.
For more details of this analysis, we refer to Gopalsamy [1992].
5.2. Oscillation criteria In this section we derive sufficient conditions for the oscillation and nonoscillation of first order neutral equations of the form k
x(t) + px(t - r) +
L qiX(t - ai) = O.
5.2.1
i=l
Some of the conditions we obtain are easily verifiable when the parameters are known. We note that the asymptotic stability of the trivial solution of (5.2.1) is not necessarily determined by the negativity of the roots of the characteristic equation. However, the oscillatory nature of (5.2.1) is determined by the roots of the characteristic equation. The following is due to Kulenovic et al. [1987b].
§5.2. 03cillation criteria
400
Theorem 5.2.1. Consider the scalar neutral differential equation (5.2.1) and assume that pER, T 2:: 0, qi > 0 and CTi 2:: 0 for i = 1,2, ... k.
Then a necessary and sufficient condition for the oscillation of all solutions of . (5.2.1) is that the characteristic equation k
).. + p)..e- AT +
L qje-
AU
,
= 0
5.2.2
i=l
has no real roots. Proof. The proof of the necessary part is simple. Suppose that every solution of (5.2.1) oscillates. IT the characteristic equation (5.2.2) has a real root )..0, then (5.2.1) will have the nonoscillatory solution
yet) = e Aot • But this contradicts the hypothesis that every solution of (5.2.1) oscillates. The proof of the sufficiency part is quite involved and therefore we shall restrict here to the special case where p E (-1,0); the reader is referred to the original article of Kulenovic et al. [1987b] for other cases where p ~ -1 and p 2:: o. Let p E (-1,0) and suppose (5.2.2) has no real roots. Let k
F()")
= ).. + p)..e- AT + L qje- AU,.
5.2.3
i=l
Then F(O) =
2::=1 qi >
o~
< ... <
0 and so F()") > 0 for every).. E R. Let us suppose Since F( 00) = F( -00) = 00, there exists a positive constant m (prove this) such that for every ).. E R , CTI
<
0'2
Uk.
k
).. + p)..e- AT + L qi e- AU,
2:: m.
i=l
Suppose (5.2.1) has an eventually positive solution y( t). Let
z(t) = yet) + py(t - r) .
5.2.4
§5.2. Oscillation criteria
401
We first show that z(t) decreases monotonically to zero and lim yet) =0. t-oo . Since
5.2.5
n
i(t) = -
:E qiy(t -
O"i) < 0,
5.2.6
i=l
limt_oo z( t) exists and there are two possibilities: (i)
lim z(t) = -00 or (ii)
t-oo
lim z(t)
t-oo
= L E (-00,00).
5.2.7
Suppose (i) holds; then yet) must be unbounded; also
yet) + py(t - I) < 0
for all large
t.
But, since p E (-1,0),
yet) < (-p)y(t - I) < (_p)2y(t - 21) etc. which contradicts the unboundedness of Yet). Thus (i) cannot hold and therefore (ii) holds. Integrating both sides of (5.2.6) on [tt, (0),
and hence y E L1 [t1' (0). It follows z E L1 [tl' (0), which together with the boundedness of z(t) (and hence that of i(t» will lead to L = O. We ask the reader to provide the extra arguments to support this statement. The conclusion limt_oo yet) = 0 is a consequence of the following due to Ladas and Sficas [1986]. Lemma 5.2.2. Let
J, g: [to, (0) J(t)
~
R be such that
= get) - pg(t - c),
t
~
to.
5.2.8
H p E (0,1), g is bounded on [to, (0) and limt_oo J(t) exists, then limt_oo get) exists. Proof of Lemma. Let limt-+ooJ(t) = points in [to, (0) which satisfy
e and let
{t n } and {sn} be sequences of
§5.2. Oscillation criteria
402
and
lim g(t n )
n-oo
= limsupg(t) = t-+oo
lim g(sn) = liminf get)
n-oo
t-oo
s
= i.
From (5.2.8),
n = 1,2;3, ... and therefore
s or i
~
~
s-i
lim g(tn - c) = - p
n ..... oo
sCi - p). Similarly we find i
~
i(l - p).
Since p E (0,1), .e
.
s<--
= i; this completes the proof of the lemma.
[]
We next observe that z(t) is differentiable and satisfies k
i(t)
+ pi(t -
r)
+ L qiZ(t -
O"i)
= 0.
5.2.9
;=1
Furthermore, if we let
wet) = z(t) + pz(t - r),
5.2.10
then one can verify that w is a twice differentiable solution of (5.2.9) and also
wet) > OJ wet) > O. Assuming p E (-1,0), we define
_ {z(t) = yet) + py(t - r) Zn (t ) Zn-1(t) + PZn-1(t - r)
if if
n=O n
= 1,2,3, ...
5.2.11
In view of the above observations, we have
5.2.12 k
Zn(t) = -
L qi zn-1(t - O"i) i=l
5.2.13
§5.2. Oscillation criteria
403
k
zn(t) + pin(t - r) +
L qiZn(t -
(7i) = O.
5.2.14
i=L
Let An be defined by 5.2.15 To complete the proof we shall derive a contradiction; note that we have supposed that yet) is an eventually positive solution and the characteristic equation has no real roots. To obtain a contradiction we establish that An has the following contradictory properties PI and P2 (such a technique has been previously exploited by Fukagai and Kusano [1983a, b] for other nonneutral equations and we have seen this technique in Chapter 1).
Pl' An is a nonempty and bounded interval of nonnegative numbers. In particular, there exist numbers .AI,.A2 independent of n such that
.AI E An and .A2
f/. An
for n
= 1,2, ...
P2 • There is a positive number fl independent of n such that ,\ E An with
.A
~
.AI ===> (.A
+ fl)
E An+l for n = 1,2,3, ...
We need the following result from Ladas et al. [1983a].
Lemma 5.2.3. Let A and a be positive constants. Assume that u( t) > 0 satisfies the inequality 5.2.16 u(t) + Au(t - a) ~ O.
Then
· 4 3 u(t - a) < Bu(t); B = (aA)2 for t ~ ; .
5.2.17
Proof of Lemma. We integrate both sides of (5.2.16) on [t-%, t] and on [t, t+~]j using the decreasing nature of u,
u(t)-u(t-%) u(t+I)-u(t)
+ +
A%u(t-a) A%u(t-I)
~
0
~
o.
§5.2. Oscillation criteria
404 Hence
A
a
a"2 u (t - a) < u(t - '2)
5.2.18
and
A a a"2u(t - 2") < u(t).
5.2.19
Combining (5.2.18) and (5.2.19) one obtains (5.2.17).
[]
We are now ready to complete the proof of Theorem 5.2.1 for p E (-1,0). Clearly Al = 0 E An for n = 1,2, ... Let us first show that An is bounded from above. From (5.2.12) and (5.2.14) we derive
with which one can show that
Integrating (5.2.13) from t - a to t we get (for a > 0) k
zn(t) - zn(t - a)
+ L qj i=l
1 t
Zn-l(S - O'j)ds
=0
t-o
and this will imply
Thus, for a
= 0' k
we get k
0= Zn(t)
+ L qiZn-l(t -
-
(1i) < Zn(t)
i=l
+ !!..Zn(t) O'k
which proves that
iJ
4
A2 = - = 3 2 (1k (1kqk is an upperbound of An. This completes the proof of Pl' We now prove Pz with J1. = m. Let A E An and set Then
§5.2. Oscillation criteria
405
and so 'Pn(t) is a nonincreasing function for any A E An satisfying (5.2.4). We note that
Zn+I(t)
+ (A + m)Zn+I(t) k
=-
L qiZn(.t -
ai)
+ (A + m)Zn(t) + peA + m)zn(t -
T)
i=I k
= e- At [ -
L qieACTi
ai) + (A
+ m)
i=I
+ peA + m)e AT
T)]
k
::; e->t [ -
s;
~ qie'·; + A + pAe'T + m 1'Pn(t)
e-At[_m
+ m]
O.
5.2.20
This establishes P2 since it follows from (5.2.20) that
A E An
==}
(A + m) E An+I ; n
= 1,2,3, ...
The two properties PI and P2 are contradictory and hence, the result of Theorem 5.2.1 follows. [] The next result provides verifiable sufficient conditions for the oscillation of all solutions of (5.2.22) below and is due to Gopalsamy and Zhang [1990].
Theorem 5.2.4. Assume the following: (i) 0 < c < l. (ii) T 2: 0, a > 0 and P 2: O. (iii) pea> 1 -
C(l + ~) . 1-c
5.2.21
Then eve.ry solution of
x(t) - cx(t - T)
+ px(t -
a)
=0
5.2.22
is oscillatory. Proof. The characteristic equation of (5.2.22) is 5.2.23
§S.2. Oscillation criteria
406
To prove the result, it suffices to show that (5.2.23) has no real roots under the assumptions of the theorem. We note that any real root of (5.2.23) cannot be positive and since f(O) = p, ,,\ = 0 is not a root. Thus, any real root of (5.2.23) can only be negative. Let us set ,,\ = -J.L and show that the equation
- f( -J.L)
= g(J.L) = 1 -
ce llr
-
(
has no positive roots when (5.2.21) holds. Define
=1-
fl(JJ)
pe
IlU
-:-;;-
h
=0
)
5.2.24
and 12 as follows:
pe llU ce llr ; 12(11) = - . 11
5.2.25
It is then sufficient to show that h(JJ) > h(J.L) for J.L > O. Note that fz has a global minimum at (~) and the minimum value is (pea). The strategy of our proof is to show the existence of a suitable curve lying between the graphs of the functions
hand
h.
One such curve is the graph of
f(ll) = (1 - c) - CTIl; J.L > O.
5.2.26
It is easy to see
f(J.L) - !t(J.L)
and hence f(JJ) > !t(Il) for JJ
=1- C= c[e llr -
Consider the value of (12 - f) at J.L =(..1-) Il
J.LT - 1] > 0 for 11 > 0
> O. From (5.2.26) and (5.2.25),
h(J.L) - f(J.L)
[h(J.L) - f(Il)]
CTJ.L - (1 - ce llr )
pe llU
= -.- + CTIl- (1 J.L
= u1a
for 1 ::; a <
5.2.27
c).
00;
= aape(l/a) + (cT/aa) -
(1- c)
(fa
5.2.28
> pa a - (1 - c).
> [(1- c)/ap], then !2(JJ) - f(ll) > 0 for 11 E (0, l/aa). It follows that, for all 11 E (0,6),12(11) - f(ll) > O. Let us now consider 11 :2: p/(l- c) and note Thus, if a
[fz(ll) - f(J.L)] >-L ~ pea + CT-- - (1- c). P
J.I
1-<:
1-
C
But by our assumption (5.2.21),
P- - (1 - c) > 0 pea + CT1-c
5.2.29
407
§5.2. Oscillation criteria showing that h(/-l) - f(J-l) > 0 for J-l
~
pI(l - c). We have shown that
It follows that (5.2.23) has no real roots and this implies all solutions of (5.2.22) are oscillatory; the proof is complete. [) We remark that the condition (5.2.21) is better than the corresponding conditions obtained by Zhang [1989] and Ladas and Sficas [1986]. For instance, in the example
"
x(t) -
~x(t -1) + ~x(t 2
4e
2) = 0
5.2.30
peO' = 1 - c = (1/2) and the results of Zhang [1989] and Ladas and Sficas [1986] do not apply for (5.2.30). But, the condition (5.2.21) can be applied to (5.2.30) in order to conclude that all solutions of (5.2.30) are oscillatory.
Corollary 5.2.5. Assume tbat tbe" real numbers c, T,
o < c < 1;
T ;:::
and
0, 0 <
0'1 ::; 0'2 ::; ••• ::; 0' m;
(t.p;u;)e> c{ 1-
1+
T
0'1, •.• ,0' m,
Pl, ... , Pm satisfy
Pi > 0, i = 1,2, ... , m
~~~Pi}.
5.2.31
Tben all nontrivial solutions of m
x(t) - cx(t - T)
+ I:PiX(t -
O'i) = 0
5.2.32
i=l
are oscillatory. Proof. Details of proof are similar to those of the previous theorem and we omit the proof. [] In preparation for the formulation of our next result, we note the following result of Ladas and Sficas [1986]. "If p, 0', T are positive constants and Q is T-periodic sucb tbat Q
C([to,oo),R +),
0'
> T, P < 1 and lim t-oo
i
t
t-(u-r)
1-p
Q(s)ds> - e
E
§5.2. Oscillation criteria
408 then, every solution of
x(t) - px(t - 7) + Q(t)x(t - (1)
= 0, t > to
is oscillatory". The next result (due to Gopalsamy and Zhang [1990J) provides an alternative and somewhat weaker condition for all solutions of
x(t) - ex(t - 7) + p(t)x(t - (1)
=0
5.2.33
to be oscillatory. Theorem 5.2.6. Assume the following:
(i)
e, 7, (1 are positive numbers, 0 < e
(ii)
p E C(R ,R+), pet + 7)
(iii)
Ro > 1=.£. e'
= pet),
< 1,
(1 ;:::
7 ;:::
O.
tER,
fLrP(S) ds = Po.
5.2.34
Then all nontrivial solutions of (5.2.33) are oscillatory. Proof. Suppose the conclusion does not hold. There exists a nonoscillatory solution x(t) which we shall assume to be eventually positive; then, there is a T > 0 such that x(t) > 0 for t 2: T. We have from (5.2.33),
d dt [x(t) - ex(t - 7)] :::; 0 for t > T
+ (1 = T 1.
Now there are two possibilities:
(i) x(t) - ex(t - 7) :::; 0 for t > Tl (ii) x(t) - ex(t - 7) > 0 for t > T1 • We first show that (i) is not possible. If (i) holds, we have for some constant
8> 0,
x(t) - ex(t - 7) :::; -8 for t > Tl and leading to
x(t) :::; -8 + ex(t - 7)
:::; -8 + e[-8 + ex(t - 27)] :::; -8[c + e 2 + ... + en]
+ en+1x(t -
(n
+ l)e).
§5.2. Oscillation criteria
409
If we let
II 'to II =
su p
tE[Tl-T,Td
I'to (t) I,
then for t 2:: Tl and sufficiently large n, 5.2.35 Since 0 < e < 1, (5.2.35) implies that x(t) will be negative and this contradiction shows that x(t) - ex(t - r) ::; 0 for t 2:: Tl is not possible. Let us then suppose x(t) - ex(t - r) > 0 for t 2:: T and define
wt= ()
x(t-r)-ex(t-2r) >l. x(t)-ex(t-r) -
5.2.36
Dividing both sides of (5.2.33) by [x(t) - ex(t - r)] and integrating,
log[w(t)] = It ( p(s)x(s t-T xes) - exes = lt p(s)[x(s - 0') t-T
0') )dS - r) - exes - 0' - r) + exes - 0' - r)] ds xes) - exes - r) t lt p(s)ex(s-O'-r) 2:: l t-T p(s)w(s)ds + t-T xes) _ exes _ r) ds.
5.2.37
Using the periodicity of p in (5.2.37),
lt xes - r) - exes - 2r) t log[w(t)] 2:: t-T p(s)w(s)ds - e t-T xes) _ exes _ r) ds l = lt p(x)w(s)ds - elt w(s): {log[x(s - r) - exes - 2r)]}ds. t-T
t-T
s
5.2.38
Let t* be a number such that t - r < t* < t and t
I
*
t-T
p(s)ds
R = -2°,
1t
p(s)ds
t*
R = ~.
2
We show that wet) is bounded above. On integrating (5.2.33) over (t*, t),
x(t) - ex(t
~ r) -
[x(t*) - ex(t* - r)]
+
t p(s)x(s - O')ds =
it-
0
§5.2. Oscillation criteria
410 which implies
x(t*)
~ ex(t* -
~ t p(s)x(s it·
r)
>
t
it·
O')ds
p(s) [x (:5 - 0') - exes - 0' - r)]ds
~ [x(t -
0') - ex(i - 0' - r)]
it
5.2.39
p(s)ds
t*
Po
= [xCi - 0') - ex(i - 0' - r)1"2 .
Integrating (5.2.33) over [t - r, t*],
x(tO) - ex(t' - r) - [x(t - r) - exit - 2r)] + l~'/(S)X(S - u)ds
=
o.
As a consequence of the previous equation,
x(t - r) - ex(t - 2r)
~
l-r
~
[x(t* - 0') - ex(t* - r - 0')] ~o
t*
p(s)[x(s - 0') - exes - 0' - r)]ds 5.2.40 •
Since x(t) - ex(t - r) is decreasing, we can combine (5.2.39) and (5.2.40) so as to have
x(t*) - ex(t* - r)
~ (x(t -
r) - ex(i -
2r)](~O)
~ [x(i* -
r) _ ex(t* _ 0' _
r)](~O 2)
~ (x(t* -
r) - ex(t* - 2r)]P! .
Thus w (t
for any i*
~
*)
=
x(t*-r)-e.x(t*-2r) 4 <x(i*) - ex(i* - r) - Pi
5.2.41
T. vVe let
lim inf wet) t-co
and note that f <
00.
=f
It follows from (5.2.38),
5.2.42
411
§5.2. Oscillation criteria We find from (5.2.43) that
log(l) ;:0: P,l - w(O(t))c
Lr :.
1
[log (x( s - r) - cx(t - 2r») ds;
+ w(B(t»clog[w(t ~ Pof + fclog(f) ~ Pof
O(t) E [t -
T,
tl
T)] 5.2.44 ~
f
since
5.2.45
1.
A consequence of (5.2.45) is that
>p ( l_c)log(f) £ _ 0 which implies
(I-C) e --
~
Po =
it
p(s)ds
t-r
and this contradicts (5.2.34); the proof is complete. Corollary 5.2.7. [f0 < c < 1, cr Po T
~
T,
[]
pet) == Po > 0 and if,
> [1 - c,B( c) ]2/ ,B( c)
where ,B( c) is a solution of
1- d
= log[e],
then every solution of
x(t) - cx(t - T)
+ Pox(t - cr) = 0
is oscillatory. Proof. Most of the details of proof are similar to those of the previous theorem and we shall be brief. We have from (5.2.44)
(1 - d) lO~(f)
~ PoT.
We define F as follows:
F(f)
= (1 -
d) log(f) f
5.2.46
§5.2. Oscillation criteria
412
and note that pI (e) = 0 leads to 1 - c1
= log(l).
It is found that j3( c) is a zero of this equation and so
13(0)
= e,
F"(j3(C»
1
< j3(c)
~ e
= -(1 + j3(C)]jj33(C) < O.
It follows that
F(j3(c» = supF(e)
= [l-cj3(c)J2jj3(c)
f;:::l
and hence (5.2.46) implies PoT ~
F(j3(c»
which contradicts our hypothesis.
(]
Theorem 5.2.8. Suppose tbe following bold: (i) c, T, cr be nonnegative numbers, 0 < c < 1,
T
2:: 0, cr > OJ
(ii) P E C(IR, R+), pet) 2:: Po > 0, t E Rj
(iii) poo-e > 1 - c ( 1 + f!!:;: ).
5.2.47
Then every solution of
x(t) - cx(t - T)
+ p(t)x(t -
cr) = 0
5.2.48
is oscillatory. Proof. We shall show that the existence of a nonoscillatory solution of (5.2.48) leads to a contradiction. Suppose y is a nonoscillatory solution of (5.2.48); we can assume that there exists aT> 0 such that yet) > 0 for all t 2:: T. (If yet) < 0 eventually the procedure is similar). One can show that nonoscillatory solutions of (5.2.48) tend to zero as t ~ 00 due to (i) and (ii). Thus we have from (5.2.48),
1 + 1 00
yet)
= cy(t -
T)
+
p(s)y(s - cr)dSj t 2:: T
+ T = to 5.2.49
00
2:: cy(t - T)
Po
yes - cr)ds,
t
> to.
413
§5.2. Oscillation criteria It is not difficult to show from (5.2.49), that
yet)
~
for large t where
=_
cy(t - r) => yet)
(10:( c») ;
II. r iO:
~
o:e-p.t
( /)
= y( to)e p.to
T
5.2.50
5.2.51
•
Define a sequence {Yn(t)} as follows:
Yo(t) == yet) t ~ to cYn(t - r) + Po ftoo Yn(s - (1)ds; Yn+l(t) = { yet) - y(to) + CYn(to - r) + Po ftC: Yn(S - (1) ds;
5.2.52
t:::; to.
It follows from (5.2.52) that
Yn+l(t) :::; Yn(t)
~
... :::; yo(t); t
~
to.
5.2.53
Furthermore from (5.2.50),
Yo(t)
~
o:e-p.t
which implies Yl(t) ~ ae-p.t leading to Yn+l (t) ~ ae-p.t, n = 1,2,3, .... Thus we have from (5.2.53),
ae-p.t :::; Yn+l(t) :::; Yn(t) :::; ... :::; yo(t), t ~ to. By the Lebesgue's convergence theorem, the pointwise limit of {Yn(t)} as n exists and 00
ae-p.t :::; y*(t)
= cy*(t -
r) + Po
1
y*(s - (1)ds
5.2.54 ~
00
5.2.55
where
y*(t)
= n-+oo lim Yn(t).
Thus, y*(t) is a nonoscillatory solution of
x(t) - cx(t - r) + Pox(t - (1) =
o.
5.2.56
But by Theorem 5.2.4, the equation (5.2.56) cannot have a nonoscillatory solution when (5.2.47) holds. This contradiction proves the result. [J We have seen that (5.2.1) can have a nonoscillatory solution when the associated characteristic equation has a real root. It is, however, desirable to obtain
414
§5.2. Oscillation criteria
verifiable sufficient conditions in terms of the parameters of (5.2.1) for its characteristic equation to have a real root. Also in certain cases, such as (5.2.43) when pet) ¢ a constant, the method of characteristic equation is not applicable. We shall now derive sufficient conditions for (5.2.1) and (5.2.43) to have nonoscillatory.solutions. We need the following lemma which combines both the Banach contraction mapping principle and Schauder's fixed point theorem. Lemma 5.2.9. (Nasbed and Wong [1969J) Let X be a Banach space; n be a bounded closed convex subset of X; A, B be maps of n into X sucb tbat Ax + By E n for every pair x, yEn. If A is a strict contraction (i.e. it satisfies tbe condition tbat for all x, yEn,
IIAx - Ayll :::; ,lIx -. yll for some" 0 :::; , < 1) and B is completely continuous (B is continuous and maps bounded sets into compact sets), tben tbe equation
Ax+Bx bas a solution in
=x
n.
Theorem 5.2.10. Assume tbat tbere exists a positive number J1. satisfying ce llr
peller
+ -J1.- < 1. -
5.2.57
Tben (5.2.22) bas a nonoscillatory solution wbicb tends to zero as t
~
00.
Proof. Let C = C([ -T, 00), IR ) denote the Banach space of all bounded functions defined on [- T, 00) with values in III = (-00,00) where T = max( 0", r)j the space C is endowed with the sup-norm. Let n be the subset of C defined by e- 1l1t
:::;
x :::; e- 1l2t j
cx(t - r) < Dx(t);
J1.1
> c
J1.2
> 0 on [-T,oo)}
< D < 1 for t
where J1.2 satisfies (5.2.57) and J.ll > J.l2' Define a map S:
S(x)(t)
= Sl(X)(t) + S2(X)(t)
~
0
n~C
5.2.58
as follows: 5.2.59
415
§5.2. Oscillation criteria where
S1(X)(t) = cx(t - T)
1
00
S2(X)(t)
=
px(s - O')ds.
It is easily seen that the integral in S2 is defined whenever x E n. It follows also from (5.2.58) that 51 is a contraction (due" to c < D < 1) and that 52 is completely continuous. The set
n is
closed, convex and bounded in C. We show that for every pair
x,y E n, For instance, we have for any x, y in
S1(X)(t)
n,
1.
+ 5 z(y)(t) S ce- JL2 (t-r) + p = e- JL2t [ceJL21'
00
e- JL2 (s-cr)ds
+ P~Z2cr 1
5.2.60
S e- JL2t where 112 by assumption satisfies (5.2.57). Also
Sl(X)(t) + 5 2 (y)(t)
~ ce- JLdt - r ) + p ~
ce- JL1 (t-r)
~
e- JL1t
provided 111 is large enough. For any x E n,
c5(x)(t - T) = c[cx(t - 2T) +
S c[Dx(t - T) +
1
00
e-JL1(s-cr)ds 5.2.61
I.:
1:
1
px(s - O")ds] px(s - 0") ds]
5.2.62
00
< D [cx(t":" T)
+
px(s - 0') ds]
= DS(x)(t). From (5.2.60) - (5.2.62), it follows that
S}(x)
Sen)
n.
+ Sz(y) E n if (x,y) E n.
n
n
Thus c By Lemma 5.2.9, the map S : -+ C has a fixed point in which is a nonoscillatory solution of (5.2.22) and the proof is complete. []
§5.2. Oscillation criteria
416
Corollary 5.2.11. Assume that one of the following holds: po-e s:; 1 - ce(r/u) (i) (ii) pTe(r1/r) s:; 1 - ceo Then (5.2.22) has a nonoscillatory solution which tends to zero.
5.2.63 5.2.64
Proof. The conclusion follows from Theorem 5.2.10 for the choices of f-L = ~ and f-L = ~ respectively in (5.2.57). []
In the equation
x(t) -
(21e)X(t - 1) + (;e )x(t -
1) = 0
the condition (5.2.63) of Corollary 5.2.11 is satisfied since po-e
= (1/2),
1 - ce( rJu) = (1/2).
This equation has a nonoscillatory solution x(t) = e- t • Theorem 5.2.12. Let c, T, 0- be nonnegative numbers, 0 < c < 1, T ~ 0, 0- > O. Let p E C(R +, IR +) and pet) -+ Po > 0 as t -+ 00. If there exists a positive number f-L satisfying
5.2.65 then
+ p(t)x(t -
x(t) - cx(t - T)
0-)
= 0
5.2.66
has a nonoscillatory solution.
Proof. Details of proof are similar to those of Theorem 5.2.10 and we will be brief. Define a map S: n -+ C([-T,oo),R) where n is defined in Theorem 5.2.10 for a suitably selected positive number T; let S be as follows: S(x)(t)
= cx(t -
T)
= Sl(X)(t)
+ /.00 p(s)x(s -
+ S2(X)(t)
(say).
o-)ds
5.2.67
417
§5.2. Oscillation criteria To show that Sl(X)(t)
+ S2(y)(t) E n
Sl(X)(t) + S2(y)(t)
for (x, y) En, we have
~ ce- Jl2 (t-r) +
1
00
p(s)e- Jl2 (S-U)ds
eJl2U = ce- Jl2 (t-r) _ 112
1
00
p(s)d(e-Jl2S) 5.2.68
t
= e-p,t [cep'T + PO::'"] ~
e- Jl2t
for all t ~ T where T is sufficiently large (we have used a limiting form of the mean value theorem of integral calculus in the last step in the derivation of (5.2.68». The other details of proof are similar to those of Theorem 5.2.12 and hence we omit them. [) Theorem 5.2.13. Assume that c, 7, a are nonnegative numbers and p E C(R+, R+); also suppose pet) ~ Po. If there exists a positive number 11 satisfying
5.2.69
then (5.2.66) has a nonoscillatory solution. Proof. Let yet) be a nonoscillatory solution of
x(t) - cx(t - 7) + Pox(t - a)
=0
which exists by virtue of (5.2.69) and Theorem 5.2.10. {xn(t),n = 0,1,2, ... } for t E [-T,oo) as follows:
5.2.70 Define a sequence
Xo(t) = yet) CXn(t - 7) + !tOO p(s)xn(s - a)ds; t>T Xn+l(t) = { roo yet) - YeT) + cXn(T - 7) + JT p(s)xn(S - 7) ds;
5.2.71
t
~
T.
Since y is a nonoscillatory solution of (5.2.70), by Theorem 5.2.10 yet) t -4 00 and hence
1
~
0 as
00
yet)
= cy(t -
~ cy(t -
7) +
1
poY(S - a)ds
j
t> T 5.2.72
00
7) +
p(s)y(s - a)ds ; t > T.
§5.2. Oscillation criteria
418
One can now show that {xn(t)} has a pointwise limit for t > T say x*(t) satisfying
x*(t) = cx*(t - r)
+
1=
p(s)x*(s - r)ds;
t >T
5.2.73
and x*(t) 2': ae- JLt for some positive numbers a and J1.. Since x* is a nonoscillatory solution of (5.2.66), the proof is complete. []
5.3. Neutral logistic equation In this section we consider the behavior of solutions of
_duet)
&
= ru(t)
[(1 _u(tK- r)) + ci& (1 _----,--u(tK-----:..r))]
5.3.1
in which c is a real number and r, r, K are positive numbers. It is shown in Pielou [1977], that a modification of the well known logistic equation
d~;t)
= rx(t)
[1 - x~) 1
leads to an equation of the form
dN(t) dt
[N(t)
- - = rN(t) 1-
+ cdN(t) ] dt K
5.3.2
where c, T, K are positive numbers; the modification itself is based on a model of Smith [19631 (for more details see Pielou [1977, p.38-40)). It is possible to consider (5.3.1) to be a generalisation of (5.3.2) incorporating a single discrete delay; it is also possible to generalise further with several discrete and continuously distributed delays. The following analysis of (5.3.1) is based on the results of Gopalsamy and Zhang [1988]. We first consider the asymptotic stability of the positive steady state K of (5.3.1). We assume that together with (5.3.1), initial conditions of the type
u(s)
=
s E [-r,O];
are provided and by a solution of (5.3.1) we shall mean a function u: [-r,oo) H [0,(0), which is absolutely continuous on compact subintervals of [-r, (0) and which satisfies (5.3.1) almost everywhere on (0,00). We leave it to the reader to
§5.3 Neutral logistic equation
419
verify that solutions of (5.3.1) corresponding to the initial conditions of the above type are positive on [0,00) and are defined on [0,00). We introduce a change of variable in the form yet) = log([u(t)/ K)) s.o that (5.3.1) transforms to
~ [y(t) + rc{ eY('-T) -
1 }] = -r [eY(t-T) - 1].
5.3.3
We write (5.3.3) as follows: d
-d [yet) t
+ rcy(t - T) + G(y(t - T»] = -ry(t - T) + F(y(t - T» .
where
G(y) = re( e Y - 1 - y) } F(y) = -r(e Y - 1 - y).
5.3.4
5.3.5
The asymptotic stability of the steady state K of (5.3.1) is now equivalent to that of the trivial solution of (5.3.4). Thus, it is sufficient for us to consider the asymptotic stability of the trivial solution of (5.3.4). In the notation of Hale [1977], (5.3.4) can be written as d
dt [DYt
+ G(Yt)] = LYt + F(Yt)
5.3.6
(where yt(fJ) = yet + fJ), fJ E [-T,O], Yt E C[-T,O]), and D,G,L,F are maps of C[-T,O] into IR defined by
Dr.p
= r.p(0) + rcr.p( -T) T
Lr.p = -rr.p(c- » G(r.p) = rc[e4' -r - 1 - r.p( -T)]
) 5.3.7
F(r.p) = _r[e4'(-r) - 1 - r.p( -T)]. One of the properties known in the literature for systems of the type (5.3.6) (5.3.7) is the following result of Hale [1977, Theorem 9.1, pp.304-305]. "If the trivial solution of the linear system
d 5.3.8 -DYt = LYt dt is exponentially asymptotically stable, then the trivial solution of the nonlinear system (5.3.6) - (5.3.7) is (locally) asymptotically stable." The next result provides sufficient conditions for the linear stability of the trivial solution of (5.3.6).
§5. 9 Neutral logistic equation
420 Theorem 5.3.1. Let
T, T, C
be positive numbers such that
(i) re < 1 2 2 (ii) TT < ,80(1 - r e ) where ,80 is the unique positive root in the interval
,80 tan,8o
+ (r / e) =
5.3.9 5.3.10 (~, 11")
of
O.
5.3.11
Then the trivial solution of (5.3.6) is linearly asymptotically stable.
Proof. The characteristic equation associated with the linear system (5.3.8) is
which on letting )..r =
{l
becomes 5.3.12
We shall first show that under the assumptions made, all the roots of (5.3.12) have negative real parts. For instance, suppose h in (5.3.12) has a zero with a nonnegative real part; let {l
= a
+ i,8,
a 2: 0, ,8
>0
be such a zero of h. It follows from (5.3.12), (a
+ i,8)[eo+ i ,8 + re]
= -rr
5.3.13
which implies that 5.3.14 A consequence of (5.3.14) is that 5.3.15 Separating the real and imaginary parts in (5.3.12), we have, -a[l
+ ree- a cos,8] =
e-a[re,Bsin,B + rrcos,B].
5.3.16
§5.3 Neutral logistic equation
421
IT rr ::; (f)(l - c2r2), it will follow from (5.3.15) that (3 :5 f. We note that the left side of (5.3.16) is nonpositive while the right side of (5.3.16) is positive and this inconsistency shows that if 5.3.17 then h in (5.3.12) cannot have a zero with a nonnegative real part. We now suppose that (5.3.12) has a root J1. We have from (5.3.16), -0:[1
+ rce- o
= o:+i{3 with 0: ~ 0 and {3 E (~, 7r).
cos (31 = e- o 1({3) = e-Orc(cos{3)F(j3)
in which
f(fJ)
= re( cos fJ) [fJ tan fJ
F({3) = {3tan{3 + (~) We shall show for {30 E
(f, 7r)
+
m1
.
5.3.18
5.3.19 5.3.20
satisfying
F({3o) = {30 tan (30
r
+ (-) c
5.3.21
that, F is increasing for ~ < j3 < (30. From this and (5.3.19), it will follow that 1({3) > 0 for ~ < {3 < (30 contradicting the nonpositivity of the left side of (5.3.18). . By direct calculation,
F'({3)
= tan{3 + {3 + f3tan 2 {3.
5.3.22
Let us show that F' ({3) > 0 on (~, 7r). We have for ~7r :5 f3 < 7r, -1 ::; tan,8 < 0 and hence from (5.3.22), F ' (f3) ~ -1 + {3 > 0; for ~ < ,8 < ~7r, tan{3 < -1; thus, tan 2 {3 > 1tan {31 implying F' ({3) > O. This contradiction shows that for all ,8 E (f, ,80), (5.3.12) cannot have a root J1. = 0: + i,8 with a nonnegative real part 2 2 2 2 0:. Since {3 :5 rr /(1 - r c ), it follows that for rr /(1 - r c ) < ,80, (5.3.12) cannot have a root with a nonnegative real part. Let us show that all the zeros of h in (5.3.12) with negative real parts are bounded uniformly away from the imaginary axis. Since h in (5.3.12) is an entire function of J1., the zeros of h cannot have a limit point on the finite part of the
§5. 3 Neutral logistic equation
422
complex plane. Suppose there exists a sequence an of zeros of h such that
/3n > 0, /3n ~ 00
+ if3n,
n = 1,2,3, ...
n = 1,2,3, ... } as n ~ 00.
5.3.23
From (5.3.12) and (5.3.23),
-I rc expan[-(an'/3 + i/3o )] 1
11 + rc exp [-('/3 an + ~ n)] I -
+~ n
and this implies 1 - rc < lim I -
n--oo
n
rTe-O:
(a;
1
+ /3~)"2
1= 0 ;
5.3.24
but (5.3.24) is impossible. Thus, all the zeros of h are uniformly bounded away from the imaginary axis. To complete the proof, we have to show that the trivial solution of the difference equation 5.3.25 is asymptotically stable. The characteristic equation associated with (5.3.25) is
1 - rce-'Xr
= O.
5.3.26
By hypothesis 0 < rc < 1 and this implies that all the roots of (5.3.26) have negative real parts and hence the asymptotic stability of the trivial solution of (5.3.25) follows by Corollary 5.3.1 of Hale [1977, p.286J. Thus, the asymptotic stability of the trivial solution of (5.3.8) follows. Therefore, by Theorem 9.1 of Hale [1977], the local (or linear) asymptotic stability of the trivial solution of the [] nonlinear system (5.3.6) follows.
Corollary 5.3.2. Let r, T1, T2 be positive numbers and c be a real number such tbat 5.3.27 + Icl) < 1.
reT
Tben (5.3.27) will imply tbe linear asymptotic stability of the trivial solution of 5.3.28
423
§5. 3 Neutral logistic equation
Proof. We shall only show that all the roots of the characteristic equation associated with the linear variational system of (5.3.28) given by
5.3.29 have negative real parts and are uniformly bounded away from the imaginary axis, ~e(A) = 0; the other details are as before. Define HI and H2 as follows:
HIP·)=A+r H2('\) = A[rce-).r2
}
+ rTl (e-).r
1
-
It is easy to see that H 1 has no zeros on the half-plane IHI(A)I> IAI on ~e(A) = O. Also IH2 (A)llRe).=o ::; IAI(rTl
5.3.30
1)/ ATI) . ~e('\)
+ rlcl) <
2:: 0 and furthermore,
IA\.
5.3.31
By Ro~che's Theorem, it will follow that all the zeros of H(A) = HI(A) + H2(A) have negative real parts. As in the proof of Theorem 5.3.1, one can show that the zeros of H in (5.3.29) with negative real parts are bounded away from~e(A) = 0 and that the trivial solution of the associated difference equation is exponentially asymptotically stable. The result follows from these details. [J The result of Corollary 5.3.2 can be generalized to the case of several delays as follows: Corollary 5.3.3. . numbers and Cj (j
(k = 1,2,3, ... ,m, j = 1,2,3, ... ,n) are positive are real numbers, tben
Ifrk,O'k,rj,Tj
= 1,2, ... , m) n
m
j=1
k=l
L rjlcj\ + L
rkO'k
<1
5.3.32
will imply the linear asymptotic stability of tbe trivial solution of 5.3.33
We remark that the results of the Corollaries 5.3.2 and 5.3.3 can be further generalized to integrodifferential equations of the form
! [yet) f' +b
k1(s)
{<.(t-,) -I} dS]
= -a
/.=
k2(S) {<.(t-,)
-
I} ds
§5.3 Neutral logistic equation under appropriate assumptions on the kernels kl and k 2 • We leave this to the interested reader. We proceed to discuss the oscillation of positive solutions of (5.3.1) about K. It is convenient to define u so that
u(t) = K[I
+ yet)], t E [-r, 00)
and this means that oscillations of u about K are equivalent to those of y about zero. In terms of y, (5.3.1) becomes
yet) = -[1
+ y(t)][ry(t -
r)
+ rcy(t -
r)]
5.3.34
and we shall be concerned with only those solutions of (5.3.34) which satisfy 1 + yet) > 0 for t > O. We first consider oscillations in the linear equation d dt [vet)
+ revet -
0")]
+ rv(t - r) = O.
5.3.35
Proposition 5.3.4. Let r, r be positive numbers; let 0" be nonnegative and let c be nonpositive such. that
riel < 1
and rer > 1 -
riel.
5.3.36
Then evezy solution of (5.3.35) is oscillatozy. Proof. Suppose the result is not true. Then, the existence of a nonoscillatory solution of (5.3.35) implies that the characteristic equation 5.3.37 has at least one real root and this root cannot be nonnegative by virtue of (5.3.36). We let ,\ = -/-L in (5.3.37) and there exists a positive number /-L satisfying 5.3.38
It is easy to see from (5.3.38) that /-L
> /-Lrlel + rel'T
which implies
(1 -
re JlT
riel) > -
~ rrej
5.3.39
/-L
but (5.3.39) contradicts the second of (5.3.36) and hence the result follows.
[]
425
§5.9 Neutral logistic equation
Corollary 5.3.5. Assume that the conditions of Proposition 5.3.4 hold. Then all solutions of d 5.3.40 dt [vet) + rcv(t - T)] + rv(t - T) = 0 are oscillatory. Proof follows from that of Proposition 5.3.4 if we let
(J
=
T
in (5.3.37).
[]
Corollary 5.3.6. Let r, T, c be positive numbers. Then there exists a nonoscillatory solution of (5.3.40) which tends to zero as t ~ 00. Proof. The characteristic equation associated with (5.3.40) is h()") = A + Arce-).r We note
h(O)
= r > 0,
+ re-).r
1 c
=
o.
5.3.41
1 c
h( -- ) = -- < O.
It follows that (5.3.41) has a real negative root corresponding to which (5.3.40) has a nonoscillatory solution which tends to zero as t ~ 00. [] Proposition 5.3.7. Let r, T be positive numbers; c be a nonpositive number. Then every bounded nonosci1latory solution of
r))
~ (log{l + yet)} + cry(t limsup ly(t)1 <
00,
t-co
= -ry(t - T) }
5.3.42
lim sup ly(t)1 <
00
t-co
satisfies lim yet) =
t-co
o.
Proof. Suppose y is an eventually positive bounded solution of (5.3.42). (We recall that we consider only those solutions which satisfy 1 + yet) > 0 for t 2 0). There are two possibilities:
(i) (ii)
+ y(t)J-/clry(t - T) > 0 log[l + y( t)] - lelry( t - T) < 0 log[l
eventually eventually.
}
5.3.43
§5. :3 Neutral logi.3tic equation
426
In case (i), log[l + yet)) - Iclry(t - r) is decreasing and bounded below; hence, there exists a 2:: 0 such that a = lim {log[l t-oo
+ yet)] - Iclry(t - r)}.
An integration of (5.3.42) on (T, (0) leads to
!roo y( s ) ds <
5.3.44
00.
By hypotheses yet) > 0 for t ;:::: T and y has a bounded derivative. Thus, y is uniformly continuous on [T, (0). By Barbalat's lemma (Lemma 1.2.2) the result follows. In case (ii), we let z(t) = log[l + yet)] ~ rlcly(t - r) and note that since z is decreasing, by the boundedness of y, it will follow as before that yet) ~ 0 as
t
~ 00.
Suppose next y is eventually negative; note that y( t)
> -1
for t ;:::: 0 implies
y is bounded. As before we have two possibilities namely
(iii) (iv)
log[l + yet)] - Iclry(t - r) > 0 log[l + yet)] - Iclry(t - r) < 0
eventually } eventually.
5.3.45
In case (iii), log[l + yet)] - Iclry(t - r) > 0 and is increasing and hence tends to a finite limit as t ~ 00 due to the boundedness of y; the remaining details are similar to the cases (i) and (ii) treated above. [] It is an open problem to remove the boundedness hypothesis in the formulation of the previous result as well as in the next one.
Theorem 5.3.8. Assume that the hypotheses of Proposition 5.3.4 hold. Then every nontrivial solution of the neutral equation d dt[log{l
+ yet)} -Iclry(t -
a)]
= -ry(t -
r)
5.3.46
is oscillatory if both y and if remain bounded on (0,00). Proof. Assume the result is not true; then there exists a nonoscillatory solution y of (5.3.46), which we shall first suppose to be eventually positive. Such a solution satisfies limt_oo yet) = O. We have by integration of (5.3.46)
1
00
log[l
+ yet)] = Iclry(t - a) + r
yes - r)ds
5.3.47
427
§5.3 Neutral logistic equation and this implies that
yet) > Iclry(t - a) + r Define a sequence
/,00 yes -
r)ds, t
~ t*-;
5.3.48
= 1,2,3, ... as follows:
5.3.49
for t E [t* - r - a, t*] . It is fOtllld from (5.3.48) - (5.3.49),
The pointwise limit of the sequence {'Pn(t)} as n
-+ 00
~
exists and if we let
t*,
then
= Iclr
a) + r
/,00
From the definition of {
'Pl(t) > Iclr'Pl(t-a)
> (lclr)2
= e-P.to:
= log(lclr) ; t = rna + to.
; -p,
0:
= eP.t°
Similarly one can show from (5.3.49) and (5.3.36) that
~
Iclr
~ Iclro:e-P.(t-u) + r 2: ae-" [Iclre'" + ~ ~
ae-P.t[lclr + rre] ae-p.t.
100
o:e-p.(s-r) ds
;e, 1 T
5.3.50
§5.9 Neutral /ogi.'3iic equation
428
In a similar way one shows that ~n(t)
> (Xe-~t
,
n
= 1,2,3, ...
and hence
r.p*(t) 2:
(Xe-~t
for t
> t*.
Thus, it follows that r.p* is an eventually positive solution of (5.3.50) satisfying
~ [r.p*(t) -Iclrr.p*(t -
0-)] = -r
5.3.51
But, (5.3.51) cannot have an eventually positive solution since the characteristic equation of (5.3.51) cannot have a real root when (5.3.36) holds; this contradiction shows that (5.3.46) has no eventually positive solution. Let us now suppose that (5.3.46) has an eventually negative solution say for t ~ t* - r - a. Then x(t) ~ as t ~ 00 and ,hence as before
x(t) <
°
°
10g[1
+ xCi)] =
Iclrx(i - a) + r
/,00 xes - r)ds ; t > t* .
5.3.52
Since -1 < xCi) < 0, it follows from (5.3.52) that
x(t) > Iclrx(t - a) + r
/,00 xes - r)ds
5.3.53
from which one can derive a contradiction as before showing that an eventually negative solution of (5.3.46) cannot exist. [] Theorem 5.3.9. Let r, r be positive numbers; let c be nonpositive and let a be nonnegative such that there exists a. positive number f-t satisfying Iclre~O'
re~T
+ -- < 1.
5.3.54
P,
Then there exists a nonoscillatory solution of
~ (Z09{1 + y(i)} -Iclry(t -
a)) = -ry(t - r).
5.3.55
Proof. Let {3 be a positive number such that
5.3.56
429
§5.9 Neutrallogi.3tic equation Define
n as follows: 5.3.57
where e is a small positive number and PI is a large positive number. Let F denote a map of n into .C([to,oo),R) defined by
yet) -Iog{l + yet)} + lelry(t - 0") + r ft yes - r)ds t > to + 0" + r y(to + 0" + r) -log{l + y(to + 0" + r)} + Iclry(to + r) +r ft';+C7+T y( s - r )ds tE[t o,to +(7+rJ OO
F(y)(t) =
5.3.58
where C([to, 00), R) is the space of all bounded continuous functions on [to, 00) endowed with the supremum norm. We shall show that F maps n into itself and that F is a contraction. First we note that y - 10g[1
+ y] > 0 for y > 0 and hence
F(y)(t) ;::: lelree-pdt-O') = ee -Ill t /e/relll 0'
;::: ee- JLlt for t
5.3.59
~ to
+ (7 + r
if PI is chosen to satisfy lelrelllO' ;::: 1. Now
/.00 ryes - r)ds ::; yet) ()y(t) + lelrj3e- p(t-O') + r /.00 j3e- p(s-T)ds 1 + ()y(t)
F(y)(t) = yet) -log[l + yet)]
+ /elry(t -
0")
+
t
= j3e-
pt { sup O,!:8y'!:f3
::; j3e-pt
( -()y() 1) . + Y
+ (lelrepO' + re
PT
P
5.3.60
)}
{j3 + (le/reIlO' + r~r) }
::; j3e- pt where 0 < () < 1. It follows F(n) c n. One can show, as in the case of the derivation of (5.3.60) that
II F(y,) -
F(V2)
11<
[.8 + (lc1re"d + r~T)] II YI -
V2
II .
5.3.61
§5.3 Neutrallogi3tic equation
430
Thus F is a contraction on n. Since n is a closed subset, F has at least one fixed point y* E n. Such a fixed point is a solution of
1
00
10g[1
+ yet)] -Ic!ry(t - 0') =
r
yes - r)ds.
5.3.62
It follows that y* is an eventually positive solution of (5.3.55).
[]
Recently Kuang and Feldstein [1991] have studied the neutral logistic equation (5.3.1) and obtained sufficient conditions for the boundedness of solutions.
5.4. A" neutral Lotka-Volterra system Application of neutral delay differential equations in modelling biodynamic processes has not been developed to any degree of maturity and this area will see more development in the future. It is expected that the following sections will be useful to those interested in such a pursuit. In this section, we derive sufficient conditions for the linear asymptotic stability of the positive equilibrium of the neutral Lotka-Volterra system
Nj(t)
= Nj(t) [r, -
t
O<'iNi(t - T'j) -
)=1
t
(3,i Ni(t - cr'i)]
)=1
5.4.1
i = 1,2.
We assume, (5.4.1) is provided with a set of initial conditions
= 'Pj(t); i = 1,2,
Nj(t) 'Pi(O) > 0;
Ni(t)
=
i
= 1,2, t E
[-tt,O]
1
'Pi E C([-tt,0],R+)nC ([-tt,0],IR+)
5.4.2
where tt = max[rij,O'ij; i,j = 1,2]. We suppose, furthermore, that (5.4.1) has a positive equilibrium N* = (Nl*' Ni), Ni > 0, Ni > 0 satisfying 5.4.3 Since solutions of (5.4.1) - (5.4.2) remain positive for t
~
0, we can let 5.4.4
and obtain for i
= 1, 2
§5.4. A neutral Lotka- Volterra system
431
It is known from Theorem 9.1 of Hale [1977, p.304), that the trivial solution of 5.4.5 is linearly asymptotically stable if the trivial solution of the linear system
5.4.6
obtained from (5.4.5), where Pij = f3ijNj , aij = aijNJ ' i,j = 1,2 is asymptotically stable. To derive sufficient conditions for the trivial solution of (5.4.6) to be asymptotically stable we proceed as follows; we first rewrite (5.4.6) in the form
~ [XI(t) + P11XI(t -
all
(111) +PI,X,(t - <11')
It t-Tll
Xl (s )ds
- al2lt X2( S)dS] t-Tn
= - [aUXI(t)
+ a I2X 2(t)] 5.4.7
~ [x,( t) + PI,XI (t -
(121) + p"x,( t - (122)
- a21lt t-T21
Xl (s )ds
- a22lt X2( S)dS] t-Tn
= - [a,lxI(t) + a"x,(t)]. We define a functional VI = VI(Xl(.) ,
X2(.
))(t) on the solutions of (5.4.7) as follows:
V, (XI(')' x,(.))(t) = [Xl(t) + P11XI(t - (711) + P12X,(t - <71') -
all
It xl(s)ds - al21t X2(S)ds]2 t-Tll t-Tn
+ [X2(t) + P21 XI(t -
a21
t
Jt-T21
Xl (S)
O'2t)
+ P22 X2(t -
0'22)
ds - a221t X2( s) ds]2 t-Tn
5.4.8
We assume in the following that all
> 0,
a22
> 0.
5.4.9
§5.4. A neutral Lotka- Volterra system
492
Calculating the rate of change of VI along the solutions of (5.4.7),
dVI dt
= 2 [ Xl(t)
-all l
+ PllXl(t -
0"11) + P12 X2(t - 0"l2)
t
Xl(S) ds - a121t
+ 2 [x2(t) + P21 X1(t - a211t
X2(S) dS] [- allx1(t) - a12 X2(t)]
t- T 12
t-Tll
.
0"2I) + P22 X2(t - 0"22)
x1(s)ds - a221t
t-T21
X2(S)dS] [- a21xl(t) - a22X2(t)].
t-T22
5.4.10 One estimates the right side of (5.4.10) and derives
dV1 dt
~ -[Xl(t) X2(t)] +
[all a12 + a21
a12 +a21 ] [Xl(t)] a22 x2Ct)
xil- an + an{lPn 1+ IPl21) + an{anTn + lal2 h2) + Ia21 1{Ip21 1+ Ip221) + Ia21 In all hI + a22T22)]
+ x~ [ - a22
+ a22(1 P21 1+ Ip22 I) + a22(1 a21 1721 + a22 7 22)
+ Ia12I(IPn 1+ IPlll) + Ia'21( all Tn + Iall h2)] +8 in which
8 = alllplllxi(t - 0"11) + alllp12Ix~(t - 0"12)
+ ail 1t
xi(s) ds
+ all Ia1211t t- T 12
t-Tll
+ 1a12IIPll Ixi(t +laI2Ia111t
x~Cs) ds
O"ll) + 1al21lpl2lx~Ct - 0"12)
Xi(s)ds+ai21t
t-Tll
x~(s)ds
t - T l2
+ a22lp21l x iCt - 0"2d + a22Ip22Ix~(t - 0"22)
+ a221 a21 11t
xiCs) ds
t-T21
+ Ia21 IIp21 Ixi(t +
a~11t
t- T 21
+ a~21t
x~Cs) ds
t-Tn
0"2t} + I a21 Ilp22Ix~(t - 0"22)
xrCs) ds
+ 1a21 la221t t- T 22
x~(s) ds.
5.4.11
§5.4. A neutral Lotka- Volterra .sy.stem
We choose another functional
V2
=
V2(Xl(,),X2(.»(t) such that
V2(XI(')' X2(.»(t)
= WI
where
WI =
and
W2 =
[au1pu 1+ 1a12l1pu
+ W2
5.4.12
I] 1~.H x~(s) ds
+ [anIP" 1+ 1a" lip" I] 1~."
x;(s) ds
+ [au1plZ 1+ 1a12 IIp12 I] 1~."
x~(s) ds
+ [ani Pnl + Iaullpnl] 1~."
x~(s) ds
(a~1 + au Ia121) LTH ([ x;(u) dU) +
433
ds
(a~1 + a221 a21 I) LT" ([ X~(U)dU) ds
LTU ([ x~(u)dU) ds + (I Ian + a~2) LT" ([ x~(uldU) ds. + (aul al21 + a12) a21
The functional V = V(Xl, X2)(t), defined by
V( Xl, X2)(t)
=
Vi (Xl, X2)(t) + V2(Xll Xz)(t)
5.4.13
;. has the following properties: if {Xl (t), X2(t)} denotes any solution of (5.4.7), then
Vex}, xz)(t)
~
0 for
t
~
0
dV(xt,xz)(t):::; -fxI(t) X2(t)] [ all dt a12 + a2l + J-llxi(t) + Jl2X~(t)
5.4.14
in which 1-'1 = [ -
au + all(lpu 1+ IPnl) + au( a","" + Ial2 !rI2)
+ la21lClp211 + IPzz /) + laZII(l a21l Tzl +aZz T 2z) + IPll I(all + Ial2!) + IpZl l(aZ2 + IaZl I) + au '"u (a" + 1al21) + 1a21 !rz,(1 a21 1+ a22l]
5.4.15
§5.4. A neutral Lotka- Volterra system
and j.l2
= [-
a22
+ a22(1 P21 I + IP221) + a22(1 a21 IT21 + an T22)
+ Ia121(lpll 1+ Ip12 I) + Ia121(allTl1 + Ia12I T12) + Ip121(all + Ial21) + Ip221(a22 + Ia21 I)
5.4.16
+ I aI2I T12(all "+ Ial2!) + a22 T22(1 a21 1+ a22)]' The foregoing preparations enable us to derive the next result: Theorem 5.4.1. Suppose the following are satisfied: (i) The quadratic form 2
L
aijXiXj
= (Xl
i,j=l
(ii)
is nonnegative for {Xl,X2} E IR X IR. aij, Pij , Tij (i,j = 1,2) satisfy all
where
j.ll
> 0,
a22 > 0,
. j.ll
< 0,
j.l2 < 0
5.4.17
and j.l2 are defined by (5.4.15) and (5.4.16) .
(iii) Iplli II ( Ip211
IP 12 I) Ip22 I
+(
allTll la211T21
5.4.18
Then the trivial solution of (5.4.1) is locally asymptotically stable in the sense that all solutions of (5.4.6) satisfy lim
t-+CXl
[xi(t) +x~(t)]
5.4.19
= 0.
Proof. We consider the Lyapunov functional V where
defined by (5.4.8) and (5.4.12) and note that V(Xl, X2)(t) tion, we have
~
O. From our prepara-
5.4.20
435
§5.4. A neutral Lotka- Volterra system
Since the quadratic form in (5.4.20) is nonnegative, (5.4.20) leads to 5.4.21 in which 5.4.22 As a consequence of (5.4.21) and the definition of V, 2
2
I Xi(t) I s ~ Ipij II Xj(t -
O";j)
t
.
I + [;, Iaij 11_r;; I Xj(S) Ids + (Vo)~ .
5.4.23
We let
mi(t) =
sup
IXi(S) I
5.4.24
sE[-Il,t]
and note that (5.4.23) implies
m1(t)] [ m2(t) where Q _ -
Since
~ Q [m1(t)] + (Vo)~ m2(t)
[1] 1
[IPll I IP121] + [ aUi'll Ip21 I Ip22 I la211i'21
5.4.25
5.4.26
IIQII < 1, we derive from (5.4.26), 5.4.27
The boundedness of XI(t), X2(t) for t 2:: 0 follows from (5.4.27) and (5.4.24). The boundedness of Xl and X2 on [0,00) is verified similarly. For instance, we let
Zi(t)=
sup
IXi(s)1
5.4.28
sE[-Il,t]
and, note from (5.4.6) that
Since IIQII < 1, we have IIPI! = II [pij] 3;2 remain bounded on [-tt, 00).
I! < 1. It follows from (5.4.29)
that Xl and
§5.4. A neutral Lotka- Volterra system
436
It is now easy to see from (5.4.21) that xHt)+xHt) E L 1 (0, 00); the boundedness of the derivatives of Xl and X2 on (0,00) will guarantee the uniform continuity of Xl and X2 on (0,00). By Barbalat's lemma (see Lemma 1.2.2), it follows that
xi(t) + x~(t)
-+
0
as
t
-+
00
and this completes·· the proof.
[]
A number of other plausible neutral differential models of population systems are formulated in the exercises. For a discussion of an n-dimensional neutral Lotka - Volterra system we refer to Gopalsamy [1992] (see also Kuang [1991]). Several results related to stability switching and absolute stability of linear neutral differential systems can be found in Datko (1978] and Freedman and Kuang [1991J.
5.5. X(t)
= AX(t) + BX(t -
7') + CX(t - I)
The method of Lyapunov functionals and Lyapunov functions has proved useful in the stability analysis of both ordinary and delay differential equations; however, application of Lyapunov technique to neutral differential equations has not yet reached any level of completeness. Results related to Lyapunov functionals for neutral differential equations of the type
for 0 < I(t) ~ 7'0, i,j = 1,2,"" n can be found in Misnik [1972) and EI'sgol'ts and Norkin [1973). Quadratic Lyapunov functions have been exploited by Khusainov and Yun'kova [1988], Li Senlin [1983, 1987] and Wu [1986) in a comparative study of the stability characteristics of the linear equation
x(t) = Ax(t) + Bx(t - I) + Cx(t - 7')
5.5.1
and the perturbed quasilinear system
x(t) = Ax(t) + Bx(t - 7') + Cx(t - I) + Q(x(t), x(t - 7'), x(t - 7')). If the system (5.5.1) is stable when 7' = 0, can we find an estimate of the delay I for the stability of (5.5.1) to continue to hold when 7' =f O. We consider this aspect briefly and refer to Khusainov and Yun'kova [1988] for other results related to nonlinear perturbations of stable systems.
§5.5. XCt) = AX(t) + BX(t - T)
+ CX(t - T)
437
It is known from the Lyapunov stability of ordinary differential equations, that if the trivial solution of
x(t)
= (1 -
C)-l [A
+ B]x(t)
5.5.2
is asymptotically stable, then for every positive definite matrix P there exists a positive definite matrix H satisfying
One can, for convenience, choose P to be the identity matrix 1 in (5.5.3) and we suppose we have done so. We denote the smallest and largest eigenvalues of H respectively by Amin(H) and Amax(H). We consider a Lyapunov function v defined by 5.5.4 where H denotes a solution of (5.5.3) with P and avO' defined by vO' = {xlv(x)
< a},
= I.
avO' = {xlv(x)
We also consider the sets vO'
= a},
a E (0,00).
The following preliminary lemmas are needed in the proof of Theorem 5.5.4 below. Lemma 5.5.1. If x(t) is a solution of (5.5.1) satisfying Ilx(t)" < 8 for to t :::; to, tben for to .< t :::; to + T, X satisfies
IIx(t)1l <
T ~
(1 + 2l1CII + IIBIIT )8e IlAIiT •
Proof. We rewrite (5.5.1) in the form
x(t)
= x(t o ) + C[x(t +
For to
T) - x(to - T)]
t [Ax(s) + Bx(s - T)] ds.
ito
< t :::; to + T, we have
IIx(tlll < 6+ 211C116 + IIAII/IIX(Slll ds + IIBlir which on application of Gronwall's inequality will lead to the assertion.
[]
§5.5. X(t)
498
= AX(t) + BX(t -
1') + GX(t - 1') for t > to
Lemma 5.5.2. If a solution xCt) of (.5 ..5.1) satisfies xCt) E ova l'
x(S)Ev a
and
II x (t)
whenever
+
for to-1':SS
IIx(t)11 < 8
for to -
l'
411GII8 +
IIAII+IIBII~ 1 -IIGII )..min(H) l'
Y
5.5.5
:S t :S to'
Proof. As before we rewrite (5.5.1) so that
x(t) = x(t - 1') + G[x(t - 1') - x(t - 2r)]
tT
[Ax(s)
+ Bx(s - r)] ds.
It follows from the above
Ilx(t) - x(t - 1')11 :SIIGllllx(t - r) - x(t - 2r)1I
+ II All
LT II x(s)1I
+ IIBII
ds
LT Ilx(s - r)1I
ds.
Sincex( s)E va and x( S - r) E va, we derive from the quadratic function estimates of v, )..min(H)lI x Il 2 :S vex) :S )..max(H)lIxll\ that 5.5.6
/lx(t) - x(t - 1')11 ::; IIGllllx(t - r) - x(t -2r)1I
J
+ (II A II + IIBII) )..mj~(H) r.
The following estimation completes the proof:
IIx(t) -
x(t - r)1I ::; II Gil [IIGllllx(t - 21') - x(t - 3r)1I
=
+ (II AII + II BII)J Am;~(H) r] + (II AII + II BII)J Am;~(H) r IIG1I211x(t - 2r) - x(t - 31')11
+ (1 + IIC11)(II AII + II BIOJ Am;~(H) r ::; IIGlin IIx(t - n1') - x(t - (n + 1)1')11 + (1 + IICII + ... + IICll n - 1 ) (II AII + II BIOJ Aml~(H) r ::; IIGlln IIx(t - n1') - x(to)1I + IIGll n ll x (t o) - x(t - (n + l)r)1I + (1 + IlCU + ... + IlCU
1
n -
)
(IIAII + IIBII)
IIAII+II B II) ~
< 411 G II8 + ( 1 -IIGII
V)..min(H) 1'.
In the next result, we estimate the derivative.
JAml~(H)
r
[]
§5.5. X(t) = AX(t) + BX(t - r)
+ GX(t -
439
r)
Lemma 5.5.3. Let x(t) be a solution of (5.5.1) satisfying
xCt) E
avo
for t
x(s)EvO for
> to + r and
to-I~s
Then
IIx(~) - x(t -1)11 < 4 11 GII(1 + IIAII + IIBII)8 l-IIGII IIAII + Il B II)2 j a + ( I-IIGII·. VAmin(H) 1
5.5.7
whenever
IIx(t)lh = max{lIx(t)l/, I/x(t)lI} < 8
for
to - r
~
t
~
to.
Proof. It follows from (5.5.1) that
IIx(t) - x(t - r)1I
Using the previous estimate for
~
IlGllllx(t - r) - x(t - 2r)1/ + IIAllllx(t) - x(t - 1)11 + IIBllllx(t - r) - x(t - 2r)lI.
Ilx(t) - x(t - 1)11,
IIx(t) - x(t - 1)11 < IIGllllx(t - I) - x(t - 2r)1I IIAII+IIBII ~ + (IIAII + IIBII) [411GII8 + I-IIGII VAmin(H) r
1 .
As before, we derive by iteration
IIx(t) - x(t - r)1I < IIGlin IIx(t - nr) - x(t - (n + I)r)1I + (1 + IIGII + ... + IIGll n- 1 ) (IIAII + IIBII) x 4 G1I8 [ 11
+ IIAII + IIBII ~ 1 -IIGII
VAmin(H)
I]
< 411GII8 + IIAII + IIBII [4 11 G1I8 + IIAII + IIBII ~Il 1 -IIGII
from which the result follows.
1 -IIGII
VAmin(H)
[]
The next result provides an estimate of the delay for the stability of the trivial solution of (5.5.1) to continue to hold for 1 =j:. 0, if it holds for 1 = O.
440
§5.5. X(t)
= AX( t) + BX(t -
r) + CX(t - T)
Theorem 5.5.4. Tfthe trivial solution of (5.5.1) is asymptotically stable forT = 0, tben tbe trivial solution is stable [or T E (0, To) wbere 1 -IICII [IIH(I _ C)-lCI"AII + IIBII 2(IIAII + IIBII) 1 - IICII
To
+ IIH(I - C)-1
Bllj-1
)..min(H) )..ma.x(H)'
5.5.8
Proof. We rewrite the system (5.5.1) as follows:
(I - C)x(t) = C(x(t - r) - x(t)]
+ (A + B)x(t)
+ B[x(t - T) - x(t)] and then convert it to the form :
x(t) = (I - C)-lC[X(t - T) - x(t)]
+ (I - C)-1 (A + B)x(t) + (I - C)-1 B[x(t - T) - x(t)]. Calculating v( x) where v( x) = x T H x, along the solutions of (5.5.1)
v(x(t))
= -xT(t)x(t) + 2x T (t)H(I - C)-lC[X(t + 2xT (t)H(I - C) -1 B [x(t -
T) - x(t)] T) - x(t)].
Estimating v( x(t)),
v(x(t)) ~ -lIx(t)1I 2
+ 211H(I - C)-lCIi (lIx(t) - x(t - r)II) IIx(t)1I + 2I1H(I - C) -1 BII (IIx(t) - x(t - r)lI) IIx(t)ll.
5.5.9
Choose {; such that 5.5.10 It will follow from Lemma 5.5.1 that any solution x satisfies
if
x(t) E va
for
to -
T ~
t
~
to + T
IIx(t)1I < {;
for
to -
T
~
t
~
to.
§5.5. X(t) = AX(t)
+ BX(t - T) + CX(t - T)
441
Suppose now there exists a t* > to + T for which x(t*) E 8v a . We have from the above Lemmas 5.5.2 and 5.5.3 that
v(x(t*)) < {[ -
~ + 2(IIH(I VCfH)
+ IIH(I -
T
+ IIBII l-IICII
5.5.11
IIAII + IIBII T ] va 1 -II C II V>'min(H)
C)-l BIl)
+ 811CII [IIH(I - C)-ICIi If
C)-lCII" AIi
(1 + ";I~~~~") + IIH(I - C)BII] 8}X(t*).
< To, then one can choose 8 in (5.5.11) small enough to satisfy v(x(t*)) < o.
Therefore, the vector x(t*) will be directed towards the interior of va. Hence, x(t) does not leave va for t > t* whenever IIx(t)11 < 8 for to - T ::; t ::; to and this
0
c~pk~t~pro~
In the next two results, we derive sufficient conditions for the linear autonomous neutral systems of the type (5.5.1) to be stable independent of the size of delay. Theorem 5.5.5. Consider the linear equation
x(t) = Ax(t) + Bx(t - T)
+ Cx(t - a),
T,a
>0
5.5.12
in which A, B, Care n x n constant matrices. Suppose there exists a differentiable function V : IRn 1-+ III satisfying the following: 5.5.13
1-11 C II> 0,
r_IIA II + II B II \. - 1-11 C II
1\7 V(x)1 = I ( 8V)T 8x I ::; )..Ixl; aV)T ( ax Ax ::;;
_txT x ,
5.5.14
5.5.15
5.5.16
442
§5.5. X(t)
= AX(t) + BX(t -
7) + CX(t -7)
If
5.5.17 then the trivial solution of (5.5.12) is exponentially asymptotically stable for all delays 7 and (J.
Proof. We have from (5.5.12),
+ Bx(t - 7) + Cx(t - 0")1 IIlx(t)l+ II B IIlx(t - 7)1+ II C
Ix(t)1 = IAx(t)
::;11
A
5.5.18 IIlx(t - 7)1·
We let met)
= sup
net) = sup Ix( s)1 s9
Ix(s)l,
s~t
and note that net) ::; Km(t).
Now d dt V ( x)
= (aV)T ax x (t ) aV)T Ax(t) + (aV)T ax Bx(t - 7)
= ( ax
:::: -exT x +
-.e
::; TV(x)
I~: I[II
+ A II
+ A II C II (
-.e ::; TV(x)
B
II
Ix(t - r) 1+
II GIl
Ix(t - r) I]
0")
5.5.19
II sup Ix(s)12 s~t
sup sE[t-(U+T),t]
+ A II
+ A II
B
+ (aV)T ax Cx(t -
C
B
II
II -1 a
(
sup
5.5.20
Vexes»~ )
sup sup
V (x ( S ) ))
5.5.21
sE[t-(U+T),t]
-.e = TV(x(t) + ;;A ( II B II + II C II K ) V(x(t) where
iT =
IX(S)I)
sE[t-(U+T),t]
sE[t-(U+T),t]
K ( a
IX(S)) (
sup sEt t-(U+T ),t]
Vexes)).
5.5.22
§5.5. X(t) = AX( t)
+ BX(t -
r)
+ GX( t -
r)
443
The result follows from (5.5.22) and (5.5.17) by virtue of Halanay's lemma (see Lemma 3.6.12 in Chapter 3) and this completes the proof. [] It is posssible to conclude that when (5.5.13) - (5.5.17) hold, all the roots of det[A + Be- zr
+ Gze-zO'] =
0
have negative real parts. The reader should try to provide an independent proof of this fact. The next result provides an alternative (and easily verifiable) set of sufficient conditions for the trivial solution of (5.5.12) to be asymptotically stable. Theorem 5.5.6. (Li-Ming Li [1988]) Suppose that the coefficient matrices A, B, G of (5.5.12) satisfy
IIGII < 1,
and
j),
(A)
+ IIBII + IIA"IIIIGII < 0 1-IIGII .
5.5.23
Then the trivial solution of (5.5.12) is asymptotically stable and there exist M ~ 1,0:: > 0 such that
for every solution x(t, <1» of (5.5.12) with
x(t) = (t), x(t) = ;Pet), t E [ - (r
+ 0'),0] and 1I<1>IIO'+r =
sup I( s )1. sE[-(O'+r),O]
Proof. We recall that the measure j),(A) of a matrix A is defined by
j),
(A)
=
lim III + hAII- 1 h-O+
h
and note that for t E [0,00),
dll~(t)1I t
_ j),(A)lI x(t)1I
= h-O+ lim -hI [llx(t + h)Il-Il(I + hA)lIlIx(t)lI]
~ h-O+ lim -hI [lIx(t + h)lI- (I + hA)X(t)lI] ~
IIBllllx(t - r)1I
+ IIGllllx(t -
0')11.
§5.5. x(t) = AX(t) + BX(t - r)
444
+ CX(t -
r)
Therefore, it follows from the above inequality
d t
-d IIx(t)1I ~ Jl(A)llx(t)II+IIBIi
sup SE[t-{CT+r),t]
II x(s)II+lIclI
sup SE[t-{CT+r),t]
II x(s)ll.
5:5.24
We have directly from (5.5.12),
o ~ lIi(t)!1
~
!I A lIllx(t)11
+ IIBII
II x (s)/I
sup SE[t-{CT+r),t]
+ IICII
sup
lI i (s)".
SE[t-(CT+r),t]
5.5.25
Define PI and P2 as follows:
t E [-(0"
Pl(t) = IIx(t)lI,
+ r), (0).
We derive from (5.5.24) and (5.5.25) that
PI(t) ~ Jl(A)Pl(t) + IIBllpl(t) + IICI/p2(t)
5.5.26
o ~ IIAllpl(t) - P2(t) + IIBllpI(t) + IICllp2(t) where
Pi(t)
=
Pi(S),
sup
i
= 1,2.
sE[ t-( CT+r) ,t]
It is a consequence of the assumption in (5.5.23), that the eigenvalues of the matrix
P where P = [
Jl(A) + IIBII
IIAII + /lBII
IICII -1
+ II CII
1
5.5.27
have negative real parts. Since the off-diagonal entries in (5.5.27) are nonnegative, the matrix P in (5.5.27) is such that, -P is an M-matrix (P is also known as a stable Metzler matrix). It is known from the theory of M -matrices (for details see Chapter 3), that there exist positive numbers a1, a2 such that 0:1
(J.t(A)
+ II B II) + D:211CII < 0
O:I(IIAII + IIBI!) + We choose positive numbers 0:0:1
0:
0:2(-1
and k such that
+ Jl(A)O:l + IIBIIO:Ieo(r+CT) + IICII0:2eo(r+CT) <
HAIIO:l
5.5.28
+ IICII) < O.
+ IIBIIO:l eO{r+CT)
- 0:2
kaje-o{r+CT)
0)
+ IIClla2eo(r+CT) < 0
> 1;
i
= 1,2.
5.5.29
§5.5. XCi)
= AX(t) + BX(t -
For a sufficiently small positive number
Wi(t) = kai
[tpi(O) + f]
€,
r)
+ CX(t - r)
445
we define
t>-(a+r).
i=1,2;
e-atj
5.5.30
)=1
It is easy. to see that
= 1,2,
i
t E
[-(a
+ r),O).
We want to prove
Pi(t) < wi(i),
i
= 1,2;
t E [0,00).
5.5.31
If (5.5.31) does not hold, then one of the following would occur; there exists a t1 > such that
°
and Pi(t)~Wi(t),
P1(t1) = Wl(t1), P1(t1)2:: Wl(tl)
t~tl,
i=1,2.
We also have
+ II B ll wl(t 1 ) + IICll w2(t1) J.l(A)W1(tt} + IIBllwl(t1 - (r + 0"» + IICllw2(t1 - (r + a»
Pl(td ~ J.l(A)Pl(td =
2
Wl(tJ)
= -kc~la[?=pj(o) + €]e-
at1
)=1
> k (1'( A)al + liB lI a l ea(r+·) + IIClla. e'>(r+')] [
= J.l(A)W1(t 1) + IIBIIWl(tl
- (r
t,
+ 0"» + IICIlW2(tI -
(Pi (0) + f)e -at, ]
(r + a»
=Pl(t 1 ) and this contradicts PI (tt) 2:: WI (tt). The other possibility is that there exists ail>
P2(t 1 )
= W2(tI)
and
Pi(t)
~
Wi(t),
°
such that
i = 1,2;
t
~ tl
.
It is found from (5.5.26) that
P2(tI) ~ I/AI/ Wl(td + I/ B llw1(tl - (r
(t,P;(O) + f) < k(tft;(O) + f)e-
+ 0"» + /lC/lW2(t1 - (r + a»
(e- at , (IiAllal
:s; k
J=1
= W2(tl)'
at1
(0"2)
+ IIBlial ea(r+.) + IIC lh ea (r+Q)) ]
446
§5.5. X(t)
= AX(t) + BX(t -
r)
+ CX(t - r)
Thus, P2(t 1 ) < W2(t 1 ) and this is a contradiction; and hence (5.5.31) follows. We also note from (5.5.30) and (5.5.31) that
t> -(r + 0"). This completes the proof.
[]
The following are examples of population model systems subject to feedback (indirect) controls. It is of interest from the viewpoint of modelling population systems to discuss the existence of positive steady states and their stability characteristics. Assume that all the parameters appearing in the following are positive and the kernels of integrals are nonnegative and normalised. We ask the reader to examine the local asymptotic stability of the positive steady states of the following equations and also examine whether delay independent stability is possible. (At this time there exists no technique for the investigation of the persistence of population systems modelled by neutral differential equations; the interested reader can try to develop methods for the investigation of persistence of the following systems).
N(t) = rN(t) u(t)
N(t) = rN(t) ti(t) N(t)
[1 - N(t; r) - o:u(t)] }
= -au(t)[l + u(t -
= -au(t) + bN(t -
r).
= rN(t) [1 _N(t -
r)
+ cN~t - 0") - au(t)] } 1 + N2 (t - 0")
= -au(t) + b1 N(t) + b2 N(t -
n
Ui(t)
+ bN(t).
[1- (N(t - r) ~cN(t - r)) - o:u(t)] } K
ti(t)
r)]
= -f3iiUi(t) + L
f3ijUj(t) j#i i = 1,2, ... , n.
r).
.
n
+ L 'YijN(t ;=1
rij)
= AX( t) + BX( t -
§5.5. X(t)
1.: + 1
N(t)=rN(t)[l-
K,(s)N(t-s)ds-
ti(t) = -au(t)
K3(S)N(t - s)ds.
b
T) + CX( t - T)
447
1.~ K'(S)N(t-S)ds-au(t)])
N(t)=rN(t)[I- (XJ K1 (s)N(t-s)ds- [ooK2 (S)(
J0
J0
N~t-~)
1 + N2 (t - s)
)dS
- a!t(t)]
1
00
u(t)
= -au(t) + b
iti(t) = -aiHi(t) +
t [" j#i
1
00
K 3(s)u(t - s) ds
+
Kg>Cs)Uj(t - s)ds +
K4(S)N(t - s) ds.
t 1.~ K~>Cs)Nj(t j=l
0
0
- s)ds
.
i=1,2, ... ,n.
~:C:~ ~:),)
-{
N,(t)
+ c,N,(t -
Tn) }] )
. [K2 + Q2Nl(t - T12) { N2(t) = N2(t) 1 + N (t _ T12) - N2(t) 1
+ c2 N. 2(t -
r22)
N,(t) = N,(t) [ K; :
1
N(t) = rN(t)(l- [N(t)]8 [N(t - T»)8
2
+
K 8 1+82
it(t)
[N(t - 0)]9
3 _
}]
.
cu(t») }
= -au(t) + bN(t).
x(t) = x(t)[r1 - al1x([tJ) - a12y([t])] yet) = y(t)[r2 - a21 x ([t]) - a22y([t])]
+ px(t)[y(t + py(t)[x(t -
r) - x(t - T)J } T) - yet - T)J.
5.6. Large scale systems
Asymptotic behavior of large scale dynamical systems described by ordinary differential equations have been considered by several authors (Bailey [1966], Michel and Miller [1977], Siljak [1978), Anderson [1979], Amemiya [1981]). Recently large scale neutral systems have been considered by Liao Xiaoxin [1986] and Zhang Yi [1988a, bJ. Besides discussing the large scale dynamics, our purpose
448
§5.6. Large scale systems
here is to introduce the reader to a stability investigation in the metric of space e(l) (for details see El'sgol'ts and Norkin [1973]). In particular, we explore the following aspect: if a nonlinear system has a dominant linear part with certain stability characteristics, then what type of nonlinear perturbations can maintain the stability of the full system. Let us consider a large scale (or composite) system described by
x(t) = F(t, x(t), x(t - T(t)), x(t - T(t)))
5.6.1
whose constituent subsystems are governed by
Xi(t) = Ai(t)Xi(t) + fi(t, x(t - r(t)), x(t - r(t)));
= 1,2, .. , Tj t 2:: to
i
5.6.2
in which the delay T is a continuous nonnegative function, 0 ::; T( t) ::; To, Ai(t), (i = 1,2, .. ,r) is an r x r real continuous matrix,
xi =
I(
(i») ,
i)
CO • X l ' ' ' ' X n'
J
r
L
F(t, 0, 0, 0)
nj = n,
= O.
j=l
The initial conditions associated with (5.6.2) are
twhere
IIcpll =
m?-x [ l:::;t:::;r
TO,
TO ~
t ::; to , i = 1,2, .. , r
to]. We define
sup
t- r o99o
(1Ii(t)1I
+ lI¢i(t)ll) ].
The exponential asymptotic stability of the trivial solution of (5.6.2) in defined as follows:
"If there exists a ). > 0 and if for any Q > 0 there exists a K such that,
Ilcpli ::; ===> II Xi(t) II + II Xi(t) II ::; K( Q
Q
)I!CPlle->.(t-t o),
i
e(l)
is
= K (Q) > 0
= 1,2, .. , r, t 2:: to,
then the trivial solution of (5.6.2) is said to be exponentially asymptotically stable in the metric of e(1)".
449
§5.6. Large scale 8ystems
We assume throughout the following, that the fundamental matrix Y associated with 5.6.3 i = 1,2, .. ,; defined by 8Yi( s, t) _ A :(t)Y;( t)
at
-
l
J
S,
i = 1,2, .. ,r
yes,s) = Ei (unit matrix) satisfies
6> 0,
t 2:: s
i
= 1, 2, .. , ;.
5.6.4
The property (5.6.4) will be called (0'1); if instead of (5.6.4), one has
t 2:: s;
i = 1,2, .. , r
5.6.5
where 6i is a nonnegative continuous function, then (5.6.5) will be referred to as (0'2). In the following we denote the spectral radius of a matrix n by pen). Theorem 5.6.1. (Zhang Yi [1988aJ) Assume that the subsystems governed by (5.6.2) satisfy the following; (i) the property (at) holds; i.e. (5.6.4) is satisfied;
II ti(t, x(t -
ret)), Xi(t - ret))) II S;
t,
( ii)
ret»~ II
+ Cijll Xi(t IIAi(t)!I~ai'
(iii)
(biill xj(t -
pen) <
1 where
bij 2::0,
Cij2::0,
ret))
II)
ai>O;
§5.6. Large scale sy.'Jtems
450
Then the trivial solution of (5.6.2) is exponentially asymptotically stable in the metric of C(l) and the stability is not conditional on the size of TO.
Proof. It is found from (iii) that n - E (where E = (eij) is the 2r x 2r identity matrix) is a stable Metzler matrix (see for instance Siljak [1973]) or (E - n) is an M -matrix (for details see Chapter 3). It will follow from the properties of M -matrices that, there exist constants ai > 0, i = 1,2, .. , 2r such that 2r
L:: aj(wij -
eij)
< 0,
i
= 1,2, .. ,2rj
5.6.6
j=l
that is
1 2r -~a'w"<1' L...J } Z} ,
i
= 1, 2, .. , 2r.
5.6.7
OJ j=l
Define hi, i
= 1,2, .. , 2r as follows: 5.6.8
5.6.9 It is found that 2r
r
h i (0)
= -a'1 [L:: 0 jb6·ij- + I
1 = -
j=l
t
L::
a ci,j-r] j--
j=r+l
6·I
2r
L::ajwjj < 1,
5.6.10
OJ j=l
5.6.11 By the continuity of the functions hi, i such that
= 1,2, .. , 2r for
there exists a number ..\ > 0,
5.6.12
451
§5.6. Large scale systems
By the variation of constants formula we have from (5.6.2),
xiit) = 1';( to, t)4>i( to) +
1:
1';( S, t)/i [s, x( s - r( s)), x( s - r( s»] ds.
5.6.13
From conditions (i) and (ii),
II Xi(t) II 5 1I
t. 1.:
e-"('-') [ biill Xj(s -
res»~ II 5.6.14
+ cijll Xj(s -
res»~ 1/ ] ds
and
/I Xi(t) lIeA(t-t o )
~
II q.1/
+ e Aro
t l' bij
j=l
I
e -(6, -A)(,-.) x j( s - r( s»
II e A( .-r(.) -'0) ds
to
t + e ATO 2:: Cij Jt. r
j=l
e-Ui";-A)(t-S)
II Xj(S -
res»~ lI eA (s-d s )-t o) ds.
to
5.6.15
Directly from (5.6.2), we derive that
II Xi(t) lIeA('-'o) 5 eA('-'o) [ aill xii t) II +
+ cijll Xi(t -
5 1I
t.
t.
(b ij II x j( t - r(t»
II
r(t))!]) ]
5.6.16
hij '-;01.9
(II x j( s) lIeA('-'o»)
+eATotcij sup (lIxj(s)lIeA(S-tO»). j=l t-To::;s::;t Define
Si(t) =
sup to-ro<s
- -
(II x -(s) IleA(S-t ») o
I
,
1
-
-
to-;:~.9 (II Xi-r( s) lIeA('-'o») , r + 1 5 i 5 2r.
I
t
~
to - roo
5.6.17
§5.6. Large scale systems
452
It is found from (5.6.15) and (5.6.17) that
II Xi(t) lIe>'(t-I,) :0 II q; II + e>.r,
[t. /!.
II Xi(t) lIe>'('-t,) :0 II q; II + e>.r,
[t,
A Sj(t)
bijSj(t) +
+ ;~1
;;'~~ S;(t)]
j~l Ci,j-rS;(t)].
5.6.18
5.6.19
Using the definition of Si,
[t /~ ASj(t) + j=r+l :t ;i,~~ Sj(t)] , II + eAro f; bi-r,jSj(t) + j~l Ci-r,j-rSj(t) ,
Si(t) ::; II q> II
+ eAro
j=l
Sj(t) ::; II
1 ::; r ::; r 5.6.20
t
Z
2r
r
]
[
r
+ 1 ::; i
We let
Set) =
5.6.21
::; 2r.
~ax
[Si(t)] , Qi
1::;1::;2r
t
~
to -
7"0
5.6.22
and note that
+ ~ Qj Ci,j-r] Set) ~ 8-->' j=l 1 j=r+l I ::; Mil
Si(t) ::; QI
lL!ll + eAro [~Qj QI
~
Q-
1
::; Mil
bij
8-->'
i = r
+ 1, r + 2, .. , 2r.
5.6.23
5.6.24
Thus, from (5.6.23) and (5.6.24),
Set) ::;
Mil q> II + hS(t)
and therefore,
Set)
s l~hll
5.6.25
§5.6. Large scale systems
453
We have from (5.6.25),
+ II Xi(t) II :s; e-'(Ho) [Si(t) + Si+,(t)]
II Xi(t) II
::; ill e-).(t-to) [Si(t)
+ Si+r]
Qi
Qi+r
::; 2M S(t)e-).(t-t o) ::;
~l~ ~ /I tP lI e -).(t-t o )
5.6.26
for t 2': to, i = 1,2, .. , r and Ai = maxl
x(t)
= F(t,x(t),x(t -
r(t»,x(t - ret»~)
5.6.27
denote a large scale system and let its constituent subsystems be governed by
Xi(t) = Ai(t)Xi(t) + fi(t, x(t -
ret»~,
x(t - r(t») , i = 1,2, .. , r, t 2': to
5.6.28
in which the delay r( t) is a nonnegative continuous function, satisfying r( t) -T 00, t - ret) -T 00 as t -T 00 and F(t, 0, 0, 0) = O. The initial conditions for (5.6.28) will be of the type
Xi(t) =
Xi(t) = ~i(t),
-00 < t ::; to
where
/I
so that
It is known that the exponential stability of the equations with unbounded delays may not be possible (see Chapter 1). We recall the following definition:
Definition. If for each
€
> 0, there exists
a 8
/I
> 0 such that i = 1,2, .. ,r,
§5.6. Large scale systems
then, the trivial solution of (5.6.28) is said to be stable in the metric of C(1). If in addition, there exists a 60 > 0 such that
then the trivial solution of (5.6.28) is said to be asymptotically stable. Theorem 5.6.2. (Zhang Yi [1988a)) Suppose tbe following conditions bold:
(i) Tbe fundamental solutions Yi( s, t) of tbe isolated subsystems i
= 1,2, .. , r
i
= 1,2, .. , r
,t ~
to
satisfy
\I Yi( s, t) II ~
e-
f
o;(u) du ,
t >s
for
and
t
as
II fi(t,X(t -
r(t)),i(t - ret)) II S
i = 1,2, .. ,r.
-* 00,
t
[bij(t)1I Xj(t - r(t)) II
+ Cij(t) II ij(t -
(ii) sup bij(t) = bij
<
= Cij
sup Cij(t)
00,
t~to
t~to
ret))
<
II]
00,
i = 1,2, .. ,r;
( iii)
(iv)
1>x l' 11: 1' p ( -
Oi(U) dU) bii(S) ds
swl]>,
i,j = 1,2, .. , r,
exp ( -
0i(U)dU)C;j(S)ds
s wl]>,
i,j = 1,2, .. , r,
p(f!) < 1 wbere
f! 1 -
(W(l») ij rxr'
n2 =
(w~:») I) rXr
- .. _ { bii
bl )
C
= (c··) rXr . 1)
-
b
+ ai,
ij,
. -I-
ZT
Z=J .
J
§5.6. Large scale systems
455
Then the trivial solution of (5.6.28) is asymptotically stable in the metric of e(l).
Proof. A consequence of condition (iv) is that there exist numbers 1,2, .. , 2r such that 1
I 2r } max { - ""' a ·w . . = h
ai
> 0,
< 1.
i =
5.6.29
j=1
By the variation of constants formula, one derives from (5.6.28),
II Xi(t) II
~ 114> II +
t, l:'xp (-[6 u) i(
dU) [ bi;(s)1I x ;(s - r(s) II 5.6.30
+ cij(s)1I :tj(s - res) II] ds; directly from (5.6.28),
II Xi(t) II
~ 114> II + adl Xi(t) II +
t,
[bi;1I Xj(t - r(t)) II 5.6.31
+ Cijll :ti(t -
ret)) ,,],
Define
Si(t)
= {SUP -
<S9 sUP-oo<s9 OO
(II ~i(S) II), (II Xi-r( s) II),
l::;i::;r r + 1 ::; i ::; 2r.
. - 00
< t < 00 5.6.32
It follows from: (5.6.30) - (5.6.32) that
II Xi(t) II
~ 114> II +
t. U>xp {-
[Oi(U) du }bi;(s) ds )S;(t)
+f (1' exp {- J.' }=r+l
to
Oi(U)du}ci,j_r(S)ds)Sj(t)
s
2r
::; II ~ II + LwijSj{t); i = 1,2, .. , rand
5.6.33
j=l
r
II :ti(t) II ::; II ~ II + L j=1
2r
bijSj(t) +
L
Ci,j-rSj(t)
j=r+l
2r
::; " ~" + L Wi+r,jSj(t) , i j=l
= 1,2, .. , r.
5.6.34
§5.6. Large scale systems
456 From (5.6.33) and (5.6.34),
2r
II
Si(t) ~
i
WijSj(t),
= 1,2, .. , 2r
j=l
which leads to
IIXi(t) II
S(t)~ l~hll
t?.to
~ ~~II
t?.to,
+ II Xi(t) II
where M=max
15:i5:2r
5.6.35 i = 1,2, .. , r
5.6.36
( 1)
-, Qli
The stability of the trivial solution of (5.6.28) in C(I) follows from (5.6.36). We now proceed to prove the asymptotic stability of the trivial solution. We let lim sup t-oo
limsup t-+oo
II Xi(t) II =
f)i
II Xi(t) II
f)i+r
It is easy to see that f)i satisfies 0 exists a tl > to such that
=
~ f)i
i = 1,2, ..
< 00,
,r.}
5.6.37
i = 1,2, .. , 2r. For any e > 0, there
5.6.38
t ?. to , i = 1,2, .. , r. By the variation of parameters formula we derive directly from (5.6.28),
IIXi(t)
II ~ II Xi(tl) lIexp( +~
l
itt(bi(u)du)
exp ( - [O;(U)dU) [b;j(s)1I Xj(S - r)s»
+ Cijll Xj(S :::; (f)j
+ e)exp
II
- res)) II] ds
(-1' O;(u)du) + f(o; + €)W;; tl
i = 1,2, .. ,r.
J=1
5.6.39
457
§5.6. Large scale systems From (5.6.28), r
II Xi(t) II::; adl Xi(t) 11+ L
[bijl! Xj(t - ret»~ II + Cijll Xj(t - ret»~ II]
j=l 2r
+ €)Wi+r,j
::; L(Oj
for
i
= 1,2, ."
r.
5.6.40-
j=l
From the definitions of Oi, there exists t2 > i 1 such that 5.6.41 Hence, from (5.6.39) and (5.6.40) 2r
OJ - € ::; €(Oj
+ €) + LCOj + €)Wij j=l
i = 1,2, .. ,r.
2r
Oi+r -
€ ::;
LCOj
5.6.42
+ €)Wi+r,j
j=l
We let
€ ~
0 and derive 2r
0i ::; L WijOj,
5.6.43
i = 1,2, .. , 2r
j=l
leading to 5.6.44 where 0 is defined by 0= max (Oi). 1~i~2r
(ti
Since h < 1, we have from 0 ::; hO that 0 = 0 and, therefore, Oi = 0, i = 1,2, .. , 2r. We can now conclude
II xiCt) II + II Xi(t) II ~ 0 This completes the proof.
as
t
~ 00,
i
= 1,2, .. ,r. []
The following result is also due to Zhang Yi [1988a] and provides a further generalization of Halanay's lemma (see Chapter 1).
§5.6. Large 3cale 3Y3tem3
458
Theorem 5.6.3. Let Pi(t), i
defined on [to -
TO,
1,2, ", r be continuous nonnegative functions
00) satisfying 5.6.45
in whichgi(t):2: k i > 0, i = 1,2, .. ,r andaij ~ 0, i =lji bij If aii + bii < 0, i = 1,2, .. , r and if all the roots of
satisfy
~e(A)
< 0, then there exist constants (:J
~ (:J [t
Pi(t)
~
Pj( S)] e-p.(t-t o ),
sup
1, Il >
°
~ to,
i
t
j=l to-To~s~to
~
0;
i,j = 1,2, .. ,r.
such that
= 1,2, .. , r.
5.6.46
Proof. The matrix (A + B) = (aij + bij)rxr is a stable Metzler matrix (i.e. -fA + B] is an M -matrix). So there exist constants a 1 > 0, i = 1,2, .. , r such that r 2)aij
+ bij)aj < 0,
i = 1,2, .. ,r.
j=l
Consider the continuous functions
Ii
defined by
r
fiCA)
= 2:(aij + bij)aj,
i
= 1, 2, .. , r
where
j=l
_ ai' J
aij j i f:. j = { a··+>... '-J U
i,j=1,2, .. ,r.
k,'· -
Note that since r
1i(0)
= I)aij + bij)aj < 0,
i
= 1,2, .. ,r
j=l
there exists a positive Il such that i = 1,2, .. ,r.
Define
i = 1,2, .. ,r.
5.6.47
459
§5.6. Large 3cale 3Y3tem3 It follows from (5.6.47) and (5.6.45) that Fi(tt) = p,Fi(t)
+ ~ell(t-to) Pi(t) ai
~ !:gi(t)Fi(t) + ~9i(t) i ai
k
t
aj [aijFj{t)
+ bij (
Fj(s))]
sup t-ro=:;s=:;t
j=l
5.6.48 We shall first show that
t ?: to , Let
M
=
t(.
Pj(S)) ,
sup
j=l
~
t
i = 1,2, .. , r.
i
to,
5.6.49
= 1,2, .. ,r.
to-ro:S;s:S;to
For any d> 1, we have
to -
TO ~
t
~
to,
t
~
to
i = 1,2, .. ,r.
5.6.50
We claim that for any
and
5.6.51
i = 1,2, .. ,r.
Suppose this is not the case; then there exist an i and tl > to such that
Fi(tl) = Md, Fj(t) ~ Md,
j
=1=
i,
j
= 1,2, .. ,r.
Thus Pi(td ~ O. But we have from (5.6.48) that
and this is a contradiction. Hence, (5.6.51) holds. Now allowing d
~
1, we get
r
t
~
to,
i = 1,2, .. ,r
§5.6. Large scale systems
460
which together with (5.6.47) leads to
Pi(t)
~ (m~xl
[t { j=l
sup to-ro~s9o
t 2= to,
i
Pj (s)}]e-l1-(t-t O )
= 1,2, .. ,r.
The proof is complete.
[]
All the above results indicate that if a system is stable without delays, then the system can continue to remain stable under certain delay dependent structural perturbations. The stability type obtained has two important features; it is delay independent and that a precise knowledge of the perturbations is not necessary, only certain estimates of the perturbations are needed. Such stability type is known as 'robust stability'. In dynamics of interacting populations, the exact type and form of interactions are rarely known; however, there exists no literature devoted to the robust stability of population systems, especially which involve time delays. It is now open to investigate the stability characteristics of systems of the form i
= 1,2, .. ,r
subject to large scale perturbations under various smallness assumptions on the delays ret) and aCt). One of the potential applications of large scale neutral systems is in mod..: elling the dynamics of compartmental systems. Recently Gyori and Wu [1991] have discussed the applications of neutral equations in the dynamics of certain compartmental systems. We have listed below some neutral differential equations modelling compartmental dynamics and the interested reader can examine the positivity of the solutions, persistence of the systems, oscillations, convergence to steady states and delay induced bifurcation to periodicity:
Xi(t) = -Xi(t)[Ai + l'i Xi(t - T;)] i
+ ~ a;jxj(t -
= 1,2"", n.
x;(t) = -Xi(t) [Ai n
+ ~ aij
+ I'i J.~r; e- 7S x;(s) ds]
1t
t-rj
e-iSxj( s) ds
j~i
i=1,2,···,n.
+ OJ
Tj)
+ "'i
1
§5.6. Large 8cale 8Y8tems
461
i = 1,2,···,n.
X, (t) = 1 + fo=
Kn(~xn(t _ s) ds -
Xl
(t) [bl
1
+ I'IXI(t - Td]
00
Xj(t) = -xiCt)[bj
+ fLjXj(t -
Ij)]
+
Kj(S)Xj_l(t - s) ds
j = 2,3"" ,no
XI(t) = -aXl(t)[X2(t - I) X2(t) = -,X2(t)
+ CX2(t -
+ aXl(t) [X2(t -
I)]
}
I) + CX2(t - I)]
X3(t) = ,X2(t).
X;(t) = -A;X;(t) [1 + x;(t - T)] k
= 1,2""
+
t.
a;jxj(kT), t E [kT, (k + 1)T)
I
and i = 1,2,,", n.
Another area of potential application of large scale systems lies in modelling the dynamical characteristics of neural networks involving time delays in neuronal response (see for instance Marcus et al. [1991]). An example of neural network modelled by a neutral system is formulated in Exercise 34.
462
EXERCISES V
1, Discuss the oscillatory and asymptotic behavior of the following scalar equations:
(1)
x(t)
+ ax(t) + bx(t I, (J
a, b, c, E R,
I,
(J
x(t)
+ ax(t) + bx(t) + ci(t) =
x(t)
T
E [0,(0), x(t)
(6)
~
=0
[0,(0).
0
= sUPsE[t-r,t}
x(s).
+ a(t)x(t) + b(t)x(t - T(t» + c(t)x(t - (J(t» = 0
a, b, c : [0,(0) ~ R j
(5)
0") = 0
E [0,(0) , a E R, b, c: [-(I + 0"),00)
a, b, c E R,
(4)
+ cx(t -
x(t) + ax(t) + ft~T b( s )x(s) ds + Itt_O' c(s )x( s) ds
(2)
(3)
I)
E [0,(0) .
T, (J :
[0,(0)
~
[0,00) .
x(t) + a(t)x(At) + b(t)X(fLt) = 0 o < A < 1, 0 < fL < 1; a, b E C(R+, R). x(t) + ax(t>') + bx(t ll )
= 0,
(7) x(t) + a It) K 1 (s)x(t - s) ds + b Io K 2 (s)x(t - s) ds oo
= O.
2. Derive sufficient conditions for the asymptotic stability of the trivial solution of the following systems; also derive conditions for the oscillation of all solutions of the systems.
(a)
Xi(t)+ Ei=l aijXj(t) + Ei=l bijXj(t-lij)+ Ei=1 CijXj(t-(Jij) = 0 aij, bij , Cij E R, i,j = 1,2, ... , n. n
Xi(t) (b)
+L j=1
n
aij(t)Xj(t)+
L bjj(t)xj(t -
Ijj(t»
}=1 n
+
L Cjj(t)Xj(t - O"ij(t» = 0 j=1
aij, bij , Cij E C(R,R), lij, O"ij E C(R+,R+), i (c)
= 1,2, ... ,n.
Exerci8e8 V
463
o < Aj < 1, 0 < Pi < 1, aij, bij, Cij E C(R+, R), (d)
i,j
= 1,2, ... , n.
Xi(t)+ 2:7=1 aij(t)Xj(t)+ 2:7=1 bij(t)Xj(t Aj )+ 2:7=1 Cij(t)X(tJLj) = 0 aij, bij, Cij E C(R+, R), 0 < Aj < 1, 0 < pj < 1, i,j = 1,2, ... ,n.
3. Discuss the asymptotic stability of the trivial solution 6f the following; also derive sufficient conditions for the oscillation of all solutions:
(1)
x(t) + ax 3 (t) + bx 3 (t - r) + cx 3 (t - 0") = 0 a, b, C E IR, r,O" E R+ .
(2)
x(t) + a(t)x 3(t) + b(t)x 3 (t - ret»~ + c(t)x 3(t - O"(t» a,b,cE C(IR+,R), r,O" E C(IR+,R+).
(3)
x(t) + a(t)x 3(t) + b(t)x3(At) + c(t)x 3 (pt) = 0 O
= O.
4. Examine the stability and oscillatory characteristics of the following nonlinear scalar equations; discuss the persistence of the positivity of the solutions . .( ) =
xt
8() [ _ (X(t - r») rx t 1 K () = 1,3,5, ... ;
. [K-X(t-r) x t = rx t () () l+cx(t-r)
cx(t - r)
+ 1+x 2 (t-r)
]8 (1)
r,K E (O,oo),c E IR. ax(t-r)] + ., 1+x 2 (t-r)
x.() t = rx ()[1 t - (x(t-r)+cx(t-r»)8] K .
(2) (3)
(4) (5)
(6) (7)
Exerci8es V
464
.( ) _ () [ _ (X(t - 8)x(>.t) x t - TX t 1 K2
+ C1X(Jlt))] '
A, Jl E (0,1).
(8)
5. Derive sufficient conditions for the asymptotic stability of the positive equilibria of the following models:
Xi(t) =Xi(t) [Ti -
t
a;;x ;(t) +
]=1
t
b;;x ;(t - T;;)
]=1
+
t
c;;x(t - 0-;;)] (1)
]=1
i = 1,2, ... ,n.
~ ~ Xi(t) =Xi(t) [ri - ~ aijXj(t) + ~ bijxj(t - Tij) ]=1
J=l
.~ ( c··x .( t - a· .) )] 1 ~ ;~(t _;: .)
+~
J=l]
J
i = 1,2, ... ,no
(2) (3)
i=1,2,···,n. 6. Assume the following:
a,b,T,a E [0,(0) (ii) a(T-a)e>(l+b) Then prove or disprove the following: "all bounded solutions of (i)
x(t) + bx(t - a) + ax(t - T) = 0 are oscillatory". Can you extend your analysis to vector - matrix systems of the type t) + B X(t - (T) + AX (t - T) = 0 ?
x(
7. Assuming (i) a,b,c,a,T E [0,(0) (T - a)e(ae CT - ceCTC) > (1 + beCTC), (ii) prove or disprove "all bounded solutions of
x(t) + bx(t - 0-) + ax(t - T) are oscillatory" .
+ cx(t) =
0
465
Exercises V
8. Suppose all positive solutions of
x(t) = TX(t)
[1 -: xU;; T)] ,
r,K,r E (0,00)
satisfy 1imt ..... oo x(t) = K. Suppose also that this logistic _equation is perturbed by the introduction of a. neutral term as follows: .( ) =
y t
()
ry t
[1 _y( tK- r) + ey( tK- r)] .
Examine whether there exists a bifurcation to periodicity induced bye: also discuss the stability of such a periodic solution if it exists. 9. Derive sufficient conditions for all positive solutions of the following to converge to equilibria:
.() ()[1 -
x t = rx t
(x(t -
ret»~ + K ex(t - aCt»~)] ;
(1)
(2) (3)
x(t) = x(t)
[ao+ a, x(t) + a2x([tJ) + a3x([t])];
(4)
x(t) = x(t) [a o + a,x(t) + a2x([AtJ) + a3X([AtJ)];
(5)
[ao+ a,x(t) + a2x([t'J) + a3x([t'])].
(6)
x(t) = x(t)
10. Investigate the oscillatory and asymptotic behavior of the following coupled systems:
ExerciJeJ V
466
11. Prove that if A, B, C are real symmetric n x n matrices and r > 0 and I B + C are positive definite then the trivial solution of
x(t) + Ax(t - r) + Bx(t) + Cx(t - r)
+ A,
=0
is asymptotically stable (see Brayton and Willoughby [1967]). Can you prove a similar result for systems like
xCt) + AxCt - r) + Bx(t) + Cx(t - r) n
Xi(t) +
L aijXj(t -
O"ij)+
j=l
n
n
j=l
j=l
=0
and
L bijXj(t) + L CijXj(t i
rij)
=0
= 1,2, . .. ,n
without assuming the symmetry of the matrices involved? Try also the above problem with B = O. 12. Investigate the local asymptotic stability of the trivial solution of the scalar equation
c[v(t) + kv(t - h)j
+
[~- glv(t) - k [~ -
gl v(t -
h)
= -v 3 (t) - kv 3 (t - h).
Discuss the possibility of the existence of delay induced bifurcation to periodic solutions and also examine the stability of such periodic solutions (for details see Brayton [1966, 1967]). 13. Prove that the trivial solution of
is asymptotically stable for all delays if and only if n
n
L Iak 1< Iao I· k=l
(for more details see Hale et.al. [1985].)
467
Exercises V
14. Examine the local asymptotic stability of the positive steady state of the following for all delays: also investigate the possibility of bifurcation to periodic solutions;
X(t) = rx(t)[l-( x(i -
T) ~cX(t - T») OJ
B E (0,00), r, K,
r E
. [(x(t-r»)8 x(t) = rx(t) 1 K
.
[K -
(1) c E R.
(0,00);
J + 1 +ci(t-r) x2(t _ r) . cx( t - r) J
x n (t - r) x(t)=rx(t) l+xn(t-r) +1+x2(t-r) .
(2) (3)
15. Derive a sufficient condition for the asymptotic stability of the positive equilibrium of the neutral logistic multiplicative scalar system
16. Prove or disprove the following: a necessary and sufficient condition for the existence of a positive solution on R of
1
00
x(t)+cx(t-r)+a
K(s)x(t-s)ds=O,
a,cER,rE(O,oo)
is that the associated characteristic equation
has a real root. 17. Derive sufficient conditions for the persistence of the neutral Lotka Volterra system
x,(t) = x,(t) [r, -
t
a,jxj(t) -
j=l
t
b,jXj(i)],
i =
1,2,···, n.
j::J;:l
Extend your result to a system of the type
Xi(t) = Xi(t) [ri -
t
J=l
aijXj(t - r) -
t
J=l
bijXj(t - r)],
i = 1,2",', n.
Exerci3e3 V
468
18. Obtain sufficient conditions for all positive solutions of the integrodifferential system
Xi(t)=Xi(t)[r i
t
-
()O KiAs)Xj(t-s)ds-
j=110
t
(>0 hij(S)Xj(t-S)ds]
j=110
i = 1,2,···,n
to remain bounded and converge to a positive equilibrium. Can you discuss the persistence of this integrodifferential system? 19. Derive the following: a necessary and sufficient condition for all solutions of
Xi(t) =
n
n
j=l
j=l
L aijXj(t - Tj) + L bijxj(t -
CTj)j
i = 1,2,"', n
to be oscillatory is that the characteristic equation
I
det )..Oij -
aije-
Arj
-
bij)..e-A/Tj]
=0
has no real roots (for more details see Arino and Gyori [1989]). 20. Discuss the oscillatory and asymptotic behavior of the following prey-predator systems with respect to their positive steady states; discuss also the possibility of stability switching and feasibility of stabilising by feedback controls: (also do the same, if in the following, one replaces H (t - T) by the term fI (t) where
fI(t)
= sUPsE[t-r,t]
H( s) ).
H(t) = H(t) [a - bP(t) (a]Hn(t - r) + a,Hn(t - r)) ]) .P(t) = pet) [ - c + (a 3 Hm(t - r)
~(t) = r H(t) [1 - (a]H(t -
r)
i/
+ a.Hm(t - r)) ].
2H
(t - r)) ]_ aH(t)P(t) }
pet) = -bP(t) + fJH(t)P(t). H(t) = rH(t)
[1-
(a]H(t -
- ap(t)(lht)
r) ~ a,H(t - r))] e-OH(t»)
= -bP(t) +.BP(t)(l- e-OH(t»).
Exercises V
[1- (aIH(t - r); ~2H(t - r)) - CX:~~;;) bP(t) [1- blP(t -;~~~~(t - r)]. TH(t) [1- (aIH(t - r); azH.(t - r)]_ cxf3H;t:c;?
H(t) = TH(t) Pet) = H(t) = Pet)
r)) -
= pet) [ - T + (CXIH(~~r17t ~~~t -
OP(t)].
fI(t) = rH(t) [K - alH(t - r) - a2H(t - r)]_ o:H(t)P(t) l+cH(t-r). f3+H(t) Pet)
=
pet) [ - f3 + (6 I H(t - :k~/~~it f3
H(t) = TH(t) [K -
Pet) =
P(t)[ - a
-r)) - a
I I I
469
6 P(t)).
al~~ ~;it-_a~~(t -
T)] _ cxH(t)P(t) }
+ b1H(t - r) + b2 H(t - r) - cP(t)].
21. Derive necessary and sufficient conditions for every solution of
~ [X(t) + a /.'''' K I ( s)x(t -
s) dS]
1.
00
+b
K 2 (s)x(t - s) ds = 0
to have at least one zero on (-00,00). Formulate your own hypotheses on a, b, K 1 , K2 (for results on the existence of positive solutions of integrodifferential equations, see Philos [1988, 1990a,b] and Ladas et al. [1991]). 22. Derive sufficient conditions for every positive solution of
.
N(t)
= rN(t)
[J::
H ( s) { N (t - s)+ eN (t - s) } ds
K
1-
1
to have at least one t* E (-00,00) such that N(t*) - K = O. 23. Derive necessary and sufficient conditions for the existence of solutions of the following equations which satisfy IXi(t)1 > 0 for t E (-00,00), i = 1,2,"" n.
~ [Xi(t) +100 Ki(S)Xi(t o
1.
00
S)dS]
+ taij J=l
0
i = 1,2", ',nj
Hij(S)Xj(t - s)ds
=0
(;) ~
Exercises V
470
+
t
j=l
aij ]."" Hij(s)xj(t - s) ds
=a
( ii)
0
i = 1,2,"', n.
24. Derive necessary and sufficient conditions for all solutions of
dn
dtn [yet) - cy(t - 7")]
to be oscillatory where 0 ~ c < 1; p > 0; odd positive integer and derive that
+ py(t (j
> 0;
0") T
~
=0 O. Assume that n is an
is a sufficient condition for all solutions of the above equation to be oscillatory. Can you derive a sufficient condition for the existence of a nonoscillatory solution? 25. Establish necessary and sufficient conditions for all solutions of
to have zeros on (-00,00) where c, a are positive constants and the kernels K11 K2 are nonnegative and integrable on [0,00). 26. Obtain a set of sufficient conditions for the neutral system of equations
i=1,2,···,n to be stable independent of the delays where stants.
Cij, aij, Tj,
0"j are all real con-
27. Derive a set of sufficient conditions for all solutions of the neutral integrodifferential system (i = 1, 2, ... , n)
Exercises V
471
to have the property of "equilibrium level crossing". State your own assumptions on the kernels H 1, H 2. 28. Can you derive sufficient conditions for all solutions of
d dt [x(t) - cx(t - r)]
+ ax3(t -
a)
=
°
to be oscillatory? Assume C E (0,1), a E [0,00), r E [0,00), obtain sufficient conditions for all solutions to satisfy lim x(t)
t-oo
29. Assume that aii, bij(i =f j), cij(i,j O(i = 1,2,···,n). Prove that if n
= 0.
= 1,2,···,n)
n
+ Llbijl + L ICijl <
aii
i#
j=l
n
n
j=l
laiil + L
Ibijl
E (0,00). Also
(1
are constants and aii <
° i=1,2,· .. ,n
+ 2: ICijl < 1
1;~
j=l
then the trivial solution of n
Xi(t)
n
= aiixi(t) + L
bijxj(t - r)
+ l: CijXj(t -
r)
j=l
i 1'i j=l
(i = 1,2,"" n) is asymptotically stable in the metric of C(1)[-r, 0]. Can you derive sufficient conditions for all solutions of n
Xi(t)
= aiixi(t -
n
r) +L bijxj(t - r)
+ LCijXj(t -
~;~
(i
r),
j=l
= 1,2,' .. , n) to satisfy lim Xi(t) = 0, i
t-oo
= 1,2""
,no
Generalize your results to a neutral system of integrodifferential equations of the form
Xi(t)=aii
1
00
o
n
la
Kii(S)Xi(t-s)ds+ Lbij.· j=l j 1'i
oo
Kij(S)Xj(t-s)ds
0
i = 1,2"" ,no
Exercise8 V
472
30. Formulate the following neutral differential equations as models of single species dynamics and examine the persistence . of the species:
N (t) = N (t)
[a - bN(t -
( i)
T) - eN (t - T) ].
N(t) = -'1'N(t)+ aNn(t - T) exp [ - ,BN(t - T)+ cN(t - T)]. N(t)
= -'1'N(t) [1 + aN(t -
+
T)]
1~;~~;:~)'
( ii)
(iii)
Can you investigate the oscillatory and convergence characteristics of the above neutral systems if the terms N(t - I) are replaced by terms such as N(t) and N([t]) where
N(t) =
sup
and [t] denotes the greatest integer in t.
N(s)
8E[t-r,t]
Generalize your models to neutral integrodifferential equations. 31. Prove the asymptotic stability independent of delay of the trivial solution in the system (for more details see Datko [1978])
dx(t)
----;It = Aox(t) + AIX(t - h) Ao
=
where
[-1 0] 0
-1'
Investigate stability switching in the above system if Al is replaced by
A = 2
[0-a
a] 0 '
a>1.
Discuss stability independent of delay in the neutral system d
-d [x(t) - Bx(t - h)] = Aox(t) + AIX(t - h), t .'
B
0
where
1]
= [ _~ ~ .
32. Discuss the possibility of stability switching in the two species competition model; consider the cases kl > 0 and kl = 0 : see (Kuang [1991]);
x(t) = rlx(t)[l - klX(t) - ax(t - ..Td - (3x(t - To) - cly(t - 12)] yet) = r2y(t)[1 - C2x(t - T3) - k2y(t - 14)].
Exercius V
473
33. Derive sufficient conditions for the stability of the equilibria in the following logarithmic population systems:
dx(t) = ()(1- IOg[x(t. - T)J :... . ~ log[x(t - T)]) dt rx t K dt K .
i = 1,2"", n.
dx(t) = () [ _ (log[X(At») _ ~ (log[X(At»))] dt x t 1 K dt K '
d~~t) =x(t)[a-b].= K,(s)[logx(t-:-s)]ds+
O
].= K2(S)~~Ogx(t-s)]ds].
34. A neural network is modelled by the neutral system U;( t) = -r;u;( t)
+
t.
[1 + a;;ui(t - r)]
aU tanh (pj
[Uj(t -
T) + ajjuj(t - T)]) + Ii
i 'Fi
i
= 1,2,· .. ,n.
Assume that aij is a symmetric matrix and r i, f3i, Ii are real numbers such that there exists an equilibrium u* = (ui, ui,'" ,u~) satisfying
~aijtanh(f3ju;)
+ Ii = riui,
i = 1,2,' .. ,n.
i:l:i
Prove or disprove the following: if aii = 0, T = 0, i = 1,2"", n, then the system is not oscillatory about u* and u* is a global attractor. Derive sufficient conditions for the system to be respectively stable, oscillatory and nonoscillatory when ajiT =f 0, i = 1,2"", n. Can you generalize this model system to one with integrodifferential equations?
REFERENCES
Abrahamson, D.L. and Infante, E.F. [1983]. A Liapunov functional for linear Volterra integrodifferential equations. Quart. AppI. Math. XLI, 35-46. Aftabizadeh, A.R. and Wiener, J. [1985]. Oscillatory properties of linear funct~onal differential equations. Appl. Anal. 20, 165-187. Akhmerov, R.R., Kamenskii, M.L, Potapos, A.S., Rodkina, A.E. and Sadovskii, B.N. [1984}. Theory of equations of neutral type. Jour. Soviet. Maths. 24, 674-719. Alexander, J.C. and Auchmuty, J.F.G. [1986J. Global bifurcations of phase locked oscillations. Arch. Rat. Mech. Anal. 93, 253-270. Alexander, J.C. [1986a]. Patterns at primary Hopf bifurcations of plexus of identical oscillators. SIAM. J. Appl. Math. 46, 199-221. Alexander, J.C. [1986b]. Spontaneous oscillations in two 2-component cells coupled by diffusion. J. Math. BioI. 23, 205-219. Amemiya, T. [1981]. On the stability of non-linearly interconnected systems. Int. J. Control 34, 513-527. an der Heiden, U. [1979]. Delays in physiological systems. J. Math. BioI. 8, 345-364. an cler Heiden, U. and Mackey, M.C. [1982]. The dynamics of production and destruction: analytic insight into complex behavior. J. Math. BioI. 16, 75101. Anderson, B.D.O. [1979]. Time delays in large scale systems. Proc. 18th IEEE Conf. Decision and Control. 655-660. Anvarinov, R. and Larinov, G.S. [19781. Volterra model with a weakly linear hereditary characteristic. Diff. Uravn. 14, 1494-1494. Araki, M. and Kondo, B. [1972]. Stability and transient behavior of composite nonlinear systems. IEEE Tans. Auto. Contr. AC-17, 537-541. Arino, 0., Gyori, 1. and Jawahari, A. [1984]. Oscillation criteria in delay equations. J. Diff. Eqns. 53, 115-123.
References
475
Arino, O. and Gyori, I. [1989]. Necessary and sufficient condition for the oscillation of a neutral diJferential system with several delays. J. Diff. Eqns. 81, 98-105. Ashkenazi, M. and Othmer, H.G. [1978]. Spatial patterns of coupled biochemical oscillators. J. Math. BioI. 5, 305-350. Atkins, G.L. [1969]. Multicompartment models for biological systems. Methuen, London. Ayala, F.J., Gilpin, M.E. and Ehrenfeld, J.G. [1973]. Competition between species. Theor. Pop. BioI. 4, 331-356. Bailey, F.N. [1966]. The application of Lyapunov's second method to interconnected systems. SIAM J. Control 3, 443-462. Bainov, D.D. and Simeonov" P.S. [1989]. Systems with impulse effect. John Wiley, New York. Banks, H.T and Mahaffy, J.M. [1978a]. Global asymptotic stability of certain models for protein synthesis and repression. Quart. Appl. Math. XXXVI, 209-221. Banks, H.T and Mahaffy, J.M. [1978b]. Stability of cyclic gene models for systems involving repression. J. Theor. BioI. 74, 323-334. Banks, H.T and Manitius, A. [1975]. Projection series for retarded functional ';;_ differential equations. J. Diff. Eqns. 18, 296-332. Barbalat, I. [1959]. Systemes d'equations differentielle d'oscillations nonlineaires. Rev. Roumaine Math. Pures Appl. 4, 267-270. Barbashin, E.A. [1970]. Introduction to the theory of stability (translated from Russian). Wolters-Noordhoff Publishing, Groningen, The Netherlands. Barbu, V. [1976]. Nonlinear semi groups and differential equations in Banach spaces. Noordhoff International Publishing, Leyden, Netherlands. Barker, G.P., Berman, A. and Plemmons, R.J. [1978]. Positive diagonal solutions to the Lyapunov equations. Lin. Multi-lin. Algebra. 5, 249-256. Bellman, R. and Cooke, K.L. [1963]. Differential difference equations. Academic Press. New York.
476
References
Berman, A. and Plemmons, R.J. [1979]. Nonnegative matrices in the mathematical sciences. Academic Press, New York. Borisenko, S.D. [1983]. Asymptotic stability of systems with impulsive action. Ukrain. Mat. Zh. 35, 144-150. Borsellino, A. and Torre, V. [1974]. Limits to growth from Volterra theory of population. Kybernetik 16, 113-118. Boucher, D.H., James, S. and Keeler, K.H. [1982]. The Ecology of Mutualism. Ann. Rev. Ecol. Syst. 13, 315 - 347. Boucher, D.H. (Ed.) (1985]. The biology of mutualism: Ecology and evolution. Croom Helm, London. Braddock, R.D and Van den Driessche [1983]. On a two lag delay differential equation. J. Austral. Math. Soc. Ser. B. 24, 292-317. Brayton, R.K and Willoughby, R.A. [1967]. On the numerical integration of a symmetric system of difference - differential equations of the neutral type. J. Math. Anal. Appl. 18, 182-189. Brayton, R. K. [19661. Bifurcation of periodic solutions in a nonlinear difference differential equation of neutral type. Quart. Appl. Math. 24, 215-224. Brayton, R.K. (1967]. Nonlinear oscillations in a distributed network. Quart. Appl. Math. 24, 289-301. Brelot, M. [1931]. Sur Ie probleme biologique hereditaire de deux especes devorante et devoree. Ann. Mat. Pura Appl. Ser. 4, 9: 57-74. Brian, M.V. [1956]. Exploitation and interference in interspecies competition. J. Anim. Ecol. 25, 339-347. Bromwich, T.J.l'a [1965]. Introduction to the theory of infinite series. Macmillan, London. Brumley, W.E. [1970]. On the asymptotic behavior of solutions of differential difference equations of neutral type. J. Diff. Eqns. 7, 175-188. Burton, T.A. and Haddock, J.R. [1976]. On the delay-differential equations x(t) +
477
References
a(t)f(x(t -ret))) 54, 37-48.
= 0 and x(t) + a(t)f(x(t -ret))) = O. J. Math.
Anal. Appl.
Burton, T.A. [1983]. Volterra integral and differential equations. Academic Press, New York. Busenberg, S.N and Travis, C.C. [1982]. On tbe use of reducible functional differential equations in biological models. J. Math. Anal. Appl. 89, 46-66. Busenburg, S.N and Cooke, K.L. [1984]. Stability conditions for linear nonautonomous delay differential equations. Quart. Appl. Maths. 42, 295-306. Cai Sui-Lin [1959]. A formula for tbe Lyapunov function of a system of linear differential equations witb constant coefficients. Acta Math. Sinica. 9, 455464. Carvalho, L.A.V., Infante, E.F. and Walker, J.A. [1980]. On tbe existence of simple Liapunov functions for linear retarded difference differential equations. Tohoku Math. J. 32, 283-297. Castelan, W.B. [1980]. A Liapunov functional for a matrix retarded difference differential equation witb several delays. Leet. Notes in Math. 799, 82-118. Castelan, W.B. and Infante, E.F. [1979]. A Liapunov functional for a matrix neutral difference differential equation witb one delay. J. Math. Anal. Appl. 71, 105-130. Chandra, J. and Scott, A.L. (Eds.) [1983]. Coupled nonlinear oscillators. North Holland Math. Studies, 80. Chang Hsueh-Ming. [1962]. On tbe stability of systems of differential equations witb time lag. Chinese Maths. 1,217-225. Chew, K.H. [1976]. Stability conditions for linear differential systems. Nanta Math. 9, 33-40. Chin Yuan-Shun. [1962]. Unconditional stability of systems witb time lags. Chinese Maths. 1, 138-155. Chow, S.N. and Mallet-Paret, J. [1977]. Integral averaging and Hopf's bifurcation. J. Diff. Eqns. 26, 112-159.
478
References
Coddington, E. and Levinson, N. (1955]. Theory of ordinary differential equations. McGraw Hill, New York. Cohen,D.S. and Neu, J.C. [1978]. Interacting oscillatory chemical reactors. Ann. N.Y. Acad. Sci. 316, 332-337. Cohen, D.S., Coustsias, E. and Neu, J.C. [1979]. Stable oscillations in single species growth models with hereditary effects. Math. Biosci. 44, 255-268. Cooke, K.L and Grossman, G. [1982]. Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl. 86, 592-627. Cooke, K.L and Wiener, J. [1984]. Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl. 99, 265-297. Cooke, K.L. [1966]. Functional differential equations close to ordinary differential equations. Bull Amer. Math. Soc. 72, 285-288. Coppel, W.A. [1965}. Stability and asymptotic behavior of ordinary differential equations. D.C. Heath, Boston. Corduneanu, C. [1971]. Principles of differential and integral equations. Allyn and Bacon, Boston, Massachusetts. Corduneanu, C. and Luca, N. [1975]. The stability of some feedback systems with delay. J. Math. Anal. Appl. 51,377-393. Crandall, M. and Rabinowitz, P. [1972]. The Hopf-bifurcation theorem in infinite dimensions. Arch. Rat. Mech. Anal. 45, 79-100. Cushing, J.M. [1977]. Integrodifferential equations with delay models in population dynamics: Lect. Notes in Biomath. 20, Springer-Verlag, Berlin. Cushing, J.M [19791. Volterra integrodifferential equations in population dynamics: in 'Mathematics in Biology' Proc. C.I.M.E., Italy. Datko, R. [1977]. The uniform asymptotic stability of certain neutral differential difference equations. J. Math. Anal. Appl. 58, 510-526. Datko, R. [1978]. A procedure for determination of the exponential stability of certain differential equations. Q. Appi. Maths. 36, 279-292.
References
479
Datko, R. [1983]. An example of an unstable neutral differential equation. Int. J. Control 38, 263-267. Dean, A.M. [1983]. A simple model of mutualism. Amer. Natural. 121, 409 - 417. Driver,R.D., Sasser, D.W. and Slater, M.L. [1973J. The equation x(t) bx( t - T) with small delay. Amer. Math. Monthly 80, 990-995.
== ax(t) +
Driver, R.D. [1977]. Ordinary and delay differential equations. Applied Math. Sciences, Springer-Verlag, New York. Edelstein - Keshet L. [1988]. Mathematical Models in Biology, Random House, New York. Eisenfeld, J. [1981J. On approach to equilibrium in nonlinear compartmental systems. pp.229 -246, Diff. Eqns. and Appl. Ecol., Epid., and Pop. Dyn., (Eds. S.N. Busenberg and K.L. Cooke), Academic Press, New York. EI'sgol'ts, L.E. and Norkin, S.B. [1973]. Introduction to the theory and application of differential equations with deviating arguments. Academic Press, New York. Fargue, D.M. [1973). Reducibilite des systemes dynamiques. C.R. Acad. Sci. Paris, Ser. B. 277, 471-473. Field, R.J. and Noyes, R.M. [1974]. Oscillations in chemical systems. IV, Limit cycle behavior in a model of a real chemical reaction. J. Chern. Phys. 68, 1877-1884. Fisher, M.E. and Goh, B.S. [1984]. Stability results for delayed-recruitment models in population dynamics. J. Math. BioI. 19, 147-156. Fox, L., Mayers, D.F., Ockendon, J.R. and Taylor, A.B. [1971]. On a functional differential equation. J. Inst. Math. Appl. 8, 271-307. Franklin, J. [1980]. New York.
Methods of mathematical economics.
Springer Verlag,
Franklin, J.N. [1968]. Matrix theory. Prentice Hall, London. Freedman, H.I. [1980]. Deterministic mathematical models in population ecology. Marcel Dekker, New York.
480
References
Freedman, H.I. and Gopalsamy, K. [1988]. Nonoccurrence of stability switcbing in systemswitb discrete delays. Canad. Math. Bull. 31, 52-58. Freedman, H.I. and Kuang, Y. [1991]. Stability switcbes in linear scalar neutral delay equations. Funkcialaj Ekvac., 34, 187-209. Fukagai, N. and Kusano, T. [1983a]. On tbe second order functional differential equations and inequalities witb deviating arguments. Monat. Math. 96, 107-118. Fukagai, N. and Kusano, T. [1983b]. Oscillation tbeory of first order functional differential equations witb deviating arguments. Annali di Matematica. 136, 95-117. Gantmacher, F.R. [1959]. The theory of matrices, Vol. II, Chelsea, New York. Gard, T.C [1980]. Persistence in food webs: Holling type food cbains. Math. Biosci. 49, 61-67. Gard, T.C. and Hallam, T.G. [1979]. Persistence in food webs: Lotka-Volterra food cbains. Bull. Math. BioI. 41, 877-891. Goel, N.S., Maitra, S.C. and Montroll, E.W. [1971]. Nonlinear models of interacting populations. Academic Press, New York. Gopalsamy, K. [1980]. Time lags and global stability in two species competition. Bull. Math. BioI. 42, 729-737. Gopalsamy, K. [1983a]. Harmless delays in model systems. Bull. Math. BioI. 45, 295-309. Gopalsamy, K. [1983b]. Stability and decay rates in a class of linear integrodifferential systems. Funkcialaj Ekvacioj 26, 251-261. Gopalsamy, K. [1983c]. Transit delays and dispersive populations in patcby babitats. Int. J. Syst. Sci. 14, 239-268. Gopalsamy, K. and Ahlip, R.A. [1983]. Time delays in n-species competition - I; Global stability in constant environments. Bull. Austral. Math. Soc. 27, 427-441.
References
481
Gopalsamy, K. [1984a]. Global asymptotic stability in Volterra's population systems. J. Math. BioI. 19, 157-168. Gopalsamy, K. [1984b]. Delayed responses and stability in two species systems. J. Austral. Math. Soc. Ser. B 25, 473-500. G~palsamy, K. [1984c].
Oscillations in linear systems of differential difference equations. Bull. Austral. 1vlath. Soc. 29, 377-387.
Gopalsamy, K. [1986aJ. Oscillations in a linear system of differential difference equations of arbitrary order. J. Math. Anal. Appl. 120, 360-369. Gopalsamy, K. [1986b]. Convergence in a resource based competition system. Bull. Math. BioI. 48, 681-699. Gopalsamy, K. [1987]. Oscillatory properties of systems of first order scalar linear delay differential inequalities. Pacific J. Math. 128, 299-305. Gopalsamy, K. and Rai, B. [1988]. Stability analysis of syncbronous oscillations in a weakly coupled integrodifferential system. J. Math. Anal. App!. 136, 369-382. Gopalsamy, K., Kulenovic, M.R.S. and Ladas, G. [1988J. limited population mode1. Appl. Anal. 31, 225-237.
Time lags in a food
Gopalsamy, K. and Zhang, B.G. [1988]. On a neutral delay logistic equation. Dyn. Stab. Syst. 2, 183-195. Gopalsamy, K., Gyori, I and Ladas, G. (1989a]. Oscillations of a class of delay equations witb continuous and piecewise constant arguments. Funkcialaj Ekvacioj, 32, 395-406. Gopalsamy, K., Kulenovic, M.R.S. and Ladas, G. [1989b]. Oscillations and global attractivity in respiratory dynamics. Dyn. Stab. of Syst. 4, 131-139. Gopalsamy, K., Kulenovic, M.R.S. and Ladas, G. [1990a]. Oscillations of a system of delay logistic equations. J. Math. Anal. App!. 146, 192-202. Gopalsamy, K., Kulenovic, M.R.S. and Ladas, G. [1990b]. On a logistic equation witb piecewise constant arguments. Diff. and Int. Eqns. 4, 215-223.
482
References
Gopalsarny, K. and Zhang, B.G. [1989]. On delay differential equations with impulses. J. Math. Anal. Appl. 139, 110-122. Gopalsarny, K. and Zhang, B.G. [1990). Oscillation and nonoscillation in first order neutral differential equations. J. Math. Anal. Appl. 151, 42-57. Gopalsamy, K. [1991]. Oscillations in Lotka Volterra systems with several delays. J. Math. Anal. Appl. 159, 440-448. Gopalsamy, K. [1992]. On a neutral Lotka Volterra system. Math. Meth. Appl. Sci. (to appear). Gopalsamy K and P. Weng [1992]. Feedback regulation of logistic growth. Int. J. Math. and Math. Sci. (to appear). Gosiewski, A. and Olbrot, A.W. [1980]. The effect of feedback delays on the performance of multivariable linear control systems. IEEE Trans. Auto. Cont. 25, 729-734. Gromova, P.S. [1967]. Stability of solutions of nonlinear equations of the neutral type in the asymptotically critical case. Math. Notes. 1, 472-479. Gromova, P.S. and Pelevina, A.F. [1976). Absolute stability of automatic control systems with time lag. Difr. Uravn. 13, 1375-1383. Gromova, P.S. and Zverkin, A.M. [1986J. On trigonometric series whose sums are continuous unbounded functions on the real axis- solutions of equations with retarded arguments. Diff. Uravn. 4, 1774-1784. Gurgula, S.L [1982]. A study of the stability of solutions of impulse systems by Lyapunov's second method. Ukr. Math. J. 34, 100-103. Gyori, 1. and Eller, J. [1981J. Compartmental systems with pipes. Math. Biosci. 53, 223-247. Gyori, 1. [1986]. Connections between compartmental systems with pipes and integrodifferential equations. Math. Model. 7, 1215-1238. Gyori, I. and Ladas, G. [1989]. Linearised oscillations for equations witb piecewise constant arguments. Diff. and Int. Eqns. 2, 123-131.
References
483
Gyori, I. and Wu, J. [1991]. A neutral equation arising from compartmental systems witb pipes. J. Dyn. Diff. Eqn~: 3, 289-311. Haddock, J.R. [1974]. On tbe asymptotic behavior of solutions of x(i) -a(t)f(x(t - ret))). SIAtvf J. Math. Anal. 5, 569-573.
=
Halanay, A. [1966J. Differential Equations. Academic Press, New York. Hale, J.K. [1977]. Theory of functional differential equations. Springer Verlag, New York. Hale, J.K., Infante, E.F. and Tsen, F.S.P. [1985]. Stability in linear delay equations. J. Math. Anal. Appl. 105, 533-555. Hale, J.K. and Sternberg, N. [1988J. Onset of cbaos in differential equations. J. Compo Phys. 77, 221-239. Harrison, G.W. [1979J. Global stability of food cbains. Amer. Natural. 114, 455-457.
Hassard, M.W., Kazarinoff, N.D. and Wan, Y.H. (1981). Theory and applications of Hopf bifurcation. London Math. Soc. Lecture Notes series, 41. Cambridge Uni. Press, Cambridge. Henry, D. [1974]. Linear autonomous neutral functional differential equations. J. Diff. Eqns. 15, 106-128. Hirsch, M.W. [1982). Systems of differential equations whicb are cooperative - I. Limit sets. SIAM J. Math. Anal. 13,167-179. Hirsch, M.W. [1983). Differential equations and convergence almost everywhere in strongly monotone semifIows. Contemporary Math. 17,267-285. Hirsch, M.W. [1984J. The dynamical systems approach to differential equations. Bull. Amer. Math. Soc. 11, 1-64. Hirsch, M.W. [1985J. Systems of differential equations tbat are competitive or cooperative II. Convergence almost everywhere. SIAM J. Math. Anal. 16, 423-439.
Hirsch, M.W. [1988a]. Stability and convergence in strongly monotone dynamical systems. J. Reine Angew. Math. 383, 1-53.
484
References
Hirsch, M.W. [1988b]. Systems of differential equations which are competitive or cooperative: III. Competing species. Nonlinearity 1, 51-71. Hofbauer, J. and Sigmund, K. [1987]. Permanence for replicator equations; pp. 70-91 in Lect. Notes in Econ. Math. Syst. 287, Springer- Verlag. Hofbauer, J. and Sigmund, K. [1988]. The theory of evolution and dynamical systems. Cambridge Uni. Press, Cambridge. Hopf, E. [1942). Abzweigung einer periodischen Losung von einer stationaren Losung eines Differentialsystems (Bifurcation of a periodic solution from a stationary solution of a system of differential equations). Ber. Math. Phys. Klasse de Sachs. Akad. Wiss. Leipzig, 94, 3-22. Howard, L.N. [1979]. Nonlinear oscillations: in "Oscillations in Biology" F.R. Hoppensteadt (Ed.) A.M.S. Lectures in AppI. Maths. 17, 1-69. Hsu, S.B. and Hubbell, S.P. [1979]. Two competing predators competing for two prey species: An analysis of MacArthur's model. Math. Biosci. 47, 143-171. Hsu, S.B. [1981]. On a resource based ecological competition model with interference. J. Math BioI. 12, 45-52. Huang Zhen Xun and Lin Xiao Biao [1982]. A problem on the stability of retarded systems. Chinese Annals of Math. 3, 115-120. Hunt, B.R. and Yorke, J.A. [1984]. When all solutions ofx(t) oscillate? J. Diff. Eqns. 53, 139-145.
= - L: qjX(t-Ti(t))
Hutchinson, G.E. [1948]. Circular causal systems in ecology. Ann. N.Y. Acad. Sci. 50, 221-246. Hutchinson, G.E. [1978]. An introduction to population ecology. Yale University Press, New Haven. Infante, E.F. and Castelan, W.B. [1978]. A Liapunov functional for a matrix differential difference equation. J. Diff. Eqns. 29, 439-451. Jacquez, J.A. [1972). Compartmental analysis in biology and medicine. Elsevier, New York.
References
485
Jansen, W. [1987]. A pennanence theorem for replicator and Lotka Volterra systems. J. Math. BioI. 21, 285-298. Jiong, R. [1991J. Asymptotic behavior of a biological model with delay, pp. 318326 in Functional Differential Equations (Eds. Yoshizawa, T. and Kato, J.), World Scientific, Singapore. Jones, G.S. [1962J. On the nonlinear differential-difference equation j( x) = - f( x1)[1 + f(x)]. J. Math. Anal. Appl. 4,440-469. Kakutani, S. and Markus, M. [1958]. On the nonlinear difference differential equation yet) = [A - By(t -'- r)Jy(t) in 'Contributions to the theory of nonlinear oscillations'. 4, 1-18. Kato, T. [1972]. Asymptotic behavior of solutions of the functional differential equation y( x) = aye AX )+by( x), in "Delay and functional differential equations and their applications" (K. Schmitt, Ed.) pp. 197-219, Academic Press. Kato, T. and McLeod, J.B. [1971]. The functional differential equation: y(x) aY(Ax) + bye x). Bull. Amer. Math. Soc. 77, 891-937.
=
Kawata, M. and Suzuki, R. [1980]. Two coupled oscillators as a model of the circadian pacemaker. J. Theor. BioI. 86, 547-575. Kawata, M., Fujita, M., Suzuki, R. and Winfree, A.T. [1982J. A three oscillator model of the human circadian system controlling the core temperature rhythm and sleep wake cycle. J. Theor. BioI. 98, 369-392. Kaykobad, M. [1985]. Positive solutions of positive linear systems. Lin. Alg. Appl. 64, 133-140. Khusainov, D.Ya. and Yun'kova, E.A. [1981]. Estimation ofmagnitude of retardation in linear differential systems with deviated argument. Ukrain. Matemat. Zh. 35, 261-264. Khusainov, D.Ya. and Yun'kova, E.A. [1988]. Investigation of the stability oflinear systems of neutral type by the Lyapunov function method. Diff. Uravn. 24, 613-621. Kirlinger, G. [1986J. Pennanence in Lotka - Volterra equations: Linked prey predator systems. Math. Biosci. 82, 165-191.
References
486
Kolmanovskii V.B and V.R. Nosov [1986]. Stability of functional differential equa-
tions. Academic Press, New York. Koplatadze, R.G. and Chanturija, T.A. [1982]. On the oscillatory and monotone solutions of first order differential equations with deviating arguments. Diff. Urav. 18, 1463-146q. Kozakiewicz, E. [1977]. On the asymptotic behavior of positive solutions of two differential inequalities with retarded argument, pp. 309-319 in Differential Equations, CoIl. Math. Soc. Janos Bolyai 15, (Ed. M. Farkas) North Holland, Amsterdam. Krasnoselskii, M.A. [1968]. Translation along trajectories of differential equations. Amer. Math. Soc. Transi. 19, Providence, Rhode Island. Krasovskii, N.N. [1963]. Stability of motion. (Translation by J. Brenner) Stanford University Press, 1963. Krikorian, N. [1979]. The Volterra model for three species predator prey systems: Boundedness and stabilty. J. Math. BioI. 7, 117-132. Krisztin, T. [1984]. Convergence of solutions of nonlinear integrodifferential equations arising in compartmental systems. Acta. Sci. Math. Szeged. 47, 471-485. Kuang, Y. [1991]. On neutral delay two species Lotka - Volterra competitive systems. J. Austral. Math. Soc. Ser. B. 32, 311-326. Kuang, Y. and Feldstein, A. [1991]. Boundedness of solutions of a nonlinear nonautonomous neutral delay equation. J. Math. Anal. Appl. 156, 293304. Kulenovic, M.R.S., Ladas, G. and Meimaridou, A. [1987a]. On oscillation of nonlinear delay differential equations. Quart. Appl. Math. XLV, 155-164. Kulenovic, M.R.S., Ladas, G. and Meimaridou, A. [1987b]. Necessary and sufficient conditions for oscillations of neutral differential equations. J. AustraL Math. Soc. Ser. B. 28, 362-375. Kuramoto, Y. [1984J. Verlag, Berlin.
Chemical oscillations, waves and turbulence.
Springer-
References
481
Ladas, G. [1979]. Sharp conditions for oscillations caused by delays. Appl. Anal. 9,95-98. Ladas, G., Sficas, Y.G. and Stavroulakis, I.P. [1983aJ. Asymptotic behavior of solutions of retarded differential equations. Proc. Amer. Math. Soc. 88, 247-253. Ladas, G., Sficas, Y.G. and Stavroulakis, I.P. [1983bJ. Necessary and sufficient conditions for oscillations. Amer. Math. Monthly 11, 637-640. Ladas, G. and Stavroulakis, I.P. [1982]. Oscillations caused by several retarded and advanced arguments. J. Diff. Eqns. 44, 134-152. Ladas, G., Philos, Ch.G. and Y.G. Sficas [1991]. Oscillations of integrodifferential equations. Diff. and Int. Eqns. (to appear). Ladas, G. and Sficas, Y.G. [1986J. Oscillations of neutral delay differential equations. Canad. Math. Bull. 29, 438-445. Ladde G.S., Lakshmikantham, V. and B.G. Zhang [1987]. qscillation theory of differential equations with deviating arguments. Marcel Dekker Inc. Landman, K.A. [1980]. Bifurcation and stability theory of periodic solutions for integrodifferential equations. Studies in Appl. Math. 62, 217-248. LaSalle, J.P. [1977a]. Stability of dynamical systems. Conference Board of Mathematical Sciences, Regional Conference Series in Mathematics, Missisipi State University. Aug. 1975, Published by SIAIvL LaSalle, J.P. [1977b]. Stability theory of difference equations, Studies in Ordinary Differential equations, Vol. 14, Studies in Mathematics (J.Hale, Ed.). Amer. Math.Soc., Providence. LaSalle, J. and Lefschetz, S. [1961]. Stability by Lyapunov's direct method. Academic Press, New York. Lefever, R. and Prigogine, I. [1969]. Symmetry breaking instabilities in dissipative systems - II. J. Chern. Phys. 48, 1695-1700. Lenhart, S.M. and Travis, C.C. [1986]. Global stability of a biological model with time delay. Proc. Amer.. Math. Soc. 96, 75-78.
488
Reference8
Levin, J.J [1963]. The asymptotic behavior of the solution of a Volterra equation. Pro. Amer. Math. Soc. 14, 533-541. Lewis, R.M. and Anderson, B.D.O. [1980a). Necessary and sufficient conditions for delay independent stability of linear autonomous systems. IEEE Trans. Auto. Contr. 25, 735-739. Lewis, R.M. and Anderson, B.D.O. [1980b). Insensitivity of a class of nonlinear compartmental systems to the introductions of arbitrary time delays. IEEE Trans. Circuits Syst. 27, 604-612. Liao Xiaoxin [1986]. Lumped iteration method for stability analysis of large scale dynamical systems of neuter type. Scientia Sinica 29, 225-240 Lim, E.B. [1976]. Asymptotic behavior of solutions of the functional differential equation xCi) = Ax('xi) + BX(i) ,'x > O. J. Math. Anal. Appl. 55, 794-806. Li-Ming Li [1988]. Stability of linear neutral delay-differential systems. Austral. Math. Soc. 38, 339-344.
Bull.
Li Senlin [1983]. The stability of superneutral functional differential equations. Scientia Sinica, Series A, 26, 1-10. Li Senlin [1987]. A V-function method for stability offunctional differential equations and its applications. Scientia Sinica 30, 449-461. MacCamy, R.C. and 'Nong, J.S.W. [1972]. Exponential stability for a nonlinear functional differential equation. J. Math. Anal. Appl. 39, 699-705. MacDonald, N. (1978). Time lags in biological models. Lecture Notes in Biomath. 27, Springer-Verlag, New York. Maeda, H. S., Kodama, S. and Ohata, Y. [1978]. Asymptotic behaviour of nonlinear compartmental systems; nonoscillation and stability. IEEE Trans. Circuits Syst. CAS 25, 372-378. Marcus, C.M., Waugh, F.R. and Westervelt, R.M. [1991]. Nonlinear dynamics and stability of analog neural networks. Physica D 51, 234-247. Marsden, J.E. and McCracken, M. [1976]. The Hopf bifurcation and its applications. Springer-Verlag, Berlin.
Reference3
489
Martin, Jr. R.H. [1976J. Asymptotic behavior of solutions to a class of nonlinear parabolic systems; Research Notes in Math. 53, (Eds. H.Brezis and J .L.Lions), Nonlinear Partial differential equations and their applications, Pitman, Boston. Martin, Jr. R.H. f1978]. Asymptotic stability and. critical points for nonlinear quasimonotone parabolic systems. J. Diff. Eqns. 30, 391-423. Martin, Jr. R.H. (1981). Asymptotic behavior of solutions to a class of quasimonotone functional differential equations, in 'Research Notes in Math. 48' (Eds. F. Kappel and W. Schappacher) Abstract Cauchy Problems and Functional Differential Equations, Pitman, Boston, 91-111. Martynuk, A.A. and Obolenski, A.Yu. [1980]. Stability of solutions of autonomous Wazewski systmes. Diff. Uravneniya. 16, 1392-1407. Matano, H. (1984). Existence of nontrivial unstable sets for equilibrium of strongly order preserving systems. J. Fac. Sci., Univ. of Tokyo 30, 645-673. May, R.M. [1973). Stability and complexity in model ecosystems. Princeton Univ. Press, New Jersey, U.S.A. May, R.M. [1975]. Biological populations obeying difference equations: Stable points, stable cycles and chaos. J. Theor. BioI. 51, 511-524. May, R.M. and Oster, G.F. [1976]. Bifurcation and dynamic complexity in simple ecological models. Amer. Natural. 110, 573-599. May, R.M. and Leonard, W.J. [1975]. Nonlinear aspects of competition between three species. SIAM J. AppI. Math., 29, 243-253. Maynard Smith J. [1974]. Models in ecology. Cambridge Univ. Press, Cambridge, England. Mazanov, A. [1976]. Stability of multi-pool models with time lags. J. Theor. BioI. 59, 429-442. Michel, A.N. and Miller, R.K. [1977]. Qualitative analysis of large scale systems. Academic Press, New York. Mikhailova, M.P. and Podgornov, V.V. [1965J. Differential inequalities for equations with lagging argument. Diff. Urav. 1, 1183-1189.
490
References
Miller, R.S (1976). Pattern and process in competition. Adv. Ecol. Res. 4, 1-74.
Misnik A.F. [1972]. The stability of systems of differential equations of neutral type. Ukr. Math. J.24, 20-29. Mori, T., Fukuma, N. and Kuwahara, M. [1982]. On an estimate of the decay rate for stable linear delay systems. Int. J. ControL 36, 95-97. Morita, Y. [1983]. Instability of spatially homogeneous periodic solutions to delaydiffusion equations. Lecture Notes in Num. Appl. Anal. 6, 107-124. Morita, Y. [1984]. Destabilisation of periodic solutions arising in delay diffusion systems in several space dimensions. Japan. J. Appl. Math. 1,39-65. Morita, Y. [1985}. Stability changes of periodic solutions to a coupled nonlinear equation with time delay. Publ. RIMS, Kyoto Univ. 21,47-74. Morita, Y. [1987J. A periodic wave and its stability to a circular chain of weakly coupled oscillators. SIAM. J. Math. Anal. 18, 1681-1698. Morita, Y. [1988J. A secondary bifurcation problem of weakly coupled oscillators with time delay. (preprint). Mulholland, R.J. and Keener, M.S. [1974]. Analysis of linear compartment models for ecosystems. J. Thoer. BioI. 44, 105 -116. Murdoch, W.W. and Oaten, A. [1975J. Predation and population stability. Adv. Ecol. Res. 9, 1-131. Murray, J.D. [1989J. Mathematical Biology. Springer - Verlag, Berlin. Myshkis, A.D. (1972J. Linear differential equations with retarded argument. (Russian) Izdatel'stvo "Nauka", 2nd Edition. Nashed, M. and Wong, J.S.W. [1969]. Some variants of a fixed point theorem of Krasnoselskii and applications to nonlinear integral equations. J. Math. Mech. 18, 767-777. Noonburg, V.W. (1971J. Bounded solutions of a nonlinear system of differential delay equations. J. Math. Anal. Appl. 33,66-76.
References
491
Nunney, L. [1985a]. Absolute stability in predator prey models. Theor. Pop. BioI. 28, 109-132. Nunney, 1. [1985b]. The effect of long time delays in predator prey systems. Theor. Pop. BioI. 27, 202-221. Nunney, L. [1985c]. Short time delays in population models: a role in enhancing stability. Ecology 66, 1849-1858. Obolenskii, A. Yu. [1983]. Stability of solutions of autonomous Wazewskii systems with delayed action. Ukr. Mat. Zh. 35, 574-579. Ohta, Y. [1981]. Qualitative analysis of nonlinear quasimonotone dynamical systems described by functional differential equations. IEEE. Trans. Circuits and Systems. CAS 28, 138-144 . . Oliveira-Pinto, F. and Conolly, B.W. [1982]. Applicable Mathematics of NonPhysical Phenomena. Ellis Horwood Limited, New York. Pandit, S.G. and Deo, S.G. [1982]. Differential systems involving impulses. Leet. Notes in Math. 954, Springer-Verlag, New York. Pavlidis, T. [1973]. Biological oscillators: Their mathematical analysis. Academic Press. Perestyuk, O.S., and Chemikova, O.S. [1984]. A contribution to the stability problem for solution of systems of differential equations with impulses. Ukr. Math. J. 36, 190-195. Peschel, M. and Mende, W. [1986]. The predator prey model: Do we live in a Volterra world. Springer-Verlag, New York. Philos, Ch. G. [1988]. Oscillatory behavior of systems of integrodifferential equations. Bull. Greek Math. Soc. 29, 131-141. Philos, Ch. G. [1990a]. (preprint)
Positive solutions of integrodifferential equations.
Philos, Ch. G. [1990b]. Oscillation and nonoscillation in integrocfjfferential equations (preprint). Pielou, E.C. [1977]. Mathematical Ecology. John Wiley and Sons, New York.
References
492
Pirapakaran, R. (19891. Impulsive integral inequalities of Gronwall - Bihari type. Pree. Int. Conf. Theory and Appl. Diff. Eqns. Vol. II, Ohio UnL Press. Plemmons, R.V [1977]. M-matrix cl1aracterisations I: Nonsingular M-matrices. Lin. Alg. Appl. 18, 175-188. Post, W.M. and Travis, C.C. [1982]. Global stability in ecological m6dels with continuous time delay. Lect. Notes Pure and Appl. Math. 67, 241-250. Qin Yuan-Xun, Liou Iong-qing and Wang Lian. [1960]. Effect of time lags on stability of dynamical systems. Scientia Sinica, IX, 719-747. Razumikhin, B.S. [1960J. Application of Liapunov's method to problems in the stability of systems with delay. Avomat. i Telemech. 21, 740-748. Ricklefs, R.E. [1973]. Ecology. Chiron Press, Newton, Mass. Rose, M.R. [1987]. Quantitative Ecological Theory. Croom Helm, London. Royden, H.L. [1963J. Real Analysis. Macmillan, New York. Rozhkov, V.I. and Popov, A.M. [1971]. Inequalities for solutions of certain systems of differential equations with large time-lag. Diff. Uravn. 7, 271-278. Sandberg, LW. [1978J. On the mathematical foundations of compartmental analysis in biology, medicine and ecology. IEEE Trans. Circuits and Syst. CAS 25, 273-279. Sattinger, D.H. [1973J. Topics in stability and bifurcation theory. Lect. Notes in Math. 309. Springer-Verlag, New York. Schoener, T.W. [1976]. Alternatives to Lotka-Volterra competition: models of intermediate complexity. Theor. Pop. BioI. 10, 309-333. Schust~r,
P., Sigmund, K. and Wolff, R. [1979]. On w limits of competition between three species. SIAM J. Appl. Math. 37,49-54.
Scudo, F.M. and Ziegler, J.R. [1978]. The golden age of theoretical ecology: 19231940. Lect. Notes in Biomath. 22, Springer Verlag, Berlin. Seifert, G. [1973]. Lyapunov-Razumikhin conditions for stability and boundedness of functional differential equations of Volterra type. J. Diff. Eqns. 14, 424-430.
References
493
Selgrade, J.F. [1980]. Asymptotic bebavior of solutions to single loop positive feedback systems. J. Diff. Eqns. 38, 80-103 Shibata, A. and Saito, N. [1980]. Time delays and chaos in two competing species. Math. Biosci. 51, 199-211. Siljak, D.D. [1978]. Large scale dynamic systems: Stability and Structure. Elsevier, North-Holland Inc., New York. Simpson, H.C. [1980]. Stability of periodic solutions of integrodifferential equations. SIAM J. Appl. Mat.h. 38,341-363. Sinha, A.S.C. and Williams, W.J. [1972]. On tbe boundedness of solutions of some systems witb retarded arguments. lnt. J. Control. 16,719-722. Slobodkin, L.B. [1980]. Growth and regulation of animal populations. Dover Publications, New York. Smale, S. [1974J. A matbematical model of two cells via Turing's equation, III "Some mathematical questions in biology V" (Ed.J. D. Cowan) A.M.S. Lectures on Mathematics in Life Sciences. 6, 15-26. Smale, S. [1976]. On tbe differential equations of species competition. J. Math. BioI. 3, 5-7. Smith, F.E. [1963]. Population dynamics in Daphnia magna. Ecology, 651 -663. Smith, H.L. [1986a]. On tbe asymptotic behavior of a class of deterministic models of cooperating species. SIAM. J. App!. Math. 46, 368-375. Smith, H.L. [1986b]. Cooperative systems of differential equations with concave nonlinearities. Nonlin. Anal. 10, 1037-3052. Smith, H.L. [1986c]. Competing subcommunities of mutualists and a generalised Kamke tbeorem. SIAM J. App!. Math. 46,856-874. Smith, H.L. [1987]. Monotone semifIows generated by functional differential equations. J. Diff. Eqns. 66, 420-442. Snow, W. [1965]. Existence, uniqueness and stability for nonlinear differential difference equations in the neutral case. N.Y.U. Courant lnst. Math. Sci. Rep. IMM-NYU 328.
494
References
Staffans, O. [1975]. Nonlinear Volterra equations witb positive definite kernels.
Pro. Amer. Math. Soc. 5, 103-108. Stech, H.W. [1979]. The Hopf-Bifurcation: A stability result and application. J. Math. AnaL Appl. 71, 525-546. Stokes, A. [1964]. On the stability of a limit cycle of autonomous functional differential equation. Contr. Diff. Eqns. 3, 121-140. Tokumaru, H., Adachi, N. and Amemiya, T. [1975J. Macroscopic stability of interconnected systems. Proc. IFAC 6th World Congress, Boston, Paper 44.4. Torre, V. [1975]. Synchronization of non-linear biochemical oscillators coupled by diffusion. BioI. Cybernetics 17, 137-144. Tsalyuk, V.Z. [1973]. On an estimate for solutions to linear differential equations with delays. Diff. Uravn. 9, 2099-2101. Turner Jr. M.E and Pruitt, K.M. [1978]. A common basis for survival, growth and autocatalysis. Math. Biosci. 39, 113-123. Turner Jr.M.E., Bradley Jr E.L., Kirk, K.A. and Pruitt, K.M. [1976]. A tbeory of growth. Math. Biosci. 29, 367-373. Vandermeer, J.H. and Boucher, D.H. [1978]. Varieties of mutualistic interaction in population models. J. Theor. BioI. 74, 549-558. Vescicik M [1984]. Oscilacia Rieseni Diferencialnycb Rovnic S oneskorenym Argumentom. Acta Math. Universitatis Comenianae, XLIV-XLV, 257-263. Vidyasagar, M. [1978]. Nonlinear systems analysis. Prentice Hall, New Jersey. Volterra, V. [1931]. Lecon sur la theorie mathematique de la lutte pour la vie. Gauthier Villars, Paris. Waltman, P. [1983]. Competition models in population biology. SIAM, Philadelphia, USA. Wang Lian [1962]. On tbe equivalence problem of differential equations and difference equations in tbe theory of stability of the first critical case. Chinese Maths. 1, 118-137.
References
495
Wangersky, P.J. and Cunningham, W.J. [1957]. Time lag in prey-predator population models. Ecology 38. 136-139. Wendi, W. and Ma Zhien [1991]. Harmless delays and uniform persistence. J. Math. Anal. Appl. 158, 256-268. Wheldon, T.E. [1975]. Mathematical models of oscillatory blood cell production Math. Biosci. 24, 289 - 305. Winfree, A.T. [1980]. The geometry of biological time. Springer-Verlag, Berlin. Winsor, C.P. [1934]. Mathematical analysis of the growth of mixed populations. Cold Spring Harb. Symp. Quant. BioI. 2, 181-187. Winston, E. [1970]. Comparison theorems for the scalar delay differential equations. J. Math. Anal. Appl. 29, 455-463. Wolin, C.L. and Lawlor, L.R. [1984]. Models of facultative mutualism: density effects. Amer. Natural. 144, 843-862. Worz-Busekros, A. [1978]. Global stability in ecological systems with continuous time delay. SIAM J. Appl. Math. 35, 123-134. Wright, E.M. [1955]. A nonlinear difference differential equation. J. Reine Angew. Math. 194, 66-87. Wu, J. [1986]. Stability of neutral functional differential equations with infinite delay. Funckcialaj Ekvacioj 29, 131-139.
Xu D.-Y. [19891. Robust stability analysis of uncertain neutral delay differential systems via differential inequality. Control Theory and Adv. Tech. 5, 301313. Yan, J. [1987]. Oscillation of solutions of first order delay differential equations. Nonlin. Anal. 1l, 1279-1287. Yamada, Y. (1982]. On certain class of semilinear Volterra diffusion equations. J. Math. Anal. Appl. 88, 433-451. Yodzis, P. [1989]. Introduction to theoretical ecology. Harper & Row. London, 1989.
496
References
Yoneyama, T. [19861. On the stability for the delay differential equation x(t) -a(t)f(x(t - ret))). J. Math. Anal. Appl. 120, 271-275.
===
Yoneyama, T. [1987]. On the ~ stability theorem for one dimensional delay differential equations. J. Math. Anal. Appl. 125, 161-173. Yoneyama, T. and Sugie, J. [1988]. Exponentially asymptotically stable dynamical systems. Appl. Anal. 27, 235-242. Yorke, J.A. [1970]. Asymptotic stability for one dimensional differential-delay equations. J. Dur. Eqns. 7, 189-202. Zhang, B.G. [1989]. Oscillation of first order neutral functional differential equations. J. Math. Anal. Appl. 139, 311-318. Zhang, B.G. and Gopalsamy, K. [1988]. Oscillation and non oscillation in a nonautonomous delay-logistic equation. Quart. Appl. Math. XLVI, 267-273. Zhang, B.G. and Gopalsamy, K. [1990]. Global attractivity in the delay logistic equation with variable parameters. Math. Proc. Camb. Phil. Soc. 107, 579-590. Zhang Yi [1988aJ. Stability of large scale neutral differential systems. Scientia Sinica (Series A) 31, 1292-1304. Zhang Yi [1988b]. The stability in the large for differential systems with infinite delay. Ann. Diff. Eqns. 4,373-380. Zhivotovskii, L.A. [1969J. Absolute stability of the solutions of differential equations with retarded arguments. (in Russian) Trudy Sem. Teor. Diff. Urav. Otkl. Arg. 7, 82 -91. Zverkin, A.M. [1962]. Theorems of existence and uniqueness for equations with deviating argument in critical case. Transactions of a seminar on the theory of differential equations with deviating arguments. 1, 37-46. Zverkin, A.M. [1968]. On the connection between boundedness and stability of solutions of linear systems with an infinite number of degrees of freedom. Diff. Eqns. 4, 196-197.
Index
~braharnson D.L.
264
absolute stability 193, 194, 217 Aftabizadeh A.R. 79 Akhmerov R.R. 393 Alexander J.C. 148 Amemiya T. 447 . almost periodic 37 an der Heiden 208, 307 Anderson B.D.O. 447 Anvarinov R. 386 Araki M. 228,230,337,363 Arino O. 37,256,468 Arzela-Ascoli 45,78 Ashkenazi M. 148 Atkins G.L. 355 Ayala F.J. 195 Bailey H.R. 447 Bainov D.D. 90 Banks H.T. 255, 298, 318, 375 BarbaJat 1. 4, 5, 30, 31, 264, 325, 396, 426,436 Barbashin E.A. 90, 215, 339 Barbu, V. 27 Barker G.P. 339 Bellman R. i, 9, 126, 188, 206, 207, 211, 310 Berman A. 295 Borisenko S.D. 90 Borsellino A. 124 Boucher D.H. 191 Braddock R.D. 109
Brayton R.K. 466 Brelot M. 200 Brian M. V. 182, 298 Bromwich T.J. 36 Brouwer fixed point theorem 360 Brumley W.E. 393 Burton T.A. 14, 31, 217, 271 Busenberg S.N. '263,373,374 eai Sui Lin 239 Carvalho L.A. V. 264 Castelan W.B. 264 Chandra J. 148 Chang Hsueh Ming 239 chaotic behavior 79, 307, 311 characteristic exponent 139, 142 Chew K.H. 222 Chin Yuan Shun 239 Chow S.N. 131 Coddington E. 127 coexistence 347 Cohen D.S. 124, 148 comparison 48,54,222,225 compartments 355,361,363,366,368, 460 competition 168, 182, 195 competitive exclusion 306 contraction 429, 430 Cooke K.L. 14,79,239 cooperation 168, 172, 182, 191, 194, 318, 326, 340 Coppel W.A. 192, 321
498
Index
Gantmacher F.R. 219, 230 Corduneanu C. 26,91 Gard T.C. 383,384 coupled oscillators 148 Gersbgorin's theorem 259,308,364 Crandall M. 138 Cushing J.M. ii, 124, 125, _131, Goel N.S. ii, 210 global attractivity 87, t10, 292, 367, 173,200,201,327 375 Datko R. 264, 393, 436, 472 global stability 55 Dean A.M. 191 Gopalsamy K. 79, 90,95, 100, 107, delay independent 60, 180, 217, 472 148, 149, 186, 196, 208, 222, 253, delay logistic 2, 55, 71, 87, 95, 116, 277,296,298,306,399405,408,418 123, 162, 173, 201, 314 Gosiewski A. 212 density dependent 1, 172, 182, 183 Gromova P.S. 217,393 difference equation 87, 88 Gurgula S.I. 90 differential inequality 32, 41, 43, 73, Gyori 1. 79, 81, 361, 366, 460 227,229,300 Driver R.D. i, 12, 18, 103 Haddock J.R. 14 Edelstein - Keshet L. ii Eisenfeld J. 388 El'sgol'ts L.E. i, 145, 448 equations with impulses 90 exploitation 196, 298 Fargue D.M. 2,336 feedback control 95, 121, 446 Field R.J. 148 Fisher M.E. 87,88 Floquet exponents 138, 156, 157 Floquet technique 138 food limi ted 107 Fox L. 36, 236 Franklin J. 45 Franklin J.N. 259,308 Fredbolm alternative 133, 141 Freedman H.I. ii, 253, 436 Fukagai N. 68,403
Halanay A. 64, 126, 133,·227 Hale J.K. i, 37, 126, 131, ,179, 187, 255, 419, 422, 466 baematopoiesis 107 Harrison G. W. 338 Hassard M. W. 127,147 Henry D. 37, 393 Hirscb M. W. 307,318,326 Hofbauer J. 347, 351, 372 Hopf E. 126, 128, 130, 131 Hopf-bifurcation 124, 125, 126, 130, 151 Howard L.N. 148 Hsu S.B. 283, 298, 384 Huang Z.X. 219 Hunt B.R. 14 Hutchinson G.E. 1, 173, 196, 201 bypercooperation 318, 324, 326 hyperlogistic 60
Index
Implicit function theorem 132, 136, 142, 157 impulsive 116, 117, 121, 356 Infrulte E.F. 264 infinite product 36 in-phase 151, 154, 155, 159 integral representation 11 interference 182, 195, 196, 298, 299 interspecific competition 183, 189, 298 intraspecific 183, 186, 298, 299 invariance principle 89
Jacquez J.A. 355 Jansen W. 372 Jiong R. 352, Jones C.S. 1
Kakutani S. 1 Kato J. 34 Kawata M. 148 Kaykobad M. 294 Khusainov D. Ya. 214, 436 Kirlinger G. 351, 353, 372 ,j(olmanovskii V.B. i,394
499
Ladde C.S. 54, 95 Landman K.A. 124, 125, 151 large scale systems 447, 448, 453 LaSalle J.P. 88, 339 Lebesgue convergence theorem 46, 413 Lefever R. 148 Lenhart S.M. 62 Levin J.J. 6 Lewis R.M. 355, 361 Liao Xiaoxin 447 Lim E.B. 235 Li Ming: Li 291, 443 limit cycle 127, 347 linear analysis 172 linear stability criteria 3 linear oscillators 37 Li Senlin 436
MacCamy R. C. 112 MacDonald N. 2, 336 Maeda H.S. 355 Marcus C.M. 369 Marsden J.E. 127, 128, 142 Koplatadze R.C. 73 Martin Jr. R.H. 318, 319, 321 Kozakiewicz E. 48 Martynyuk A.A. 387 Krasnoselskii M.A. 318 Matano H. 326 Krasovskii N.N. i, 126 matrix measure 257, 356, 443 Krikorian N. 339,385 May R.M. ii, 79,307 Krisztin T. 361, 367 Maynard Smith J. ii Kuang Y. 430,436,472 Mazanov A. 355 Kulenovic M.R.S. 67, 399, 400 mean diagonal dominance 330 Kuramoto Y. 148 Michel A.N. 447 ~adas G. 18,20,38,68,401,403,407 Mikhailova M.P. 321 Miller R.S. 298 469
500
Misnik A.F. 436 M-matrix 227-230, 232-235, 317, 337 Mori T. 222 Morita Y. 148 Mulholland R.J. 355 Murdoch VV.VV. 196 Murray J.D. 11 mutualism 172, 191, 318 Mysbkis A.D. 54
Index
positive definite 61, 219, 333, 336 positive feedback 179, 186 positivity condition 26 Post W.M. 336, 337 Qin Yuan Xun
239, 285
Razumikhin B.S. 218 respiratory model 107 Ricklefs R.E. 2 robust stability 460 414 Nashed M. negative feedback 9, 60, 174, 180, 186 Rose M.R. ii Rouche's Theorem 11,365,423 neural networks 369, 473 Routh-Hurwitz 150, 204, 248, 393 neutral equations 393 Royden H.L. 46 neutral logistic 418 Rozhkov V.I. 212, 237 neutral Lotka Volterra system 430, 467 Sandberg I. W. 321,355 No on berg V. W. 348 Sattinger D.H. 136, 138, 154 N unney 1. 203 Schauder-Tychonoff 45, 76, 78, 414 Obolenskii A. Yu. 374 Ohta Y. 318, 321 Oliveira-Pinto ii Pandit S.G. 90 Pavlidis T. 148 Perestyuk O.S. 90 Perron-Frobenius 230, 295 persistence 347, 348, 351, 352, 353, 460 Peschel M. 318 Philos Ch.G. 469 piecewise constant 78 Pielou E.C. 418 Pirabakaran R. 122 Plemmons R.V. 61,337
Schoener T. W. 196 Schuster P. 383 Scudo F.M. ii Seifert G. 32 Selgrade J.F. 318 Shibata A. 307, 311 Siljak D.D. 447,450 simple stability criteria 263 Simpson H.C. 125 Sinha A.S. C.· 32 Slobodkin L.B. ii Smale S. 148, 307 Smith H.L. 318, 321 Smith F.E. 418 Snow W. 393 spectral radius 230, 295, 449
Index
stability switches 193, 208, 239 Staffans O. 112 Stech H. W. 131 Stokes A. 137, 138, 139 strongly positive 26, 27, 29, 62 synchronous 154, 155 Tokumaru H. 225,227,229,317 Torre V. 166 transport delays 365 Tsalyuk V.Z. 212 Turner Jr.M.E. 318 Unbounded delay
30,34, 453
Vandermeer, J.H. 191 variation of constants 19, 451, 455 Vescicik M. 75 Vidyasagar M. 257 Volterra V. 124 Waltman P. ii Wang Lian 239 Wangersky P.J.
201
"Vendi W.
501
352
Wheldon T.E. 279 Winfree A. T. 148 Winsor c.P. 196 Winston E. 51 Wolin C.L. 191 Worz Busekros A. 148, 327, 336, 378 Wright E.M. 1 "Vu J. 436
XU D.Y. 274 Van J. 45 Yamada Y. 62 Yodzis P. ii Yoneyama T. 14,20 Yorke J.A. 14
Zhang B.G. 19,76,407 Zhang Yi. 447, 449, 454, 457 Zbivotovskii L.A. 284 Zverkin A.M. 393, 394
