Global convergence dynamics of almost periodic delay Nicholson's blowflies systems

We take into account nonlinear density-dependent mortality term and patch structure to deal with the global convergence dynamics of almost periodic delay Nicholson’s blowflies system in this paper. To begin with, we prove that the solutions of the addressed system exist globally and are bounded above. What’s more, by the methods of Lyapunov function and analytical techniques, we establish new criteria to check the existence and global attractivity of the positive asymptotically almost periodic solution. In the end, we arrange an example to illustrate the effectiveness and feasibility of the obtained results.


Introduction
There has been a growing concern that the dynamic model plays an important role in many fields including biology system, financial and economic network, physics, and engineering technology [1,11,13,15,16,22,41]. In order to describe the oscillatory fluctuations of the laboratory population of the Australian sheep blowfly Lucilia cuprina, Gurney et al. [7] proposed the following delay Nicholson's blowflies equation: Biologically, N(t) represents the size of sexually mature adults at time t, the per capita adult death rate with density-independent value equals δ, T D is the generation time from eggs to sexually mature adults, and P denotes the maximum possible per capita daily egg production rate. The birth function gets the maximum reproduction value with N = 1 N D . The delay Nicholson's blowflies equation offers a suitable benchmark for describing a 'humped' relationship between future recruitment and current population as it presents abundant dynamics characteristics, such as global attractivity, complex oscillations and even chaotic behaviour [2,9,10,14,21,23,25,29,31,36,39,42].
In biological system, the stability and instability of population-dynamics process are essentially influenced by the interaction of trophodynamic interactions among individuals [26]. Considering that the lethal fighting or cannibalism is usually inevitable, it is more reasonable to introduce a nonlinear (density-dependent) mortality term to the population model [28]. In addition, some cases of patchiness caused by diffusion instability occur in natural populations [27]. Naturally, by introducing nonlinear density-dependent mortality term and patch structure, the following revised Nicholson's blowflies systems with the Rickers type birth function and the harvesting strategy Type II (cyrtoid) were proposed in the pioneering works [3,19,34], where i ∈ Q = {1, 2, . . . , n}, in ith patch, is the birth function involving maturation delays τ ij (t) and gets the maximum reproduce value with x i (t − τ ij (t)) = (1/γ ij (t)); for i, j ∈ Q and j = i, (a ij (t)x j (t)/(b ij (t) + x j (t))) denotes cooperative connection weight of the populations ith patch and jth patch. Please refer to [5,17,35]. The periodic phenomenon in population dynamics has become a hot topic in recent years, yet there is almost no phenomenon that is purely periodic, and the almost periodic phenomenon is obviously more common [4,18,24]. Consequently, the almost periodic problems for delay Nicholson's blowflies equation and its variants have been intensively investigated in [12,33,37,40]. In particular, if there exists a positive constant M > κ such that the following conditions hold: γ ij (t)M ≤ κ, for all t ∈ R, i ∈ Q, j ∈ I = {1, 2, . . . , m} , inf the authors in [20] built the existence and global stability of almost periodic solutions for system (2). Unfortunately, conditions (4)-(6) have considerable limitations and are not consistent with the actual biological significance. Just as shown in [33,40], for a better biological interpretation, it may be a good choice to replace (4)-(6) with the following relaxed conditions: sup inf It is a great pity that appears to be a contradiction, and more details can be found in page 497 of [20]. Furthermore, since l i > 0 has not been proved in [20], the above contradiction is not clear. Similarly, l i > 0 has not been proved successfully in page 189 of [40] where the author used lim t→+∞ N(π(t)) = 0 and π(t) > t 0 , j ∈ I, Obviously, the above certification process requires the following statement: Sparked by the above reasons and discussions, we try to search a novel proof to investigate the existence and global attractivity of the positive asymptotically almost periodic solutions for system (2) under weaker conditions (8)- (11). In particular, we will correct the abovementioned mistakes. The rest of the proposed work is furnished as follows: In Section 2, some necessary definitions are listed. Further, some basic assumptions and three beneficial lemmas needed in this paper are given. The main results with the existence and global convergence of asymptotically almost periodic solutions are established in Section 3. A numerical example and its computer simulation are provided to illustrate the effectiveness of the acquired results in Section 4. At last, conclusions are drawn in Section 5.

Preliminary results
Throughout this paper, it will be assumed that there exists t 0 > t 0 such that, for i ∈ Q, j ∈ I, which is a weaker condition than inf t∈R γ ij (t) ≥ 1 that adopted in [20,37]. As usual, we also define |x| = (|x 1 |, . . . , |x n |) and ||x|| = max i∈Q |x i | for and the collection of all bounded and continuous functions from J 1 to J 2 is denoted by BC(J 1 , J 2 ).

Definition 2.1 (see [6,43]):
If there exists a number l > 0 such that [t, t + l] P = ∅ (t ∈ R), then we say that the subset P of R is relatively dense in R. If for any > 0, the set T(u, ) = {δ : |u(t + δ) − u(t)| < , ∀t ∈ R} is relatively dense, then u ∈ BC(R, J) is said to be almost periodic on R.

Definition 2.2 (see [6,43]):
If there exist an almost periodic function h and a continuous function g ∈ W 0 (R + , J) such that u = h + g, then we say that u ∈ C(R + , J) is asymptotically almost periodic. For J ⊆ R, we use AP(R, J) to present the set of the almost periodic functions from R to J. We label AAP(R, J) as the set of all asymptotically almost periodic functions. What's more, according to [6,43], AP(R, J) should be a proper subspace of AAP(R, J). Hereafter, we assume that a ii , b ii , γ ij ∈ AAP(R, (0, +∞)), a ij (i = j), b ij (i = j), β ij , τ ij ∈ AAP(R, R + ) and To proceed further, we need to introduce a nonlinear almost periodic differential system: It will be considered the following admissible initial value conditions (IVC): Lemma 2.1: Denote x(t; t 0 , ϕ) as a solution of (14) with respect to the IVC (15). Suppose that there is a positive constant M > κ such that (8), (10) and the following inequality hold. Then x(t) = x(t; t 0 , ϕ) exists on [t 0 , +∞), and there is t ϕ ∈ [t 0 , +∞) such that Proof: First, we claim that where [t 0 , η(ϕ)) is the maximal right existence interval of x(t). Otherwise, one can choose i 0 ∈ Q andt i 0 ∈ (t 0 , η(ϕ)) to satisfy that For t ∈ [t 0 ,t i 0 ), from the fact that which is a contradiction and results the above statement. Now, we demonstrate that Suppose that x(t) is unbounded on [t 0 , η(ϕ)). Then, we can choose i * ∈ Q and a strictly monotone increasing sequence {ζ n } +∞ n=1 such that and then It follows that there exists n * > 0 satisfying According to sup u≥0 ue −u = 1/e, it follows from (14) and (19) that, for all n > n * , which is absurd and suggests that x(t) is bounded on [t 0 , η(ϕ)). By Theorem 2.3.1 in [8], one can easily show that η(ϕ) = +∞. Hereafter, we validate that (17) is true.
By the fluctuation lemma (please see [30], Lemma A.1.), one can find a sequence t * k +∞ k=1 such that From the almost periodicity of (14), we can select a subsequence of {k} k≥1 , still denoted by {k} k≥1 , such that for all j ∈ Q, q ∈ I, the limits lim k→+∞ a h i L j (t * k ), ) exist. Furthermore, by taking limits, we have from (16) and (20) that which entails that L < M, and there exists t * 0 ≥ t 0 such that Next, we show that l > 0. By way of contradiction, we assume that Let (22), one can choose i * * ∈ Q and a strictly monotone increasing sequence {ξ n } +∞ n=1 such that and then lim n→+∞ ω i * * (ξ n ) = +∞.
Similar to the proof of Lemma 2.1, we state the following Lemma 2.2 directly.

Lemma 2.3:
Suppose there is a positive constant M > κ such that (8), (10), (11) and (16) hold. In addition, if x(t) = x(t; t 0 , ϕ) is a solution of (14) with respect to the IVC (15), then, for any > 0, one can pick a relatively dense subset P of R with the below property: for each δ ∈ P , there exists T = T(δ) > 0 satisfying Proof: With the help of Lemma 2.1, (11) and the fact that a g ij , b g ij , β g ij ∈ W 0 (R + , R + ), one can pick positive constants T 1 > max 0, t ϕ and ζ such that, for all t ≥ T 1 , i ∈ Q, which implies there are two positive constants η > 0, λ ∈ (0, 1], such that for i ∈ Q, Define and Again from Lemma 2.1, one can see that x(t) is bounded and the right side of (14) is also bounded. It follows from (31) that x(t) is uniformly continuous on R. Therefore, ∀ > 0, we can choose a sufficiently small constant * > 0 such that from Furthermore, for * > 0, from the uniformly almost periodic family theory (please see Corollary 2.3 in page 19 of [6] ), one can choose a relatively dense subset P * of R such that hold for δ ∈ P * , t ∈ R, i ∈ Q, j ∈ I. Denote P = P * , for any δ ∈ P . From (33) and (34), we gain and where i ∈ Q. Let i t be such an index that Then, for all t ≥ 0 , we have From (17), (37) and the inequalities where x, y ∈ [κ, M], i ∈ Q, j ∈ I, j = i, and we obtain Let It is easy to see that e λt u(t) ≤ E(t), and E(t) is non-decreasing. Now, the remaining proof will be divided into two steps.
Step one. If E(t) > e λt u(t) for all t ≥ 0 , we assert that In the contrary case, one can pick 1 which contradicts the fact that E(β * ) > e λβ * u(β * ) and it proves the above assertion. Then, we can select 2 > 0 satisfying Step two. If there exists ς ≥ 0 such that E(ς ) = e λς u(ς ) , just from (41) and the definition of E(t), we obtain which leads to For any t > ς with E(t) = e λt u(t) , by the same method as that in the derivation of (45), we can show e λt u(t) < ε 2 e λt , and u(t) < ε 2 .
Furthermore, if E(t) > e λt u(t) and t > ς, one can pick 3 ∈ [ς , t) such that which, together with (45) and (46), suggests that With a similar reasoning as that in the proof of Step one, one can entail that which, together with (17), follows that Finally, the above discussion infers that there existsˆ > max {ς , 0 , 2 } obeying that which finishes the proof.

Main results
In this section, we will use the method of Lyapunov function and analytical techniques to present our main results on the existence and global attractivity of the positive asymptotically almost periodic solutions.
Proof: Let ϕ be an initial function of (15), and denote the solution of (14) with respect to For all t ∈ R, i ∈ Q, we can define where t q q≥1 ⊆ R is a sequence. Then for all t + t q ≥ t 0 , i ∈ Q. By using a similar proof as in Lemma 2.3, one can pick t q q≥1 such that Employing Arzala-Ascoli Lemma, together with the fact that the function sequence v(t + t q ) q≥1 is uniformly bounded and equiuniformly continuous, one can choose a subsequence t q j j≥1 of t q q≥1 , such that v(t + t q j ) j≥1 (for the sake of convenience we shall still use v(t + t q ) q≥1 ) uniformly converges to x * (t) = (x * 1 (t), x * 2 (t), . . . , x * n (t)) on any compact set of R. Let '⇒' be 'uniformly converge'. Then, from Lemma 2.1, for all t ∈ R, i ∈ Q, we have and , as q → +∞, on any compact set of R. Thus, for i ∈ Q, combing with (49), (50) and (52), on any compact set of R, it is easy to prove that v i (t + t q ) q≥1 uniformly converges to Making full use of the uniform convergence function sequence properties, it is obvious that x * (t) is a solution of (14) and for all t ∈ R, i ∈ Q, According to the conclusion of Lemma 2.3, ∀ > 0, one can select relatively dense subset P with the following properties: ∀δ ∈ P , there is that is to say, x * (t) is a positive almost periodic solution of (14). Now, we reach the point to show that all solutions of (2) converge to x * (t). Let x(t) be any solution of (2) corresponding to the initial function ϕ satisfying (15), Then For any > 0, combing the global existence and uniform continuity of x(t) with the fact that a g ij , b g ij , β g ij , γ g ij , τ g ij ∈ W 0 (R + , R + ), one can select a constant T * * ϕ > max T 1 , t * ϕ such that the following inequality holds: Set and let i t be such an index that e λt |y i t (t)| = ||e λt y(t)||.

Remark 3.1:
It is easy to check that all results corresponding to (14) in [20,37] are special cases of this paper. Specifically, when n = 1, the assumption (10) is weaker than which plays a fundamental role in the recent paper [40]. In a word, our results are an extension and a useful supplement of papers [20,37,40].

An example
In order to verify the advantage of the above theoretical results, an illustrative numerical simulation is performed in this section.
Note κ ≈ 1.342276, κ ≈ 0.7215355. Let M = 1.31, and by some simple calculations, it is easy to verify that all the conditions of Theorem 3.1 are satisfied. Therefore, all solutions of system (58) are asymptotically almost periodic function on R + , and converge to a same almost periodic function as t → +∞. These conclusions are verified by the following numerical simulations in Figure 1.

Remark 4.1:
It is worth noting that system (58) is not almost periodic, and the following inequalities: do not meet the requirements of conditions (1.3) and (1.5) in [20,37].
To the best of our knowledge, few authors have considered the asymptotically almost periodic dynamics of Nicholson's blowflies model with both nonlinear density-dependent mortality term and patch structure. We only find [3,5,12,[17][18][19][20][32][33][34][35]37,38,40] in the literature. However, all results in these papers can not be used to imply that all solutions of (58) converge to the almost periodic solution.

Conclusions
In the present paper, the issue of asymptotic almost periodicity of Nicholson's blowflies systems with nonlinear density-dependent mortality term and patch structure is investigated. The positivity, global existence and boundedness of the initial value problem on the addressed system have been shown, and the existence of the positive asymptotically almost periodic solution and its global attractivity have been established by applying Lyapunov function and analytical techniques. In particular, a numerical example is provided to illustrate these analytical conclusions. It is worth noting that our conditions are very easy to test in practice by a simple algebraic method, and the method used in this paper provides a possible approach for studying the asymptotic almost periodic dynamics of other population systems with asymptotic almost-periodic environments.