Delay Effect in the Nicholson's Blowflies Model with a Nonlinear Density-dependent Mortality Term

This paper is concerned with a class of non-autonomous delayed Nicholson's blowflies model with a nonlinear density-dependent mortality term. Under proper conditions, we prove that the positive equilibrium point is a global attractor of the addressed model with small delays. Moreover, some numerical examples are given to illustrate the feasibility of the theoretical results.


Introduction
Recently, based on that marine ecologists are currently constructing new fishery models with nonlinear density-dependent mortality rates, the following Nicholson's blowflies model with a nonlinear density-dependent mortality term N (t) = −D(N(t)) + PN(t − τ)e −N(t−τ) , ( was proposed in L. Berezansky et al. [1].Here function D(x) might have one of the following forms: D(N) = aN b+N or D(N) = a − be −N with positive constants a, b > 0. The detailed biological explanations of the parameters of (1.1) can be found in [1,13].Furthermore, (1.1) and its generalized equations have been extensively studied, and this extensive study has produced a lot of progress on the existence and stability of positive equilibrium point, positive periodic solutions, and positive almost periodic solutions, see more details in [2][3][4]8,[11][12][13]18].In particular, the author in [9] established several criteria on the global asymptotic stability of zero equilibrium point for the following Nicholson's blowflies model with a nonlinear density-dependent mortality term: N (t) = −a + be −N(t) + m ∑ j=1 β j N(t − τ j (t))e −γ j N(t−τ j (t)) , ( Email: wanminxiong2009@aliyun.com where a, b, β j , γ j are positive constants, τ j (t) ≥ 0 is a bounded and continuous function, and j ∈ J = {1, 2, . . ., m}.
On the other hand, the effect of delay on the asymptotic behavior of population models can reveal the essential characteristics of time delay in practical problems, and it has attracted extensive attention in [5,6,15,17].It is worthy mentioned that there have been few papers concerning the effect of delay on the dynamical behavior of the delayed Nicholson's blowflies model with a nonlinear density-dependent mortality term.
Motivated by the above works, the effect of delay on the dynamical behavior of the delayed Nicholson's blowflies model with a nonlinear density-dependent mortality term attracted our attention.In this paper, we aim to provide a criterion to guarantee that all solutions of (1.2) converge to the positive equilibrium point, which entails that (1.2) is global attractive under the small delays.In fact, one can see the following Remark 2.2 and Remark 3.1 for details.
In what follows, we designate r = max 1≤j≤m sup t∈R τ j (t), C = C([−r, 0], R) be the continuous functions space equipped with the usual supremum norm • , and let It will be always assumed that there exists at least one positive constant N such that which is a positive equilibrium point of (1.2).

Main result
In this section, we establish some sufficient conditions on the global asymptotic stability of positive equilibrium point for (1.2).

W. Xiong
According to (1.2), we get Then, taking limits gives us that Remark 2.2.From Lemma 2.1, it is not difficult to see that (1.2) and (1.4) is uniformly permanent.Moreover, N is also a solution of (1.2) and (1.4), and For simplicity, denote N(t; t 0 , ϕ) by N(t).Now, we show the global attractivity of N by the following three propositions: Proof.Clearly, there exists T > t 0 such that In order to prove Proposition 2.3, it suffices to show that lim sup t→+∞ x(t) = 0. Again by way of contradiction, we assume that lim sup t→+∞ x(t) > 0. By the fluctuation lemma [16, Lemma A.1], there exists a sequence {t k } k≥1 such that In view of (2.2), we can choose K > T to satisfy that for all t > K, j ∈ J, which, together with the fact that xe −x decreases on [1, +∞), implies that By taking limits, (1.5) and (2.3) lead to β j e −γ j N = 0, a contradiction.Hence, lim sup t→+∞ x(t) = 0.This completes the proof.Proof.Obviously, we can choose T > t 0 such that Next, we prove that lim inf t→+∞ x(t) = 0. Otherwise, lim inf t→+∞ x(t) < 0. Again from the fluctuation lemma [16, Lemma A.1], there exists a sequence { tk } k≥1 such that tk → +∞, x( tk ) → lim inf t→+∞ x(t), x ( tk ) → 0 as k → +∞.
From (2.1), we can choose K * > T to satisfy that which, together with the fact that xe −x decreases on [1, +∞), implies that By taking limits, (1.5) and (2.4) give us that 0 ≥ −a + be −(lim inf t→+∞ x(t β j e −γ j N = 0, a contradiction and hence lim inf t→+∞ x(t) = 0.This completes the proof.
Now, in order to prove Proposition 2.5, it suffices to show that that λ = µ = 0. Again from the fact that y(t) oscillates about zero, we can choose a strictly monotonically increasing sequence {q n } n≥1 to satisfy that q n > r, lim n→+∞ q n = +∞, y(q n ) = 0 for all n = 1, 2, . . ., and such that in each interval (q n , q n+1 ) the function y(t) assumes both positive and negative values.For any positive integer n, let t n , s n ∈ (q n , q n+1 ) such that Then, and (2.9) Subsequently, we assert that for each positive integer n, there is and y(t) > 0 for all t ∈ (T n , t n ). (2.10) In the contrary case, given a positive integer n, we have which, together with (1.5), (2.2), (2.6), (2.8) and the fact that xe −x decreases on [1, +∞), tells us that This is a contradiction and proves (2.10).Similarly, we can prove that for each positive integer n, there is y(S n ) = 0, and y(t) < 0 for all t ∈ (S n , s n ). (2.11) For any ε > 0, (2.9) implies that there exists a positive integer n * such that min{t n * , s n * } − 2r > t 0 , and (2.12) Thus, In view of (1.3), (2.2), (2.10), (2.12) and (2.13), integrating (2.6) from T n to t n , we find Letting n → +∞ and ε → 0 + , (2.5) and (2.14) give us that Furthermore, from (1.3), (2.2), (2.6), (2.11) and (2.12), we obtain According to the proof in Theorem 4.1 of [17], one can show λ = µ = 0.This ends the proof.
By Propositions 2.3, 2.4 and 2.5, we have the following result.implies that condition (2.5) is not satisfied when the delays in (1.2) is sufficiently large.

An example
In this section, we will give an example to verify the correctness of our main results obtained in previous section.Considering the following Nicholson's blowflies model with a nonlinear density-dependent mortality term: If we choose τ = 0.1, it is straight to check that (3.1) satisfies (1.3) and (2.5).It follows from Theorem 2.6 that the positive equilibrium point 2 is a global attractor of (3.1).Fig. 3.1 supports this result with the numerical solutions of system (3.1) with different initial values.Moreover, if we choose τ = 10, then, (3.1) does not satisfy (2.5), we give the numerical simulations in Fig. 3.2 to show that 2 is no longer a global attractor of (3.1).This implies that a small delay does not affect the asymptotic behavior of system (3.1), and large delay will cause the complex dynamic behavior of this system.Remark 3.1.The effect of delay on the asymptotic behavior plays an important role in describing the dynamics of population models [10,14].Thus it has been extensively studied by many scholars in recent decades.In this article, we first studied the effect of delay on the asymptotic behavior of Nicholson's blowflies model with a nonlinear density-dependent mortality term.By means of the fluctuation lemma and some differential inequality technique, delay-dependent criteria are obtained for the global attractivity of the considered model.The sufficient condition, which is easily checked in practice, has a wide range of application.This implies that the obtained results in this article are completely new and extend previously known studies to some extent.In addition, the method in this paper can be applied to study the effect of the delay on the asymptotic behavior for some other dynamical systems.Also, it is natural to ask whether the delay affects the dynamical behavior of the addressed systems involving time-varying delays and time-varying coefficients.We leave this as our future work.

Theorem 2 . 6 . 5 )
Suppose that (2.5) holds, then the positive equilibrium point N of (1.2) is a global attractor.naturally holds under the sufficiently small delay, and the positive equilibrium point N is a global attractor of (1.2) with the small delays.Moreover,