EXISTENCE AND GLOBAL ATTRACTIVITY OF PERIODIC SOLUTION OF A MODEL IN POPULATION DYNAMICS

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1 T)e Vol.12 No.4 ACTA MATHEMATICAE APPLICATAE SINICA Oct., 1996 EXISTENCE AND GLOBAL ATTRACTIVITY OF PERIODIC SOLUTION OF A MODEL IN POPULATION DYNAMICS WENG PEIXUAN (~.w_) LIANG MIAOLIAN (:~5"h~:) (Department off South China Normal University, Guangzhou , China) Abstract We establish sufficient conditions for the existence and global attractivity of the nonnegative periodic solution of an integro-differential equation which can be taken as a generalization of a model in describing the dynamics of Nicholson's blowflies. In particular, when the coefficients of the equation are constants, the result obtained reveals the threshold property of the equation. Key words. Global attractivity, existence, periodic solution, population dynamics 1. Introduction This article investigates the existence and global attractivity of the nonnegative periodic solution of the integro-differential equation dn(t) fo dt = -f(t)n(t) + a(t) K(s)N(t - s)e -~(*)N(t-8) ds, t >_ O, (1.1) where a(t), 13(t) and f(t) are positive continuous periodic functions on [0, +co) with a positive period w; K(s): [0, co) ~-~ [0, oo) is assumed to be piecewise continuous, nonincreasing eventually and normalized such that oo K(s) = (1.2) ~ ds 1. The equation (1.1) is a generalization of the delay equation dn(t) dt = -fin(t) + an(t - -~3y(t-r), t > O, (1.3) which was used by Gurney et al [~1 in describing the dynamics of Nicholson's blowflies. Here N(t) denotes the size of the population at time t; a is the maximum per capita daily egg production rate; 1//3 is the size at which the population reproduces at its maximum rate; ~, Received July 12, Revised October 19, 1994.

2 428 ACTA MATHEMATICAE APPLICATAE SINICA Vol.12 is the adult death rate per capita daily and ~" is the generation time. Equations (1.1) and (1.3) are both models of population dynamics with delays in production, and the periodicity of a(t),~(t),7(t) is based on the consideration of the periodic environment (e.g. seasonal effects of weather, mating habits etc.). If one chooses K(s) = lf(s - r), a(t) ~ c~, /5(t) =/5, 7(0 = 7, where 5(. ) is Dirac Delta function, a, f~, 7 are positive constants, then (1.1) will simplify to (1.3). 2. Estimates of solutions of (1.1) In what follows, we always use the following expressions: te[0,w] 80= m~. {~(t)}, te[o,w] 70= mi~ {7(t)}, te[o,~] ao= max {o~(t)}; t~[0,w] ~0= max 7o= max {7(t)}. te[0,~] Note from the positivity of a(t),~(t) and 7(t) that ao > 0,/3o > 0,70 > 0. Together with (1.1), we shall consider a continuous initial function ~ so that N(s) = ~(s) > 0, s e (-oo,0], ~(0) > 0. (2.1) One can show that solutions of (I.I) and (2.1) remain positive and exist for t _> 0, but we omit it. We introduce the following Lemma 2.1 without proof. Lemma 2.1. Assume that N(t) is any positive solution of (1.1) with the initial function ~ such that Then N (t) satisfies a ~n, se(-oo,0]. o < ~(8) _< 7o&---~ = 0 < N(t) < n for t E R. For convenience, we introduce a function h(x) as follows: we say that h(x) has the following properties: (i) %~{h(~)}=h(~)=_ ~,~. h(x)==e -~%, x > O. 1 (ii) h(z) is increasing for z E (0, ~) and decreasing for x E (~,+oo); (m) h(o) = o, h(=) -, 0 as x -, +oo; (iv) for any x > 0, there exists ~:0 < x so that h(xo) <_ h(x ) and h(z) > h(xo), x E (=o,= ). If ~_a ~o > 1, then we define A = ~ 1 In ~. a In view of the property (iv) of h(z), one can choose a constant a > 1 such that No- zx A a < min n,

3 No.4 PERIODIC SOLUTION IN POPULATION DYNAMICS 429 and No e-#on <_ n e -~%0, h(x) ~_ h(no) for z E [No, no]. (2.3) Lemma 2.2. Assume that Nit ) is any positive solution of (1.1) and ~ > 1. Then there exists a T > 0 such that 1 h ao Nit) > No = a~ o -~ for t > T, (2.4) where a > 1 is a constant satisfying (2.2) and (2.3). Proof. Let N(t) be any positive solution of (1.1). If (2.4) is not true, then there are two possibilities: (1) there is a T1 > 0 so that Nit ) < No for t _> TI; (2) Y(t) oscillates about No. We want to show that both (1) and (2) lead to contradictions, thus (2.4) is true. Suppose that the first alternative holds. Consider a function It is not difficult to see that g(~) = _~o + ao exp(-~%), x > 0. g(a)=o and g(x)>o for ze(0,a). (2.5) Since N(t) E (0, No) C (0, A) for t > T1, we have g(no) = -7 + ao exp(-/~ N0) > 0. Thus one can choose a if > 0 such that 0 < rl = K(s) ds < 1, ao~exp(-~no) > o. Afo~ By using the positivity of N(t), we derive from (1.1) that (2.6) (2.7) dn(t)dt >- - 7 N(t) + ao fo K(s)N(t - s) exp { -/~ N(t - s)} ds >_ - 7 N(t) + ao /o K(s)N(t - s)exp { - ~ N(t - s)} ds. There are three sub-possibilities: (a) N(t) is decreasing eventually; (b) there is a sequence {tn} such that t,~ > 0, tn ~ oo as n -~ oo and N(tn) (n = 1, 2,--. ) are local minimalms; (c) N(t) is increasing eventually. Due to the property (ii) of h(z), (2.7) and (2.8), it is easy to show that (a) is impossible. Now we discuss case (b). Let B = llm inf N(t) > 0. If B = 0, from the definition of lower $--*oo limit and the positivity of N(t), one can find a subsequence {t~u} of {t,~} such that and (2.8) N(t,~u) = min{n(t) t 0 < t < tn, } (2.9) lira N(tn~) = 0. Choosing some tn= E {t,~k} so that t,,. > T1 + a, then we have k---*oo N(t,..) = min{n(t)it~. - a < t < t~.}. (2.10)

4 430 ACTA MATHEMATICAE APPLICATAE SINICA Vol.t2 We derive from (2.8), (2.10), (2.7) and the monotonicity of h(x) that d 0 =~-~N(t~.~) > g(tn.~) [- ~,0 ao~exp{_zon(t~m)}] which is impossible. Thus we obtain >_g(t~)( a0~exp{-/3 N0}) > 0, and (I) implies that B < No. Noting that B = lim inf Y(t) = lim inf g(t~) > 0, (2.11) t ---*OO n-'~oo -7 S + ao~?b exp{-/3ob} _> B( ao~? exp{-fl No}) > 0, (2.12) this, together with the continuity of x(- ~/o + ao~exp{_f~ox}), means that we can find a small o > 0 such that B - So > 0 and -~/ (B eo) + aorl(b - 0)exp { - ~ (B - Co)} > 0. (2.13) For such an eo > 0 we know from the definition of lower limit and (2.11) that there exist a T3 > T1 + cr and a subsequence {t,~} of {tn} that N(t)>B-eo for t>t3 and N(t,~)<B+ o for k=1,2,.-.. (2.14) Note that B - 0 < N(t) <_ No < ~o, t > :I'3, and h(x) is increasing in (0, ~). We derive 'from (2.8), (2.13) and (2.14) that 0 N(tn,) >_ -~/ N(t~) + ao // K(s)N(t~ k - s)exp { - fl N(t~ - s)} ds _> - "/ (B + eo) + ao~/(b - Eo) exp { -/~ (B - ~o} > 0 (2.15) if t~ k > T3 + 0". This is a contradiction. Thus case (b) is impossible. Finally, for case (c) we suppose that there is a T4 > 0 such that N(t) is increasing for t > T4. From (1) we have that lim N(t) exists and t---+~ 0 < C zx lim N(t) < No. (2.16) By a derivation procedure similar to that in (2.13) we have C - cl > 0 and -7 C + aorl(c - el) exp { - 13 (C - 1)} > 0 (2.17) for some small 1 > 0. For such an 1 > 0, there exists a T5 ~ T4 such that 1 C - el < N(t) < C < No < ~"-6 for t >_ :/"5. (2.18) Thus we get dn(t._.~) > _.yon(t) + ao dt - j~o ~ K(s)N(t - s) exp { - ~ N(t - s)} ds _> - 7 C + aorl(c - 1) exp { - fl (C - el)} > 0, (2.19)

5 No.4 PERIODIC SOLUTION IN POPULATION DYNAMICS 431 for t >_ T5 + a from (2.8), (2.17), (2.18) and the monotonicity of h(x). (2.19) implies that lim N(t) = +c~, which contradicts (1). Thus case (c) is false. In summary, we conclude from the above discussion that case (1) is impossible. Next, suppose N(t) oscillates about No. Then there exists a sequence {tn}, tn > 0, lim tn --- +oo such that N(tn) ~ No (n = 1, 2,.-. ) are local minimums. Furthermore, t ---*oo we have E ~ lim ira N(t) = lira ira N(t~) < No. t-'*oo n-~oq On the other hand, we can show by an argument similar to that in Lemma 2.1 that there is a T~ > 0 such that N(t) <_ n o for t _> :/"6, which together with (2.2), (2.3) and the properties (ii), (iv) of h(x) can lead to a conclusion that h(n(t - s)) > h(n(t)) (2.20) if there is some t satisfying t T6 + a and N(t) = min{n(s) lt- a < s < t} and N(t) < No. Now we can show from (2.8) and (2.20) that E > 0 by an argument similar to that in situation (1)-(b) which shows that B > 0, but we omit the details. Replacing B, T3 by E, T~ (T7 :> T6) from (2.11) to (2.15) respectively, we obtain a contradiction which says that case (2) is impossible. Thus we complete the proof. 3. Existence of Nonnegative Periodic Solution of (1.1) The following lemma changes the existence problem of periodic solution of (1.1) into the same problem of another equation with finite delay. Lemma 3.1. Any w-periodic solution P(t) of (1.1) is also an w-periodic solution of the following equation dn(t) fo ~ dt =-v(t)n(t)+a(t) H(s)Y(t-s)exp{-fl(t)g(t-s)}ds, t>0, (3.1) where H(s) = E K(s + rw), s e [0, w], and vice verse. r~0 Lemma 3.2. There exists an w-periodic solution P(t) of (3.1) such that 0 _< P(t) < n o for t > 0. A proof of a lemma similar to Lemma 3.1 can be found in [5]. The proof of Lemma 3.2 is completed by showing that solutions of (3.1) are uniformly bounded and uniformly ultimately bounded (for instance see [3, p.120]), and by using Theorem 37.1 in [41. Lemma 3.3. Assume that ~ > 1. Then there exists a positive w-periodic solution P*(t) of (3.1) such that No <_ P*(t) <_ n o for t >_ 0. Proof. We have from (3.1) that dn(t)dt >- -7 N(t) + ao ~o ~ H(s)N(t - s) exp { -/3 N(t - s)} ds. (3.2) In proving Lemma 2.2, by replacing ~7, ~, K(s) by 1, w, H(s) respectively, we can obtain that there is a T > 0 such that all positive solutions of (3.1) satisfy Y(t) > No for t > T. (3.3)

6 432 ACTA MATHEMATICAE APPLICATAE SINICA Vol.12 On the other hand, one can show that No < N(t) < n o for t > 0 (3.4) if the initial function (s) satisfies No _< _< n for s e o]. (3.5) That is to say, Ix = [N0,n ] is an invariant set of equation (3.1). Now we derive from (3.3) and (3.4) that all of the positive solutions of (3.1) are uniformly ultimately bounded with a lower bound No. Combining this with the conclusion of Lemma 3.2, we know that (3.1) must have a positive periodic solution P*(t) lying in/1 = [N0,n ]. The proof is complete. Now we can state the following theorems from Lemma Theorem 3.4. There exists a nonnegative w-periodic solution P(t) of equation (1.1) such that 0 _< P(t) _< n o for t > 0. Theorem 3.5. Assume that a0/7 > 1. Then there exists a positive w-periodic solution P*(t) of (1.1) such that No <_ P*(t) <_ n o for t >_ Global Attractivity In this section we investigate the global attractivity of the periodic solution P(t) or P*(t). Theorem 4.1. Assume that foo sk(s) ds < oo (4.1) and 7o > a. Then all positive solutions N(t) of (1.1) satisfy lira N(t) (4.2) t---*oc Proof. Suppose that N(t) is any positive solution of (1.1). Define a Lyapunov functional V(t) = V(N)(t) as follows: v(t) = IN(t)l + K(s) ~(u + s){n(u)l exp { - ~(u + s)n(u)} d ds. (4.3) $ Calculating the upper right derivative of V along the positive solutions of (1.1), we can obtain the conclusion of the theorem, but we omit the details. Remark. Under the assumption of Theorem 4.1 we know that N(t) - 0 is the only periodic solution of (I.1), i.e., (1.1) has no any positive periodic solution. Theorem 4.2. Assume that (4.1) holds and "to (4.4) -~ < ao < a < -~, where F = max{e -2, 1 -f~0n0}. Then there exists a unique positive periodic solution P*(t) of (1.1) such that any positive solution N(t) of (1.1) satisfies lira [N(t) - P'(t)] = 0.' t--*~

7 No.4 PERIODIC SOLUTION IN POPULATION DYNAMICS 433 Proof. Let Y(t) = N(t) - P*(t), t e R. It is found from (1.1) that Y(t) satisfies dy(t) fo ~ dt = - 7(t)Y(t) + a(t) K(s) [N(t - s) exp { - ~(t)n(t - s)} - P*(t - s)exp { - fl(t)p*(t - s)}] ds, t > O. Consider a Lyapunov functional V(t) = V(Y)(t) as follows: v(t)--iy(t)l + - We obtain D+V(t) <- 7(t)lY(t)l + fo~g(s)~(t + s)in(t)exp { - P'(t) exp { - #(t + s)p*(t)}[ ds + s)n(t) =- 7(t)lY(t)l + K(s)~(t+s) le~p{-o(t,s)}(!-a(t,s))l l~(t + s)y(t)[ ds, (4.5) where O(t, s) lies between/3(t + s)p*(t) and t3(t + s)n(t). We have from the conclusion of Lemma 2.2 that N(t) > No for t >_ T, where T > 0 is as in Lemma 2.2. Thus O(t, s) > &No for t _> T, s > 0. (4.6) Define a function f(x) as follows: f(x) =e-=(l --x), x>0. It is known that (i) max{f(z)}=l, mjn{f(x)}=-e 2, o < y(x) _< i - x for x e [0, 1]; =>_o =>_o (~) f(~) is decreasing for ~ e [0, 2]; (iii) f(x) is increasing for z > 2. Since /(&No) = exp{-/3ono}(1 - &No) < 1 - &No when/30n0 < 1, by using the properties of f(x) we obtain [f(x)l < max{e -2, 1 -/3oN0} _A F (4.7)

8 434 ACTA MATHEMATICAE APPLICATAE SINICA Vol.12 when x _>/~ono. We derive from (4.5)-(4.7) that D+V(t) <_ -70[Y(t)[ + a F[Y(t)[ = (-70 + a F)[Y(t)[ < 0 for t > T. By a derivation procedure similar to that in the proof of Theorem 4.1, we can obtain lim Y(t) = O. t~oo Thus the proof is complete. Corollary 4.3. Assume that fo sk(s)ds < oo and a(t) =- a, ~(t) =_ ~, 7(t) -= ~, where a, fl and q, are positive constants. Then the following conclusions hold. (i) All positive solutions N(t) of (1.1) satisfy lim Y(t) = O, when ~/> a; (ii) All positive solutions N(t) of (1.1) satisfy lira N(t) = P*, ~ --* o0 when ~/< a < 9, where P* = ~ In ~, f - max {e -2, 1 - ~ In ~ }, constant a > 1 satisfying!a In < rain I,, -a exp 1 + In _< It can be seen from Corollary 4.3 that for the case of constant coefficients and infinite delay, the result of this paper reveals the threshold property of the equation: If a certain quantity involving the coefficients is negative, the origin is stable; while when the same quantity becomes positive, a unique globally attractive positive equilibrium appears and persists as long as another expression of the coefficients is negative. It is well known that the system dn(t) dt = -Tn(t) is a dissipative system. Hence our result is of significance that a delay induced Hopf type bifurcation to a positive equilibrium is possible and this can be used for devising population control and stabilization strategies. A more extensive generalization of (1.3) involving continuously distributed delays over an unbounded interval is the equation of the form dn(t) = -~(t)n(t) + a(t) g(s)a(n(t - s), t) ds, t > O, dt - where G(x, t) denotes a suitable growth rate function, but we do not discuss it in this paper. References [1] C. Corduneanu. Integral Equations and Stability of Systems. Academic Press, New York, [2] W.S.C. Gurney, S.P. Blythe and R.M. Nisbet. Nicholson's Blowflies Revisited. Nature, 1980, 287: [3] Li Senlin and Wen Lizhi. Functional-differential Equations. Hunan Press of Science and.technology, Chsngsha, 1986, p.120 (in Chinese). [4] T. Yo6hizawa. Stability Theory by Lyapunov Second Method. The MathematiCal Society of Japan, [51 K. Gopalsamy. Globally Asymptotical Stability in a Periodic Integrodifferential System. Tohoku Mathematical Journa/, 2nd Set., 1985, 3: