Nonh,warAnnl~ws, Thror): Mrthodr & Ap~~lrcamns. Vol. 25, No 2. pp. 187-195, 19Y5 CopyrIght D 1995 Elsevm Science Ltd

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0362-546X(94)00199-5

ON THE STABILITY OF A DELAY DIFFERENTIAL POPULATION MODEL

MING-PO CHENTQ, J.S. YU$l]S, X. Z. QIAN$]] and Z. C. WANG+]] tInstitute of Mathematics, Academia Sinica, Nankang, Taipei 11529, Taiwan; and

$Department of Applied Mathematics, Hunan University, Changsha, Hunan 410082, People’s Republic of China

(Received 1 October 1993: received in revised form 14 June 1994; received forpublication 17 August 1994)

Kq): words and phrases: Population model, uniformly stable, global attractivity

1. INTRODUCTION AND PRELIMINARIES

The delay differential equation

[

x(t - 7) x’(t) =dt) 1 - k 1 ) t 2 0,

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, t20. (3)

with the initial condition for any t,, 2 0

i

y(t) = #dt) 2 - 1, t E [to - l,t”l, 4Ea[tO-1,4)1, L-l,=)) and da()) > -1.

(4)

It is easy to see that (3)-(4) has a unique solution y(t) =y(t; t,, 41 which exists on [to - 1,~) and which is greater than - 1, for all r r t,,.

§The work of M.-P. Chen was supported by the NSC of Taiwan under Grant NSC-82-0208-M-1-145. lIThe work of J.S. Yu. X.Z. Qian and Z.C. Wang was supported by the NNSF of China. lIAuthor to whom correspondence should be sent.

187

18X MING-PO CHEN etal.

Definition 1. The zero solution of (3) is uniformly stable if, for any E > 0, there exists a c%E)>O such that t,>O and Il~Il=~up~,,-,~~~,, l+(s)1 < S imply ly(t;t,, $>I < l for all 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 < (Y I + 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” < $ for all t 2 0,

then the zero solution of (3) is uniformly stable.

(5)

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,

then the solution of (3)-(4) tends to zero as t + m?

(6)

Problem 2. Is it true that, if condition (5) is replaced by

J

I a(s) ds 5 (Y() < ; for t r 1,

I- I (7)

then the zero solution of (31 is uniformly stable?

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.

Then every oscillatory solution of (3)-(4) tends to zero as t + 0~.

THEOREM 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.

(8)

(9)

On the stability of a delay differential population model 189

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 equation

y’(t) + (1 + 2e’)-‘(1 + e’-‘)-’ [l +y(t)ly(t - 1) = 0, t20

which satisfies (61, but has a positive solution y(t) = 1 + eP’ which does not tend to zero as t -+ J;, due to the invalidity of (9).

2. PROOF 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), u = -1iminf y(t). t--t= t +r

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<y(t)<u+E. (10)

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

u = lim sup y<&* >, u = -1iminf y(s,*). t--)x t-+x

In view of (3) and (101, we have

and

y’(t) I(0 + e)(Y(t)(l +y(t>,, tzT+l

y’(t) 2 -(a + E)(Y(f)(l +y(t)>, t~T+l.

First integrating (11) from t,* - 1 to t,*, we have

I

1: en<1 +y(t,*)) I (U + E) a(s)ds I ;<u + E).

t; I

As this is true for every n = 1.2,, . . , it follows that

y(t) < e3/2(u+c) - 1, fortkt;.

(11)

(12)

(13)

190 MING-PO CHEN er al.

Similary, by integrating (12) from s,* - 1 to sz, we have

y(t) 2 e-3/2(u+~) - 1, for t 2s:.

Next for s,* - 2 5 t 5 s,* - 1, integrating (12) from t to s,* - 1, we get

s; -1

or

-!,(l +y(t)> 2 - CL4 + El I

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

y(t) 5 e3/2(~+~xs~-l --I) _ 1, for t E [sx - 2,s: - 11.

Similarly, we have, for t,* - 2 5 f 5 t,* - 1, that

y(t) > e-3/2(ii+eXr:-l -I) _ 1.

Let 7~ [O, 11 be an arbitrary constant. Then by (10) and (16), we have

t, FE (1 +y(t,*)) = -

/ a(s>y(s - l)ds

I, - 1

(14)

(15)

(16)

=-/ 1: -7 a(s>y(s - 1) ds - / t.’ (~(s)y(s - 1) ds t; - 1 1; - 7

I gu + E)(l - 7) + I

1; a(s)(l _ ,(-3/2!4L1+EX1,*-S))dS

f; - 7

L

5 g/J + E)(l - T) + ; /

‘” (1 _ ,(-3/2X~7d0-d)ds

f - 7

= $(u + E)(l - E) + + -L-(1 _ e(-3/2xu+c)7 )

which implies

fn,<l -u> I gu + E)(l - 7) + + A(’ _ e(-J/2Xu+cb ). (17)

Similarly, by using (10) and (15), we obtain

fn,<l - u) 2 - $(u + E)(l - 7) + ;T- --!&,3/2cu++ -1). (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 of a delay differential population model 191

In (20), let 7 = (2/3u)/“(l + u), then 0 < 7 < 1 and (20) becomes

-fn’,<l - u) I ;24 - d(l + UVJl + u) + 1

(1 -s)dsdu,

That is = 11 + $2.

In the following, we prove

In fact, if iv 2 -fn(l - v), then in (19) let T= -(2/3v)/,(l - u), we have

(21)

(22)

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

/,<l +u) I g - A (1 -e-(3/2)“)

1

/

(3/2)P =-

V (1 - eP”)ds

0

1

/

( 3 / 2 ) i’ ‘jj (s - is’ + +s’> ds

0 9 9 27

= gv - xv* + i&.

We can easily find that v r 0.5 by the assumption +v < -e,Cl - v). However, on the other hand, for 0.5 5 v < 1 we have

9 9 27 1 -- 8 ~vfiZiiuz<l-~u

which can be proved by finding out the solution of the inequality

27 mu2 -TV++

192 MING-PO CHEN pr al

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

i

a = 1 _ ,-h-&/6) (23)

b = ea-(u’/6) _ 1

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),

b2 = eat -(of/b) - 1 . . . . . . . . . . . . . . . . . .

a, = 1 - e-h,-ch;/6)

7

b n+i= e%-(~i/6) - 1, n = 1,2,3 ,....

Then by (21) and (22), we have

a,=l-e --u -(u’/6) > u

b, = e o,-(of/f~) - 1 2 eL;-(~‘/6) _ 1 > u = b

13

a, = 1 - e-bz-(b;/6) 2 1 _ e-h,r(h,/6) =a,.

In general, by mathematical induction we can prove

u la, Ia, 2 .*. <a, IU,,, < 1,

u=b,~b,I..._<b,~b,_,<e~/‘-l. & Set

Then by (24),

u(, = lim b a*% It’

u(, = lim a,. n-X

u,j = 1 _ e-W4/6), u,) = e v,,-(G/h) - 1 3

(24)

which shows that (23) has a positive solution a = u”, b = u,). By (23) we see that the equation in a

f(a) :=/‘Jl -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 193

< 1 I;* I ,-(0*,‘3)+u*-(0**,‘6)+ f,-(~;*/3)+l~*-(u*‘/6)~*

I 4*+(u*‘/6) + e(2/3xu*-(u*‘/6)J + +*e(2/3Xu*-(u*‘/6))

z-e u* *@*‘/6) + (1 + u*)2/3 + $*(I - u*)2/3

< - 1+ u* + k4*2 + ;(u* + +u**)*) + 1+ $u* + i(1-t $u*ju* - ( 6

= hu *2-r 2 (u* + $d*2)* < 0,

which contracts the fact that f’(u*> 2 0. This contradiction shows that u = u = 0 and so the proof is complete. n

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.

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.

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 ’ P. (25)

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 - 111 I pq /

*4- 1 a(s) ds, t E [t4 - l,t*l.

l-1

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

Thus

p=y(t*) = y’(s) ds < pncu(s), pq2 a(s)

194 MING-PO CHEN et al.

If rl/I:~, cu(s)ds _< 1, then

‘? f4 y(t,) I

/ pqa(s) ds <

/ pqa(s) ds s p,

c4 - I ‘$ - I

which is a contradiction. If 77 j& , a(s) ds > 1, choose q E (0,l) such that

Then

/

f4 77 a(s)ds = 1.

1,-l-q

+d l”,+, 6’,’ a(s)a(u)duds 4

r4

I I

P zprl: cY(s)a(u)du ds

f‘$- I +q s-l

‘4

J J

I -m2 a(s)a(u)du ds

r&-l+9 t,-lf9

= p(Tp - ;) < p($ - ;) = p.

Which contradicts the assumption that y(t2) = p. Thus, this lemma is proved. n

Proof sf 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 <t, I t,, Iy(~,>l= p, y’(t,)y(t,) > 0, ly(t,>l> p, ly(tI <p for all t E [t, - l,t,] and 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

and

y(t) 5 (1 + 6)e”’ - 1

1 y(t) 2 1+6e -“‘-12 -(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).