On the stability of a delay differential population model

Nonh,warAnnl~ws,

Thror):

Vol. 25, No 2. pp. 187-195, 19Y5 CopyrIght D 1995 Elsevm Science Ltd Printed in Great Bntam All rights reserved 0362-%6X/95 $9.50 + .OO

Mrthodr

& Ap~~lrcamns.

0362-546X(94)00199-5

ON THE STABILITY OF A DELAY DIFFERENTIAL POPULATION MODEL MING-PO $Department

(Received

CHENTQ, J.S. YU$l]S, X. Z. QIAN$]]

tInstitute of Mathematics, of Applied Mathematics,

1 October

1993: received

Kq): words and phrases:

Academia Sinica, Hunan University, China in revised form Population

The delay differential

Nankang, Taipei 11529, Taiwan; and Changsha, Hunan 410082, People’s Republic

14 June 1994; received forpublication model,

1. INTRODUCTION

and Z. C. WANG+]]

uniformly

AND

stable, global

17 August

of

1994)

attractivity

PRELIMINARIES

equation x’(t)

=dt)

1 -

x(t

k

[

7)

1 20, )

t

called Hutchinson’s equation, is a single species population growth model, where r, r and k are positive constants. This equation has been studied by many authors; see for example Cuningham [l], Gopalsany [2], Kakutani and Markus [3], Kuang [4], May 151, Wright [6], and Zhang and Gopalsamy [7]. By making the change of variable y(t) = (.x(Tt))/(k) - 1 and a =rr, equation (1) is reduced to the canonical form y’(t) + (Y[l

+y(t)ly(t - 1) = 0,

t

2 0.

(2)

In this paper we consider the case where (Y is a positive continuous function of t (then so is r),

y’(t) + cx(t)[l +y(r)ly(t - 1) = 0, with the initial

condition

(3)

for any t,, 2 0

y(t) = #dt) 2 - 1, i

t20.

4Ea[tO-1,4)1,

t E [to - l,t”l, and L-l,=))

It is easy to see that (3)-(4) has a unique solution y(t) =y(t; and which is greater than - 1, for all r r t,,. §The work of M.-P. Chen was supported by the NSC of Taiwan lIThe work of J.S. Yu. X.Z. Qian and Z.C. Wang was supported lIAuthor to whom correspondence should be sent. 187

under Grant by the NNSF

da()) > -1.

(4)

t,, 41 which exists on [to - 1,~)

NSC-82-0208-M-1-145. of China.

MING-PO CHEN etal.

18X

Definition 1. The zero solution of (3) is uniformly stable if, for any E > 0, there exists a l+(s)1 < S imply ly(t;t,, $>I < l for all c%E)>O such that t,>O and Il~Il=~up~,,-,~~~,, t 2 t,,. For equation (21, it is known that the zero solution is uniformly stable if cx< n/2. However, it was shown in [81 that this result does not hold in the case of time-dependent a(t). The following is due to Wright [6], see also Kuang [4]. THEOREM

1. Let 0 < (YI + then the solution of (2)-(4) tends to zero as f -+ m.

The following result is taken from Sugie [8]. THEOREM

2. If there exists a constant c+, > 0 such that a(t) I a” < $

then the zero solution of (3) is uniformly

for all t 2 0,

(5)

stable.

The following first open problem is proposed by Kuang [4, open problem 4.21, and the second one is a natural problem replacing (5) by an interval condition on a(t). Problem 1. Is it true that, if

0 < a(t) < ;

for t 2 0,

(6)

then the solution of (3)-(4) tends to zero as t + m? Problem 2. Is it true that, if condition

(5) is replaced by

I a(s) ds 5 (Y()< ; J I- I then the zero solution of (31 is uniformly stable?

for t r 1,

(7)

Solving the above two problems is the main goal of this paper. We say that a function y(t), t 2 a for some a E R, is oscillatory if there exists a sequence {t,} satisfying t, > a, t,, -+ ~0 as it ---)x and y(t,,) = 0, n = 1,2,3, . . . . Otherwise, we say that y(t) is nonoscillatory. We now

state our main results. THEOREM

3. Assume that 0 < a(t) I ;

for t20.

(8)

Then every oscillatory solution of (3)-(4) tends to zero as t + 0~. 4. Assume that (8) holds and that z a(s) ds = m. /0 Then every solution of (3)-(41 tends to zero as t + ~0.

THEOREM

(9)

189

On the stability of a delay differential population model THEOREM 5. Assume that (7) holds. Then the zero solution

of (3) is uniformly

stable.

Observing the above three theorems, theorem 3 shows that the answer to problem 2 is true, and theorem 4 shows that the answer to problem 1 is positive under (9). However, the following example shows that problem 1 may not be true without (9). Example 1. Consider the delay differential

y’(t)

+ (1 + 2e’)-‘(1

equation

+ e’-‘)-’

[l +y(t)ly(t

which satisfies (61, but has a positive solution t -+ J;, due to the invalidity of (9). 2. PROOF

- 1) = 0,

t20

y(t) = 1 + eP’ which does not tend to zero as

OF THEOREMS

Proof of theorem 3. Let y(t) be an oscillatory solution of (3)-(4). It is easy to see that y(t) is bounded above and bounded below away from - 1. (See the proof of lemma 2.1 of Chapter 4 in [4].) Define

u = limsup y(t), t--t=

u = -1iminf t +r

y(t).

Then 0 I u < 1 and 0 I u < SC.Let E be an arbitrary positive constant such that, for t 2 T = T(E) > 0 -Lri-e
(10)

Now let it,,] be a sequence of zeros of y(t) which satisfy t, 2 T + 2, t, --, ~0 as n + = and tzn], and y(t) I 0 for t E [t2n,t2n+,], it = 1,2,3 ,.... Also, let t,* E t,,_, I t I t2,,], ~($1 = min(y(t): (t**-l&J, s,* E&f tzn+, 1 be such y(r,*) = max(y(t): t,, I t 2 t,, + ,]. Then for n = 1,2,. . , , y(t,* )> 0 and y ‘
y(t) 2 0 for t E [t,,-,,

u = lim sup y<&* >,

u = -1iminf t-+x

t--)x

y(s,*).

In view of (3) and (101, we have y’(t) I(0

+ e)(Y(t)(l

tzT+l

+y(t>,,

(11)

and y’(t) 2 -(a + E)(Y(f)(l First integrating

t~T+l.

+y(t)>,

(12)

(11) from t,* - 1 to t,*, we have en<1 +y(t,*))

I (U + E)

1: I t; I

a(s)ds

I ;
As this is true for every n = 1.2,, . . , it follows that y(t)

< e3/2(u+c)

- 1,

fortkt;.

(13)

MING-PO

190

Similary, by integrating

CHEN

er al.

(12) from s,* - 1 to sz, we have y(t) 2 e-3/2(u+~)

Next for s,* - 2 5 t 5 s,* - 1, integrating

- 1,

(12) from

for

2s:.

t

(14)

to s,* - 1, we get

t

s; -1 -!,(l

+y(t)>

2 - CL4+ El

u(s)ds

It

L - ;(u + ,)(s,* - 1 -t),

or y(t) 5 e3/2(~+~xs~-l Similarly,

we have, for

t,*

- 25 f5

t,*

y(t)

--I)

_ 1,

for

t

E [sx - 2,s: - 11.

(15)

- 1, that > e-3/2(ii+eXr:-l

-I)

_ 1.

(16)

Let 7~ [O, 11 be an arbitrary constant. Then by (10) and (16), we have FE (1 +y(t,*))

= -

t, / I, - 1

=-/

1: -7

a(s>y(s - l)ds

a(s>y(s - 1) ds -

/

t; - 1

1;

I gu + E)(l - 7) +

I f;

a(s)(l

g/J

+

E)(l

-

T)

+

L

; /

= $(u + E)(l - E) + +

_ ,(-3/2!4L1+EX1,*-S))dS

- 7 ‘”

5

(~(s)y(s - 1) ds

t.’

1; - 7

(1

_

,(-3/2X~7d0-d)ds

f-7 -L-(1

_ e(-3/2xu+c)7 )

which implies fn, I gu + E)(l - 7) + + Similarly,

A(’

_ e(-J/2Xu+cb ).

(17)

by using (10) and (15), we obtain fn,
-1).

--!&,3/2cu++

(18)

The proof of this theorem will be complete if one can show that u = u = 0. Assume that, for the sake of contradiction, that u > 0 and 0 < u < 1. As E > 0 is arbitrary, (17) and (18) imply /,(l

+ U) I Gj~(l - 7) + +T- A(1 - e(-3/2)ur),

(19)

and -f*(l

-u)

i tU(l-

7) - +r+ $(e(3”‘/2)-

1).

(20)

On the stability

In (20), let 7 = (2/3u)/“(l

of a delay differential

population

model

191

+ u), then 0 < 7 < 1 and (20) becomes -fn’,
+ UVJl

+ u) + 1

(1 -s)dsdu, = 11+ $2. That is (21) In the following,

we prove (22)

In fact, if iv 2 -fn(l

- v), then in (19) let T= -(2/3v)/,(l

- u), we have

If sv < -Y, (1 - v), then in (19) let 7 = 1, we have /,
+u)

I g - A (1 -e-(3/2)“) =-

V

‘jj

(3/2)P

1

1

/

(1 - eP”)ds 0

(3/ 2)i’ /0

(s - is’ + +s’> ds

9 9 = gv - xv* We can easily find that v r 0.5 by the assumption hand, for 0.5 5 v < 1 we have

27 + i&. +v < -e,Cl

9 9 27 1 -~vfiZiiuz
-TV++

- v). However,

on the other

192

MING-PO

Thus we also have that (22) holds. Now we prove that the following

CHEN

pr al

system of equations a

=

1

,-h-&/6)

_

b = ea-(u’/6)

(23)

_

1

i

has a positive solution in the region 0 < a < 1, b > 0. In fact, one can define two positive numbers sequences {a,} and {b,,) as follows

b, =u, a, = 1 - e-h,-(b;/6),

-1

b2 = eat -(of/b) .

.

.

a,

.

.

=

.

1

b n+i=

.

.

-

.

.

.

.

.

.

.

.

.

(24)

.

e-h,-ch;/6) 7

e%-(~i/6)

-

n = 1,2,3 ,....

1,

Then by (21) and (22), we have --u -(u’/6)

a,=l-e

b, = e o,-(of/f~) a,

=

In general, by mathematical

1

-

1

Ia,

u

2

eL;-(~‘/6)

_

1

>

u

=

b 13

e-bz-(b;/6)

induction u la,

-

>

2

1

_

e-h,r(h,/6)

=a,.

we can prove 2 .*.
IU,,,

< 1,

u=b,~b,I..._
u(, = lim a,.

a*% It’

n-X

Then by (24), u,j = 1 _ e-W4/6), which shows that (23) has a positive solution in a

f(a) :=/‘Jl

u,)

=

e

v,,-(G/h)

-

1

3

a = u”, b = u,). By (23) we see that the equation

-a) + ea-(a’/6)

- 1 + ;(e~-(h6)

- 1)’ = 0

has a positive root a = LJ(,E (0,l). Since f(O) =f’(O) =f”(O) = 0 and f”‘(0) < 0, it follows that f(u) < 0 for sufficiently small a > 0. Also f(1 - ) = --. If we let U* E (0,l) be the least positive root of f(a) = 0, then f’(~*) 2 0. Set K* = e”x’-(U*‘/h) - 1. Then U* = 1 - e-“*--(u*‘/6), and f’(u*)

= -&

+ (1 - 4 j,l,*-Cl~*‘%

+ &cl _ ~je’*-(l”,6)(ei.*~(i’*.,6~

_ 1)

On the stability of a delay differential population model <

1 I;*

I

4*+(u*‘/6)

z-e

I ,-(0*,‘3)+u*-(0**,‘6)+

u*

f,-(~;*/3)+l~*-(u*‘/6)~*

+ e(2/3xu*-(u*‘/6)J

*@*‘/6)

+

(1

+

+ u*)2/3

+ ;(u* -< - ( 1+ u* + k4*2 6

193

+

+*e(2/3Xu*-(u*‘/6))

$*(I

-

+ +u**)*)

u*)2/3

+ 1+ $u* + i(1-t

$u*ju*

= hu *2-r 2 (u* + $d*2)* < 0, which contracts the fact that f’(u*> proof is complete. n

2 0. This contradiction

shows that u = u = 0 and so the

Proof of theorem 4. In view of theorem 3, it suffices to prove that every nonoscillatory solution of (3)-(4) tends to zero. This follows the proof of [4, Chapter 4, lemma 2.1). The proof is complete. n

In order to prove theorem 5, we first prove the following lemma. 1. Assume that (7) holds. Let n E (1,2) be a constant satisfying an < + and let y(t) be a solution of (3)-X4) on [to - l,=] such that y(t,) = 0 for some t, r t, + 1. Then, for any p < n - 1, ly(t)l I p for t E [to - 1, t,] implies /y(t)1 5 p for all t 2 rr.

LEMMA

Proof Suppose that it is not true. Then there exists t, > t, such ly(t,)l= p, (y(t2 + T)I > p for a sufficiently small r> 0 and /y(t)1 I p for t, I t 2 t,. We assume y(t2) = p > 0. Since the proof is similar in the case y(t2) = -p < 0. Hence, there exists a t, E (t2, t2 + T) such that y’(t3) > 0

and

YW

(25)

’ P.

From equation (31, it is easy to prove that there exists t, > t, such that y’(t4) = 0 and y(t4) > p. Clearly, t, < t2 + 1 and y(t4 - 1) = 0. Since ly(t)l I p for all t E [to - 1, t2],

Iy’(t>l I a(t>ly(t - 1Nl + ly(tN 5 p7p(t),

t4t4-1,tJ

and, hence,

ly(t - l>l = ly(t, - 11 -y(t - 111I pq Consequently,

*4- 1 / l-1

a(s) ds,

for all y E [t4 - 1, t2],

Thus p=y(t*)

=

y’(s) ds <

pncu(s), pq2 a(s)

t E [t4 - l,t*l.

194

MING-PO

et al.

CHEN

If rl/I:~, cu(s)ds _<1, then ‘? y(t,)

which is a contradiction.

I

f4

/ c4- I

pqa(s) ds <

/ ‘$- I

pqa(s) ds s p,

If 77j& , a(s) ds > 1, choose q E (0,l) such that f4 77 /

a(s)ds

= 1.

1,-l-q

Then

+d zprl:

l”,+, 4 r4

6’,’ P

I f‘$-I +q I s-l -m2

a(s)a(u)duds

cY(s)a(u)du

‘4

I

Jr&-l+9

Jt,-lf9

ds

a(s)a(u)du

ds

= p(Tp - ;) < p($ - ;) = p. Which contradicts the assumption that y(t2) = p. Thus, this lemma is proved.

n

Proofsf theorem 5. Let 7) E (1,2) be such that q~y < $. Then for every E E (0, q - 1) we choose a 6 = L?(E) > 0 so small that Consider the solution y(t) =y(r; t,, 4) of (3)-(4) with 11$11< 6. Suppose that ly(t,)I > p for some t, > t,. Then it follows from 6 < p that there exist t, and t, such that t,, < t2 l= p, y’(t,)y(t,) > 0, ly(t,>l> p, ly(tI

p for all t E [t2, t,]. We suppose that y(t) > 0 for t, < t I t,, the case y(t) < 0 is similar and the proof is omitted. Since y’(r,>y(t,) > 0, from lemma 2.1 of Sugie [8, p. 1801, it is easy to see that y(t) must have a zero point t, E (ti - 1, t,). First, for t E [to, t,, + 11 we have

Ice, (1 +y(t))‘l i

6a(t)

On the stahility

of a delay differential

population

model

19s

and, hence,

therefore,

we have y(t)

5 (1 + 6)e”’

- 1

and 1

-“‘-12

y(t) 2 1+6e

-(1+6)d”‘+l.

That is ly(t>l 2 (1 + G)e*‘-

1< p

for tE[tO,tO+

11.

Similarly, for fO + 1 2 r I: t, + 2, we can show that [y(f)1

< (1 + 8)easea((1+‘)e”sP’)-

1 =p.

Therefore, we have t, > t, + 2 and, hence, t, > t, - 1 > t,, + 1. Therefore, ly(t)l I p holds for t E [to - 1, t,]. Thus by lemma 1, we have ]y(t)l I p for all t 2 t,, which contradicts the assumption that ]y(t,)l > p. Hence if ]]c#]] < 6, then ]y(t; t,, +I I p < E for all t 2 t,. The proof is complete. n It now remains an open problem whether the condition in theorem 3 can be replaced by f a(s) ds I ; / ,- 1 or not. REFERENCES 1, CUNNINGHAM W. J., A nonlinear differential-difference equation of growth, Proc. nafn. Acad. Sci. U.S.A. 40, 708-713 (1954). 2. GOPALSAMY K., Stability and Oscillation in Delay Differential Equations of Population Dynamics. Kluwer Academic Publishers, Boston (1992). 3. KAKUTAN S. & MARKUS L., On the nonlinear difference-differential equation y’(t) = (A - By(t - T))y(t), Control Theor. Nonlinear Ox. 4, l-18 (1958). 4. KUANG Y., Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993). 5. MAY R. M., Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (19731. 6. WRIGHT E. M., A nonlinear difference differential equation, /. reine angew. Math. 194, 66-87 (1955). 7. ZHANG B. G. & GOPALSAMY K., Global attractivity in the delay logistic equation with variable parameters, Proc. Cumb. Phil. Sot. 107, 579-590 (1990). 8. SUGIE J., On the stability for a population growth equation with time delay, Proc. R. Sot. Edinb. 120A, 179~184 (1992).