New Results on Almost Periodic Solutions for a Nicholson's Blowflies Model with a Linear Harvesting Term

In this paper, we study the global dynamic behavior of a non-autonomous delayed Nicholson's blowflies model with a linear harvesting term. Under proper conditions , we employ a novel argument to establish a criterion on the global exponential stability of positive almost periodic solutions for the model. Moreover, we also provide a numerical example to support the theoretical results.


Introduction
Since the exploitation of biological resources and the harvest of population species are commonly practiced in fishery, forestry, and wildlife management, the study of population dynamics with harvesting is an important subject in mathematical bioeconomics, which is related to the optimal management of renewable resources (see [3,7,13]).Recently, Berezansky et al. [2] presented the following Nicholson's blowflies model x (t) = −δx(t) + px(t − τ)e −ax(t−τ) − Hx(t − σ), δ, p, τ, a, H, σ ∈ (0, +∞), (1.1) where Hx(t − σ) is a linear harvesting term, x(t) is the size of the population at time t, p is the maximum per capita daily egg production, 1 a is the size at which the population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ is the generation time.Furthermore, the authors in [1,12,15,16] extended (1.1) to non-autonomous equations with periodic time-varying coefficient and delays, and they also established some criteria to guarantee the existence of positive periodic solutions for these generalized models by applying the method of coincidence degree and the fixed-point theorem in cones.L. Berezansky et al. [2] formulated an open problem: what can be said about the dynamic behavior of (1.1).

Aiping Zhang
It is well known that the focus in theoretical models of population and community dynamics must not be only on how populations depend on their own population densities or the population densities of other organisms, but also on how populations change in response to the physical environment.To consider almost periodic environmental factors, it is reasonable to study the non-autonomous Nicholson's blowflies model with almost periodic coefficients and delays.Most recently, some sufficient conditions were obtained in [10,11,14] to ensure the local existence and exponential stability of positive almost periodic solution for the non-autonomous Nicholson's blowflies model with a linear harvesting term.However, as pointed out by Liu [8], it is difficult to study the global dynamic behavior of the Nicholson's blowflies model with a linear harvesting term.So far, there is no literature considering the global exponential stability of positive almost periodic solutions for (1.1) and its generalized equations.Thus, it is worthwhile to continue to investigate the global dynamic behavior of positive almost periodic solutions for non-autonomous Nicholson's blowflies model with the linear harvesting term.
Motivated by the above discussions, the main purpose of this paper is to establish some criteria for the global dynamic behavior of positive almost periodic solutions for a general Nicholson's blowflies model with the linear harvesting term given by where a, H, σ, γ j : R → (0, +∞) and β j , τ j : R → [0, +∞) are almost periodic functions for j = 1, 2, . . ., m. Obviously, (1.1) is a special case of (1.2) with constant coefficients and delays.
For convenience, we introduce some notations.In the following part of this paper, given a bounded continuous function g defined on R, let g + and g − be defined as It will be assumed that ) be the continuous functions space equipped with the usual supremum norm • , and let We also consider admissible initial conditions We denote by x t (t 0 , ϕ)(x(t; t 0 , ϕ)) an admissible solution of the admissible initial value problem (1.2) and (1.4).Also, let [t 0 , η(ϕ)) be the maximal right-interval of the existence of x t (t 0 , ϕ).
Since the function 1−x e x is decreasing on the interval [0, 1] with the range [0, 1], it follows easily that there exists a unique κ ∈ (0,

Preliminary results
In this section, we shall first recall some basic definitions, lemmas which are used in what follows.
Definition 2.1 ([4, 5]).A continuous function u : R → R is said to be almost periodic on R if, for any > 0, the set T(u, ) = {δ : |u(t + δ) − u(t)| < for all t ∈ R} is relatively dense, i.e., for any > 0, it is possible to find a real number l = l( ) > 0 with the property that, for any interval with length l( ), there exists a number δ = δ( ) in this interval such that |u(t From the theory of almost periodic functions in [4,5], it follows that for any > 0, it is possible to find a real number l = l( ) > 0, for any interval with length l( ), there exists a number δ = δ( ) in this interval such that for all t ∈ R and j = Then, there exist two positive constants K 1 and K 2 such that 2) hold.Suppose that there exists a positive constant M such that and
(2.6) Lemma 2.5.Suppose (2.2), (2.4) and (2.5) hold, and Moreover, assume that x(t) = x(t; t 0 , ϕ) is a solution of equation (1.2) with initial condition (1.4) and ϕ is bounded continuous on [−r, 0].Then for any > 0, there exists l = l( ) > 0, such that every interval [α, α + l] contains at least one number δ for which there exists N > 0 satisfying Proof.Define a continuous function Γ(µ) by setting Then, we have which implies that there exist two constants η > 0 and λ ∈ (0, 1] such that (2.10) By Lemma 2.2, the solution x(t) is bounded and which implies that the right side of (1.2) is also bounded, and x (t) is a bounded function on [t 0 − r, +∞).Thus, in view of the fact that x(t) ≡ x(t 0 − r) for t ∈ (−∞, t 0 − r], we obtain that x(t) is uniformly continuous on R. From (2.1), for any > 0, there exists l = l( ) > 0, such that every interval [α, α + l], α ∈ R, contains a δ for which (2.12) Then, for all t ≥ N 0 , we get ( In summary, there must exist N > max{t * 0 , N 0 , t 2 } such that |u(t)| ≤ holds for all t > N. The proof of Lemma 2.5 is now complete.

Main results
In this section, we establish sufficient conditions on the existence, uniqueness, and global exponential stability of positive almost periodic solutions of (1.2).
Proof.Let v(t) = v(t; t 0 , ϕ v ) be a solution of equation (1.2) with initial conditions satisfying the assumptions in Lemma 2.5.We also add the definition of v(t) with v(t) ≡ v(t 0 − r) for all t ∈ (−∞, where {t k } is any sequence of real numbers.By Lemma 2.3, the solution v(t) is bounded and which implies that the right side of (1.2) is also bounded, and v (t) is a bounded function on [t 0 − r, +∞).Thus, in view of the fact that v(t) ≡ v(t 0 − r) for t ∈ (−∞, t 0 − r], we obtain that v(t) is uniformly continuous on R.Then, from the almost periodicity of a, H, σ, τ j , γ j and β j , we can select a sequence {t k } → +∞ such that for all j, t.
Since {v(t + t k )} +∞ k=1 is uniformly bounded and equiuniformly continuous, by the Ascoli-Arzelà lemma and diagonal selection principle, we can choose a subsequence {t k j } of {t k }, such that v(t + t k j ) (for convenience, we still denote by v(t + t k )) uniformly converges to a continuous function x * (t) on any compact set of R, and Almost periodic solutions of Nicholson's blowflies model 9 Now, we prove that x * (t) is a solution of (1.2).In fact, for any t ≥ t 0 and ∆t ∈ R, from (3.3), we have where t + ∆t ≥ t 0 .Consequently, (3.5) implies that Secondly, we prove that x * (t) is an almost periodic solution of (1.2).From Lemma 2.5, for any ε > 0, there exists l = l(ε) > 0, such that every interval [α, α + l] contains at least one number δ for which there exists N > 0 satisfing Then, for any fixed s ∈ R, we can find a sufficiently large positive integer  Calculating the upper left derivative of V(t) along the solution y(t) of (3.9), we have    This completes the proof of Theorem 3.1.

An example
In this section, we present an example to check the validity of our results obtained in the previous sections.Remark 4.2.We remark that the results in [6,[8][9][10][11]14] and the references therein cannot be applied to prove the global exponential stability of positive almost periodic solutions for (4.1).This implies that the results of this paper are new and they complement previously known results.In particular, in this present paper, we employ a novel proof to establish some criteria to guarantee the global dynamic behavior of positive almost periodic solutions for nonautonomous Nicholson's blowflies model with the linear harvesting term.