Exponential Convergence of a Non-autonomous Nicholson's Blowflies Model with an Oscillating Death Rate

This paper is concerned with a non-autonomous delayed Nicholson's blowflies model with an oscillating death rate. Under proper conditions, we employ a novel argument to establish a criterion on the global exponential convergence of the zero equilibrium point for this model. The obtained result improves and supplements existing ones. We also use numerical simulations to demonstrate our theoretical results.


Introduction
The Nicholson's blowflies model x (t) = −ax(t) + βx(t − τ)e −γx(t−τ) , (1.1) was used in Gurney et al. [6] to describe the periodic oscillation in Nicholson's classic experiments [11] with the Australian sheep blowfly, Lucilia cuprina.Here β is the maximum per capita daily egg production rate, 1 γ is the size at which the blowfly population reproduces at its maximum rate, a is the per capita daily adult death rate, and τ is the generation time.As a classical model of biological systems, model (1.1) and its modifications have also been later used to describe population growth of other species, and thus, have been extensively and intensively studied by many researchers (see, e.g., [3] and the references therein).
When the model is used to describe the population dynamics with periodically varying environment, the coefficients and delays in the model are usually periodically time-varying.

Z. Long
Therefore, (1.1) has been frequently generalized into the following non-autonomous Nicholson's blowflies model: where m is a given positive integer, a : R → R and β j , γ j , τ j : R → [0, +∞) are bounded and continuous functions, and j = 1, 2, . . ., m.In particular, there have been extensive results on the problem of the convergence and persistence of model (1.2) in the literature.We refer the reader to [1,3,4,[7][8][9][10] and the references cited therein.Moreover, in these known results in [1,3,4,[6][7][8][9][10][11], we find the following condition that the coefficient function a(t) in the death rate is not oscillating, i.e., a(t) > 0 for all t ∈ R, has been adopted as fundamental for the considered dynamic behaviors of (1.1) and (1.2).However, as pointed out in [2,12], equations with oscillating coefficients appear in linearizations of population dynamics models with seasonal fluctuations, where during some seasons the death or harvesting rates may be greater or lesser than the birth rate, and therefore, it is more reasonable to assume that the death rate in (1.2) is oscillating.This motivates us to establish criteria on the global exponential convergence of the zero equilibrium point for (1.2) without condition (1.3).
The remaining of this paper is organized as follows.In Section 2, we give a lemma, which tells us that some kinds of solutions to (1.2) are bounded and permanent.This result plays an important role in Section 3 to establish the global exponential convergence for (1.2) with an oscillating death rate.The paper concludes with an example to illustrate the effectiveness of the obtained results by numerical simulation.
is continuous and defined on [−τ + t 0 , ) with t 0 , ∈ R and t 0 < , then, for all t ∈ [t 0 , ), we define x t ∈ C, in which x t (θ) = x(t + θ) for all θ ∈ [−τ, 0].Given a bounded continuous function g defined on R, let g + and g − be defined as According to the biological interpretation of (1.2),only positive solutions are meaningful and therefore admissible.Consequently, the initial conditions are given by a(v)dv , and integrating it on [t 0 , t], by virtue of (2.1), we have for all t ∈ [t 0 , η(ϕ)).
Observe that which is a contradiction and proves (2.4).

Main result
We are now in a position to establish new criteria on the global exponential convergence of the zero equilibrium point for (1.2) with an oscillating death rate.
Otherwise, there exists T > t 0 , such that On the other hand, in view of (2.3) and (2.5), we have where L = K ϕ e λt 0 .The proof is complete.

An example
In this section, we give an example and its numerical simulations to demonstrate the result obtained in Section 3.
Example 4.1.Consider the following Nicholson's blowflies model with an oscillating death rate: x (t) = − (8 + 10 cos 2000t)x(t) where a(t) = 8 + 10 cos 2000t,  Remark 4.2.To the best of our knowledge, no results on the dynamics of (1.2) with an oscillating death rate have been reported up to now and we also mention that none of the results in the references [1-4, 7-10, 12] can be applied to (4.1), which implies that the obtained results in the present paper are completely new and extend previously known results to some extent.
Then (4.1) satisfies all the conditions in Theorem 3.1.It follows that all solutions of (4.1) with initial conditions in (2.1) converge to the zero equilibrium point as t → +∞.This fact is verified by the numerical simulations in * (u) − a(u))du ≤ M, for all t, s ∈ R and t − s ≥ 0.