Asymptotically almost periodic dynamics on delayed Nicholson-type system involving patch structure

This paper explores a delayed Nicholson-type system involving patch structure. Applying differential inequality techniques and the fluctuation lemma, we establish a new sufficient condition which guarantees the existence of positive asymptotically almost periodic solutions for the addressed system. The results of this article are completely new and supplement the previous publications.


Introduction
As we all know, periodicity is important in real surroundings and the world, but almost periodicity is always more accurate, more realistic, and more general than periodicity when adding varied environmental factors. In comparison with periodic effects, almost periodic effects are more frequent in lots of real world applications [1][2][3][4]. In particular, the existence and global stability of almost periodic solutions for the famous scalar Nicholson's blowflies equation x (t) = -a(t)x(t) + m j=1 β j (t)x tτ j (t) e -γ j (t)x(t-τ j (t)) (1.1) and the Nicholson's blowflies systems with patch structure (t)x i (t-τ ij (t)) , i ∈ Q := {1, 2, . . . , n}, (1.2) have been extensively investigated in previous studies [5,6] and [7], respectively. It is easy to know that scalar Nicholson's blowflies Eq. (1.1) is a special case of Nicholson's blowflies system (1.2), where x i (t) denotes the density of the ith-population at time t, a ij (t) (i = j) is the rate of the population moving from class j to class i at time t, a ii (t) is the coefficient of instantaneous loss (which integrates both the death rate and the dispersal rates of the population in class i moving to the other classes), β ij (t)x i (tτ ij (t))e -γ ij (t)x i (t-τ ij (t)) is the birth function, β ij (t) is the birth rate for the species, 1 γ ij (t) is the ith-population reproducing at its maximum rate, and τ ij (t) is the generation time of the ith-population at time t. For the feedback function xe -x and its derivative 1-x e x , the author in [8] pointed out that there exist two fixed positive numbers κ andκ such that It is worth noting that the global exponential stability of almost periodic solutions of (1.1) has been shown in [5,6] under the restriction that the almost periodic solution exists in a small interval [κ, κ] ≈ [0.7215355, 1.342276], and the global exponential stability of (1.2) has been established in [7] where the authors adopted the restraint that the almost periodic solution exists in a small domain Obviously, the above restriction and restraint do not accord with the biological significance of the population models.
On the other hand, γ ij (t) ≥ 1 for all t ∈ R, i ∈ Q, j ∈ I := {1, 2, . . . , m}, (1.5) has been made as the crucial assumption in [5][6][7]. It should be mentioned that the stability of a class of delayed nonlinear density-dependent mortality Nicholson's blowflies model has been investigated in [9][10][11][12] without assumption (1.5), when the maximum reproducing rate is not limited (i.e. 1 γ ij (t) maybe sufficiently large). However, there is no research work on the global exponential stability of almost periodic solutions for Nicholson's blowflies Eq. (1.1) without assumption (1.5) and avoiding [κ, κ] as the existence interval for almost periodic solutions. In particular, to the best of our knowledge, there has not yet been research work on the global stability of almost periodic solutions of Nicholson's blowflies systems with patch structure and nonlinear density-dependent mortality terms when the almost periodic solutions do not belong to the above domain (1.4).
Regarding the above discussions, in this paper, without adopting [κ, κ] × · · · × [κ, κ] n as the existence domain of almost periodic solutions, we establish the existence and global exponential stability of positive almost periodic solutions for Nicholson's blowflies systems involving patch structure and nonlinear density-dependent mortality terms. Our results improve and complement some existing ones in the recent publications [5][6][7]12], and its effectiveness is demonstrated by some numerical examples. This paper is organized as follows: In Sect. 2, some necessary definitions, lemmas, and assumptions are presented. In Sect. 3, the existence and global attractivity of positive asymptotically almost periodic solutions are demonstrated by virtue of some differential inequalities and analytic techniques. To verify our theoretical results, a numerical experiment is carried out in Sect. 4. Conclusions are drawn in Sect. 5.

Preliminary results
Throughout this paper, it will be assumed that and let BC(J 1 , J 2 ) be the set of bounded and continuous functions from J 1 to J 2 .
Definition 2.1 (see [1,2]) A subset P of R is said to be relatively dense in R if there exists a constant l > 0 such that [t, t + l] ∩ P = ∅ (t ∈ R). u ∈ BC(R, J) is almost periodic on R if, for any > 0, the set T(u, ) = {δ : |u(t + δ)u(t)| < , ∀t ∈ R} is relatively dense.
Definition 2.2 (see [1,2]) u ∈ C(R + , J) is asymptotically almost periodic if there exist an almost periodic function h and a continuous function g ∈ W 0 (R + , J) such that u = h + g.
For J ⊆ R, we denote the set of almost periodic functions from R to J by AP(R, J). The set of asymptotic almost periodic functions will be represented by AAP(R, J). In addition, AP(R, J) is a proper subspace of AAP(R, J) [1,2]. Remark 2.1 (see [1,p. 64,Remark 5.16]) The decomposition given in Definition 2.2 is unique.
Proof From Lemmas 2.1 and 2.2, we can designate i l , i L ∈ Q such that By the fluctuation lemma [14, Lemma A.1], one can select a sequence {t * k } +∞ k=1 satisfying Now, we show that l > 0. By way of contradiction, we assume that for each t ≥ t 0 . From (2.15), we can choose i * * ∈ Q and a strictly monotone increasing sequence {ξ n } +∞ n=1 such that lim n→+∞ ξ n = +∞, and then According to (2.17), one can find that there exists n * * > 0 such that, for n > n * * and j ∈ I, and which together with (2.16) and the fact that lim inf t→+∞ β i * * j (t) > 0 gives Letting n → +∞, it follows from (2.10) and (2.18) that which is a contradiction. Hence, l > 0. Furthermore, from the asymptotically almost periodicity of (1.2), we can select a subsequence of ) exist for all j ∈ Q, q ∈ I. In addition, from (1.2) and (2.14), we have which, together with the definitions of δ and A, entails that Finally, we show that l > κ γ -. Again from the fluctuation lemma [14, Lemma A.1] and the asymptotically almost periodicity of (1.2), we can pick a sequence {t * * k } +∞ k=1 such that and Consequently, according to (2.20) and (2.21), we gain together with (2.12) and (2.22), we obtain Proof From (2.1), (2.2), (2.10), (2.11), (2.12) and the definition of asymptotically almost periodic function, one can easily find that and δ = 1 .
Then, by applying a similar argument as Lemma 2.3, we can obtain which proves Lemma 2.4.

Lemma 2.5
Let assumptions adopted in Lemma 2.3 hold, and x h (t) = x h (t; t 0 , ϕ) be a solution of equation (1.2) h and (2.4). Then, for any > 0, we can choose a relatively dense subset P of R with the property that, for each δ ∈ P , there exists T = T(δ) > 0 satisfying Proof According to the fact we have This means there exist two constants η > 0 and λ ∈ (0, 1] such that, for i ∈ Q, (2.27) and (2.28) In view of Lemma 2.4, one can see that x h (t) and the right-hand side of (1.2) h are bounded. It follows from (2.27) that x h (t) is uniformly continuous on R. Therefore, for any > 0, we can choose a sufficiently small constant * > 0 such that where t ∈ R, i ∈ Q, j ∈ I. Furthermore, for * > 0, from the uniformly almost periodic family theory in [2, p. 19, Corollary 2.3], one can choose a relatively dense subset P * of R such that Denote P = P * for any δ ∈ P , from (2.29) and (2.30), we have and where i ∈ Q. Let i t be such an index that Then, for all t ≥ Λ 0 , we have It is obvious that e λt u(t) ≤ E(t), and E(t) is nondecreasing. Now, the remaining proof will be divided into two steps.
Step one. If E(t) > e λt u(t) for all t ≥ Λ 0 , we assert that In the contrary case, one can pick Λ 1 > Λ 0 such that E(Λ 1 ) > E(Λ 0 ). From the fact that we can find that there exists β * ∈ (Λ 0 , Λ 1 ) such that which contradicts the fact that E(β * ) > e λβ * u(β * ) and proves (2.36). Then we can select (2.37) Step two. If there exists ς ≥ Λ 0 such that E(ς) = e λς u(ς) , from (2.35) and the definition of E(t), we have With a similar proof in step one, we can entail that which together with (2.41) leads to Finally, the above discussion infers that there existsΛ > max{ς, Λ 0 , Λ 2 } obeying that which finishes the proof of Lemma 2.5.

Main result
We also define for all t + t q ≥ t 0 , i ∈ Q. By using the proof similar to Lemma 2.5, we can choose {t q } q≥1 such that From Arzela-Ascoli lemma and the fact that the function sequence {v(t + t q )} q≥1 is uniformly bounded and equi-uniformly continuous, we can choose a subsequence {t q j } j≥1 of {t q } q≥1 such that {v(t + t q j )} j≥1 (for convenience, we still denote it by {v(t + t q )} q≥1 ) uniformly converges to a continuous function x * (t) = (x * 1 (t), x * 2 (t), . . . , x * n (t)) on any compact set of R. Then, from Lemma 2.4, we have and on any compact set of R, where "⇒" denotes "uniformly converge". Thus, (3.2), (3.3), and (3.5) produce that {v i (t + t q )} q≥1 uniformly converges to on any compact set of R. According to the properties of the uniform convergence function sequence, we obtain that x * (t) is a solution of (1.2) h and From Lemma 2.5, for any > 0, we can choose a relatively dense subset P of R with the property that, for each δ ∈ P , there exists which implies that x * (t) is a positive almost periodic solution of (1.2) h . Next, we show that all the solutions of (1.2) converge to x * (t) as t → +∞. Let x(t) be an arbitrary solution of system (1.2) with initial value (2.4). Define y(t) = x(t)x * (t) and Then For any > 0, in view of the global existence and uniform continuity of x and the fact that a g ij , β and let i t be such an index that e λt y i t (t) = e λt y(t) .
According to (3.4) and Lemma 2.3, one can find T ϕ,x * > T * * ϕ such that Combined with (2.34) and (3.7), we gain Then, from (2.26) and (3.8), by employing the argument of Lemma 2.5, we know that there is a constant T ≥ T ϕ,x * such that It follows from the uniqueness of the limit function that (1.2) h has exactly one positive almost periodic solution x * (t). The proof is complete.
Then, we will establish the existence and global exponential stability of the almost periodic solution of (1.2) h . To do this end, we first show the following proposition. Proof We only need to validate the case that lim sup t→+∞ f (t) = sup t∈R f (t), since the other case that lim inf t→+∞ f (t) = inf t∈R f (t) can be proved similarly. Define It is easy to see that B ≤ A. We claim Otherwise, B < A, let ε 0 = A-B 8 , from the definition of upper limit, there exists T = T(ε 0 ) > 0 such that According to the definition of the upper bound, one can take t 0 ∈ R to satisfy that Furthermore, there exists a constant l = l(ε 0 ) > 0 such that, ∀[α, α + l] ⊂ R with α ∈ R, one can pick τ ∈ [α, α + l] satisfying that Letting α = Tt 0 and τ ∈ [Tt 0 , Tt 0 + l] leads to which is contrary to the fact that f (t) < B + ε 0 < A -2ε 0 for all t ≥ T. This finishes the proof of Proposition 3.1.

Theorem 3.2 Suppose that, for i ∈ Q, j ∈ I,
(3.14) and δ = 1 It follows from Lemma 2.4 that there is M ϕ,x * > t 0 such that Together with (3.11), we obtain where i ∈ Q, j ∈ I. With the help of (3.13), we can choose λ ∈ (0, 1] such that Now, we define the Lyapunov functional as follows: With the help of (3.16) and (3.18), we get In the sequel, we prove that, for all t > M ϕ,x * , Otherwise, there exist K * > M ϕ,x * andî ∈ Q such that It follows from (3.19), (3.20), and (3.22) that which is a contradiction. Thus (3.21) holds, and it follows that This completes the proof of Theorem 3.2.

Some numerical simulations
In this section, we give two examples with simulations to demonstrate the feasibility and the validity of our theoretical results.

Conclusions
In this paper, we combine the Lyapunov function method with the differential inequality method to establish some new criteria ensuring the existence and attractivity of positive asymptotically almost periodic solutions for a class of delayed Nicholson's blowflies systems with patch structure. The assumptions adopted in this present paper are different from some previously known literature. Numerical simulations have been given to illustrate the obtained results. The approach presented in this article can be used as a possible way to study the asymptotically almost periodic patch structure population models such as neoclassical growth model, Mackey-Glass model, epidemical system or age-structured population model, and so on. We leave this as our future work.