Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term

This paper mainly investigates a class of almost periodic Nicholson's blowflies model involving a nonlinear density-dependent mortality term and time-varying delays. Combining Lyapunov function method and differential inequality approach, some novel assertions are established to guarantee the existence and exponential stability of positive almost periodic solutions for the addressed model, which generalize and refine the corresponding results in some recent published literatures. Particularly, an example and its numerical simulations are given to support the proposed approach.


1.
Introduction. Many problems in the fields of physics [1,2], mathematical biology [3,4,5,6,7,8,9] and control theory can be attributed to study of the nonlinear differential equations, especially it is almost periodicity because there is almost no phenomenon that is purely periodic [10,11,12]. Consequently, the qualitative theory of differential equations involving almost periodicity has been the new world-wide focus. In particular, more attention has been paid to the existence and global stability of almost periodic solutions for delayed Nicholson's blowflies equation and and its variants [9,13,14,15,16,17,18,19,24,25]. For example, some sufficient conditions ensuring the global exponential stability of positive periodic solutions and almost periodic solutions of classical Nicholson's blowflies equation with time-varying delays have been established in [10] and [11,12] respectively. Furthermore, the global exponential stability of positive almost periodic solutions for Nicholson's blowflies models involving nonlinear density-dependent mortality terms has been obtained in [14,15,16,17,18,19,20]. The results included in each paper of [10,11,12,14,15,16,17,18,19,20] gave an answer to the open problem: Find global stability conditions for the positive periodic solution of delayed non-autonomous Nicholson's blowflies equation, which was proposed by Berezansky et al. [21].
It should be pointed out that, to a large extent, all results involving the global exponential stability of periodic solutions and almost periodic solutions for delayed non-autonomous Nicholson-type equations established in [10,11,12,14,15,16,17,18,19,20] are based on that the solutions exist in a smaller interval [κ, κ] ≈ [0.7215355, 1.342276], where Obviously, the existence of periodic solutions and almost periodic solutions is restricted to [κ, κ] will inevitably impose many constraints on mathematical modeling. To the authors' best knowledge, no existing work has discussed the global stability of periodic solutions for Nicholson's blowflies equation when the existence interval of periodic solution exceeding [κ, κ], such studies are however important for us to understand the dynamical characteristics of population models. Motivated by the above discussions, in this paper, our purpose is to study the existence and global exponential stability of almost periodic solutions of the following Nicholson's blowflies model with a nonlinear density-dependent mortality term: where x(t) denotes the population of sexually mature adults at time t, a, b, β j , γ j : R → (0, +∞) and τ j : R → [0, +∞) are almost periodic functions, and j ∈ S = {1, 2, · · · , m}. The definition on almost periodic function can be found in [1,3]. For more details on biological explanations of the coefficients of (1.1), we refer the reader to [12,14,15,16,17,18,19,20,21]. The significance of this paper is as follows. Firstly, based on differential inequality techniques, a novel proof of the positivity of solutions of delayed non-autonomous Nicholson's blowflies model with a nonlinear density-dependent mortality term is given. Secondly, by using the fluctuation lemma, we establish the boundedness interval on all solutions of (1.1) without involving [κ, κ]. Thirdly, a almost periodic solution for system (1.1) is given in bounded interval without adopting [κ, κ], which is global exponential stable. For simplicity, we introduce the following notations: Then, we introduce the initial value conditions of (1.1) as follows: Let x(t; t 0 , ϕ) be a solution of the initial value problem (1.1) and (1.2), and [t 0 , η(ϕ)) be the maximal right-interval of existence of x t (t 0 , ϕ). Particularly, the existence and uniqueness of x(t; t 0 , ϕ) is straightforward in [22].
2. Main results. Before proceeding, the following two lemmas will be introduced.

>0.
This contradiction suggests that x(t) > 0 for all t ∈ [t 0 , η(ϕ)). In order to demonstrate the boundedness of x(t), define Evidently, which, together with the fact that sup With the help of (2.1) and which is a contradiction and reveals the boundedness of x(t).
respectively. Regard to the boundedness of a(t), b(t), β j (t), γ j (t) and x(t − τ j (t)), we can select a subsequence of {k} k≥1 , still denoted by {k} k≥1 , such that exist for all j ∈ S. Consequently, it follows from (2.4) and (2.5) that ,

ALMOST PERIODICITY ANALYSIS FOR A DELAYED NICHOLSON'S BLOWFLIES 3341
which yield .
This finishes the proof of Lemma 2.1.
Proof. According to (2.6), it is easy to see that there exists t * 0 ≥ t 0 such that Then, H(u, v) is a continuous function, and which implies that there exist two constants η, λ ∈ (0, 1] such that For any ε ∈ (0, min{η, S − }), it follows from Lemma 2.1 that there exists T ϕ > t * 0 such that which implies that the right side of (1.1) is also bounded, and x (t) is a bounded function on [t 0 − σ, +∞). Thus, with the help of the fact that Bε, for all t ∈ R. (2.11) Then, for all t ≥ Q 0 , we get Note the following inequalities and
Step one. If E(t) > e λt |u(t)| for all t ≥ Q 0 , we assert that In the contrary case, one can pick which contradicts the fact that E(β * ) > e λβ * |u(β * )| and proves the above assertion. Then, we can select Q 2 > Q 0 satisfying

CHUANGXIA HUANG, HUA ZHANG AND LIHONG HUANG
Step two. If there exists Q * ≥ Q 0 such that E(Q * ) = e λQ * |u(Q * )|, we can have from (2.13) and (2.20) that For any t > Q * satisfying E(t) = e λt |u(t)|, by the same method as that in the derivation of (2.20), we can show e λt |u(t)| < εe λt , and |u(t)| < ε. With a similar reasoning as that in the proof of Step one, we can entail that which, together with (2.22), follows that Finally, the above discussion infers that there existsQ > max{Q * , Q 0 , t 2 } obeying that which finishes the proof of Lemma 2.2.
3. Global exponential stability of almost periodic solutions. Combined with Lemmas 2.1 and 2.2, we can have the following theorem: Theorem 3.1. Assume that all assumptions of Lemmas 2.1 and 2.2 are satisfied. Then, (1.1) has a globally exponentially stable positive almost periodic solution x * (t). Moreover, there exist constants K ϕ,x * and t ϕ,x * such that |x(t; t 0 , ϕ) − x * (t)| < K ϕ,x * e −λt for all t > t ϕ,x * .

Proof.
Let v(t) = v(t; t 0 , ϕ v ) be a solution of equation (1.1) with initial conditions satisfying the assumptions in Lemma 2.2. We also add the definition of where {t k } is any sequence of real numbers. For any ε ∈ (0, min{η, S − }), by Lemma 2.1 , we can choose t ϕ v > t 0 such that which together with the boundedness of v (t) and the fact that is uniformly continuous on R. Then, from the almost periodicity of a, b, 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 Arzala-Ascoli Lemma and diagonal selection principle, we can choose a subsequence {t kj } of {t k }, such that v(t + t kj )(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 Now, by similar lines used in the proof of Theorem 3.1 in [21], we can prove that x * (t) is a positive almost periodic solution of (1.1), which, together with the arbitrariness of ε, implies that (3.5) Finally, we prove that x * (t) is globally exponentially stable. Let x(t) = x(t; t 0 , ϕ) and Obviously, .

Remark 1.
Note that x * (t) ≥ 3 does not belong to [κ, κ] ≈ [0.7215355, 1.342276], and γ 1 (t) = γ 2 (t) = 1 2 does not satisfy the following condition: γ j (t) ≥ 1, for all t ∈ R, j ∈ S, which has been considered as fundamental for the considered periodicity and almost periodicity of delayed non-autonomous Nicholson's blowflies models in [10,11,12,14,15,16,17,18,19,20]. Hence, all the results in these above mentioned references cannot be applicable to show the global exponential stability on the positive almost periodic solution of (4.1). Whether or not our approach and method in this paper are available to study the global stability of almost periodic solutions for other delayed Nicholson's blowflies models (1.1) in the case that the almost periodic solution does not belong to [κ, κ] ≈ [0.7215355, 1.342276], it is an interesting topic and we leave this as our future research.