Existence and exponential stability of positive almost periodic solution for Nicholson’s blowflies models on time scales

In this paper, we first give a new definition of almost periodic time scales, two new definitions of almost periodic functions on time scales and investigate some basic properties of them. Then, as an application, by using a fixed point theorem in Banach space and the time scale calculus theory, we obtain some sufficient conditions for the existence and exponential stability of positive almost periodic solutions for a class of Nicholson’s blowflies models on time scales. Finally, we present an illustrative example to show the effectiveness of obtained results. Our results show that under a simple condition the continuous-time Nicholson’s blowflies model and its discrete-time analogue have the same dynamical behaviors.


Introduction
To describe the population of the Australian sheep-blowfly and to agree with the experimental data obtained in [1], Gurney et al. [2] proposed the following delay differential equation model: x ′ (t) = −δx(t) + px(t − τ )e −ax(t−τ ) , where p is the maximum per capita daily egg production rate, 1/a is the size at which the blowfly population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ is the generation time.Since equation (1.1) explains Nicholson's data of blowfly more accurately, the model and its modifications have been now refereed to as Nicholson's Blowflies model.The theory of the Nicholsons blowflies equation has made a remarkable progress in the past forty years with main results scattered in numerous research papers.Many important results on the qualitative properties of the model such as existence of positive solutions, positive periodic or positive almost periodic solutions, persistence, permanence, oscillation and stability for the classical Nicholsons model and its generalizations have been established in the literature [3,4,5,6,7,8,9,10,11,12].For example, to describe the models of marine protected areas and B-cell chronic lymphocytic leukemia dynamics that are examples of Nicholson-type delay differential systems, Berezansky et al. [13] and Wang et al. [14] studied the following Nicholson-type delay system: , where α i , β i , c ij , γ ij , τ ij ∈ C(R, (0, +∞)), i = 1, 2, j = 1, 2, . . ., m; in [15], the authors discussed some aspects of the global dynamics for a Nicholson's blowflies model with patch structure given by In the real world phenomena, since the almost periodic variation of the environment plays a crucial role in many biological and ecological dynamical systems and is more frequent and general than the periodic variation of the environment.Hence, the effects of almost periodic environment on evolutionary theory have been the object of intensive analysis by numerous authors and some of these results for Nicholsons blowflies models can be found in [16,17,18,19,20,21,22,23].
Besides, although most models are described by differential equations, the discrete-time models governed by difference equations are more appropriate than the continuous ones when the size of the population is rarely small, or the population has non-overlapping generations.Hence, it is also important to study the dynamics of discrete-time Nicholson's blowflies models.Recently, authors of [24,25] studied the existence and exponential convergence of almost periodic solutions for discrete Nicholson's blowflies models, respectively.In fact, it is troublesome to study the dynamics for discrete and continuous systems respectively, therefore, it is significant to study that on time scales, which was initiated by Stefan Hilger (see [26]) in order to unify continuous and discrete cases.However, to the best of our knowledge, very few results are available on the existence and stability of positive almost periodic solutions for Nicholson's blowflies models on time scales except [27].But [27] only considered the asymptotical stability of the model and the exponential stability is stronger than asymptotical stability among different stabilities.
On the other hand, in order to study the almost periodic dynamic equations on time scales, a concept of almost periodic time scales was proposed in [28].Based on this concept, almost periodic functions [28], pseudo almost periodic functions [29], almost automorphic functions [30], weighted pseudo almost automorphic functions [31], weighted piecewise pseudo almost automorphic functions [32] and almost periodic set-valued functions [33] on on time scales were defined successively.Also, some works have been done under the concept of almost periodic time scales (see [34,35,36,37,38,39,40,41]).Although the concept of almost periodic time scales in [28] can unify the continuous and discrete situations effectively, it is very restrictive.This excludes many interesting time scales.Therefore, it is a challenging and important problem in theories and applications to find new concepts of periodic time scales ([42, 43, 44, 45, 46]).
Motivated by the above discussion, our main purpose of this paper is firstly to propose a new definition of almost periodic time scales, two new definitions of almost periodic functions on time scales and study some basic properties of them.Then, as an application, we study the existence and global exponential stability of positive almost periodic solutions for the following Nicholson's blowflies model with patch structure and multiple time-varying delays on time scales: where t ∈ T, T is an almost periodic time scale, x i (t) denotes the density of the species in patch i, b ik (k = i) is the migration coefficient from patch k to patch i and the natural growth in each patch is of Nicholson-type.
For convenience, for a positive almost periodic function f : T → R, we denote f + = sup t∈T f (t), f − = inf t∈T f (t).Due to the biological meaning of (1.2), we just consider the following initial condition: where θ = max This paper is organized as follows: In Section 2, we introduce some notations and definitions which are needed in later sections.In Section 3, we give a new definition of almost periodic time scales and two new definitions of almost periodic functions on time scales, and we state and prove some basic properties of them.In Section 4, we establish some sufficient conditions for the existence and exponential stability of positive almost periodic solutions of (1.2).In Section 5, we give an example to illustrate the feasibility of our results obtained in previous sections.We draw a conclusion in Section 6.

Preliminaries
In this section, we shall first recall some definitions and state some results which are used in what follows.
A point t ∈ T is called left-dense if t > inf T and ρ(t) = t, left-scattered if ρ(t) < t, right-dense if t < sup T and σ(t) = t, and right-scattered if σ(t) > t.If T has a left-scattered maximum m, then T k = T \ {m}; otherwise T k = T.If T has a right-scattered minimum m, then T k = T \ {m}; otherwise T k = T.
A function f : T → R is right-dense continuous provided it is continuous at right-dense point in T and its left-side limits exist at left-dense points in T. If f is continuous at each right-dense point and each left-dense point, then f is said to be continuous function on T.
For y : T → R and t ∈ T k , we define the delta derivative of y(t), y ∆ (t), to be the number (if it exists) with the property that for a given ε > 0, there exists a neighborhood If y is continuous, then y is right-dense continuous, and if y is delta differentiable at t, then y is continuous at t.
Let y be right-dense continuous.If Y ∆ (t) = y(t), then we define the delta integral by The set of all regressive and rd-continuous functions r : T → R will be denoted by R = R(T) = R(T, R).We define the set Lemma 2.1.( [47]) Suppose that p ∈ R + , then (i) e p (t, s) > 0, for all t, s ∈ T; (ii) if p(t) ≤ q(t) for all t ≥ s, t, s ∈ T, then e p (t, s) ≤ e q (t, s) for all t ≥ s.Definition 2.1.[48] A subset S of R is called relatively dense if there exists a positive number L such that [a, a + L] ∩ S = φ for all a ∈ R. The number L is called the inclusion length.Definition 2.2.[28] A time scale T is called an almost periodic time scale if The following definition is a slightly modified version of Definition 3.10 in [28].
is relatively dense for all ε > 0 and for each compact subset S of D; that is, for any given ε > 0 and each compact subset S of D, there exists a constant l(ε, S) > 0 such that each interval of length l(ε, S) contains a τ (ε, S) ∈ E{ε, f, S} such that τ is called the ε-translation number of f .

Almost periodic time scales and almost periodic functions on time scales
In this section, we will give a new definition of almost periodic time scales and two new definitions of almost periodic functions on time scales, and we will investigate some basic properties of them.Our new definition of almost periodic time scales is as follows: Definition 3.1.A time scale T is called an almost periodic time scale if the set Proof.By contradiction, suppose that there exists a t 0 ∈ T such that for every τ ∈ Π \ {0}, On the other hand, since t 0 ∈ T and −τ + τt 0 ∈ Π, t 0 − τ + τt 0 ∈ T. This is a contradiction.
Therefore, for every t ∈ T, there exists a τ ∈ Π \ {0} such that t ± τ ∈ T. Hence, T is an almost periodic time scale under Definition 2.2.The proof is complete.
Throughout this section, E n denotes R n or C n , D denotes an open set in E n or D = E n , S denotes an arbitrary compact subset of D.
From [28], under Definitions 2.2 and 2.3, we know that if we denote by BUC(T × D, R n ) the collection of all bounded uniformly continuous functions from T × D to R n , then where AP (T × D, R n ) are the collection of all almost periodic functions in t ∈ T uniformly for x ∈ D. It is well known that if we let T = R or Z, (3.1) is valid.So, for simplicity, we give the following definition: Let T be an almost periodic time scale under sense of Definition 3.
is relatively dense for all ε > 0 and for each compact subset S of D; that is, for any given ε > 0 and each compact subset S of D, there exists a constant l(ε, S) > 0 such that each interval of length l(ε, S) contains a τ (ε, S) ∈ E{ε, f, S} such that This τ is called the ε-translation number of f .Remark 3.2.If T = R, then T = R, in this case, Definition 3.2 is actually equivalent to the definition of the uniformly almost periodic functions in Ref. [48].If T = Z, then T = Z, in this case, Definition 3.2 is actually equivalent to the definition of the uniformly almost periodic sequences in Refs.[49,50].
For convenience, we denote by AP (T × D, E n ) the set of all functions that are almost periodic in t uniformly for x ∈ D and denote by AP (T) the set of all functions that are almost periodic in t ∈ T, and introduce some notations: Let α = {α n } and β = {β n } be two sequences.Then β ⊂ α means that β is a subsequence of α; α and α and β are common subsequences of α ′ and β ′ , respectively, means that and for some given function n(k).We introduce the translation operator T , ) and is written only when the limit exists.The mode of convergence, e.g.pointwise, uniform, etc., will be specified at each use of the symbol.
Similar to the proofs of Theorem 3.14, Theorem 3.21 and Theorem 3.22 in [28], respectively, one can prove the following three theorems.
then for any ε > 0, there exists a positive constant L = L(ε, S), for any a ∈ R, there exist a constant η > 0 and α According to Definition 3.2, one can easily prove Similar to the proofs of Theorem 3.24, Theorem 3.27, Theorem 3.28 and Theorem 3.29 in [28], respectively, one can prove the following four theorems.
By Definition 3.2, one can easily prove Theorem 3.9.Let f : R → R satisfies Lipschitz condition and ϕ(t) ∈ AP (T), then f (ϕ(t)) ∈ AP (T).Definition 3.3.[42] Let A(t) be an n × n rd-continuous matrix on T, the linear system is said to admit an exponential dichotomy on T if there exist positive constant k, α, projection P , and the fundamental solution matrix X(t) of (3.2), satisfying where Similar to the proof of Lemma 2.15 in [42], one can easily show that Lemma 3.2.Let a ii (t) be an uniformly bounded rd-continuous function on T, where a ii (t) > 0, −a ii (t) ∈ R + for every t ∈ T and min 1≤i≤n {inf t∈T a ii (t)} > 0, then the linear system admits an exponential dichotomy on T.
According to Lemma 3.1, T is an almost periodic time scales under Definition 2.2, we denote the forward and the backward jump operators of T by σ and ρ, respectively.Lemma 3.3.If t is a right-dense point on T, then t is also a right-dense point on T.
Proof.Let t be a right-dense point on T, then Since σ(t) ≥ t, t = σ(t).The proof is complete.
Similar to the proof of Lemma 3.3, one can prove the following lemma.Lemma 3.4.If t is a left-dense point T, then t is also a left-dense point on T.
For each f ∈ C(T, R), we define f : T → R by f (t) = f (t) for t ∈ T. From Lemmas 3.3 and 3.4, we can get that f ∈ C( T, R).Therefore, F defined by is an antiderivative of f on T, where ∆ denotes the ∆-derivative on T.
Set Π = {τ ∈ Π : t ± τ ∈ T}.We give our second definition of almost periodic functions on time scales as follows.
is relatively dense for all ε > 0 and for each compact subset S of D; that is, for any given ε > 0 and each compact subset S of D, there exists a constant l(ε, S) > 0 such that each interval of length l(ε, S) contains a τ (ε, S) ∈ E{ε, f, S} such that This τ is called the ε-translation number of f .Remark 3.4.Since T is an almost periodic time scales under Definition 2.2, under Definition 3.2, all the results obtained in [28] remain valid when we restrict our discussion to T.
In the following, we restrict our discuss under Definition 3.4.Consider the following almost periodic system: where A(t) is a n × n almost periodic matrix function, f (t) is a n-dimensional almost periodic vector function.Similar to Lemma 2.13 in [42], one can easily get Lemma 3.5.If linear system (3.2) admits an exponential dichotomy, then system (3.3) has a bounded solution x(t) as follows: where X(t) is the fundamental solution matrix of (3.2).
By Theorem 4.19 in [28], we have Lemma 3.6.Let A(t) be an almost periodic matrix function and f (t) be an almost periodic vector function.If (3.2) admits an exponential dichotomy, then (3.3) has a unique almost periodic solution: where X(t) is the restriction of the fundamental solution matrix of (3.2) on T.
From Definition 3.2 and Lemmas 3.5 and 3.6, one can easily get the following lemma.
Lemma 3.7.If linear system (3.2) admits an exponential dichotomy, then system (3.3) has an almost periodic solution x(t) can be expressed as: where X(t) is the fundamental solution matrix of (3.2).

Positive almost periodic solutions for Nicholson's blowflies models
In this section, we will state and prove the sufficient conditions for the existence and exponential stability of positive almost periodic solutions of (1.2).Throughout this section, we restrict our discussion under Definition 3.4. Set In the proofs of our results of this section, we need the following facts: There exists a unique ς ∈ (0, 1) such that 1−ς e ς = 1 e 2 (ς ≈ 0.7215354) and sup = 1 e 2 .The function xe −x decreases on [1, +∞).Lemma 4.1.Assume that the following conditions hold. .
Then the solution x(t) = (x 1 (t), x 2 (t), . . ., x n (t)) of (1.2) with the initial value ϕ ∈ C{A 1 , A 2 } satisfies Proof.Let x(t) = x(t; t 0 , ϕ), where ϕ ∈ C{A 1 , A 2 }.At first, we prove that where [t 0 , η(ϕ)) T is the maximal right-interval of existence of x(t; t 0 , ϕ).To prove this claim, we show that for any p > 1, the following inequality holds By way of contradiction, assume that (4.2) does not hold.Then, there exists i 0 ∈ {1, 2, . . ., n} and the first time t 1 ∈ [t 0 , η(ϕ)) T such that Therefore, there must be a positive constant a ≥ 1 such that In view of the fact that sup u≥0 ue −u = 1 e and ap > 1, we can obtain which is a contradiction and hence (4.2) holds.Let p → 1, we have that (4.1) is true.Next, we show that To prove this claim, we show that for any l < 1, the following inequality holds By way of contradiction, assume that (4.4) does not hold.Then, there exists i 1 ∈ {1, 2, . . ., n} and the first time t 2 ∈ [t 0 , η(ϕ)) T such that Therefore, there must be a positive constant c ≤ 1 such that which is a contradiction and hence (4.4) holds.Let l → 1, we have that (4.3) is true.Similar to the proof of Theorem 2.3.1 in [51], we easily obtain η(ϕ) = +∞.This completes the proof.
Then system (1.2) has a positive almost periodic solution in the region Proof.For any given ϕ ∈ B, we consider the following almost periodic dynamic system: Since min 1≤i≤n {c − i } > 0, t ∈ T, it follows from Lemma 3.2 that the linear system x ∆ i (t) = −c i (t)x i (t), i = 1, 2, . . ., n admits an exponential dichotomy on T. Thus, by Lemma 3.7, we obtain that system (4.5) has an almost periodic solution x ϕ = (x ϕ 1 , x ϕ 2 , . . ., x ϕn ), where

Define a mapping T : B
For any ϕ ∈ B * , by use of (H 2 ), we have and we also have Therefore, the mapping T is a self-mapping from B * to B * .Next, we prove that the mapping T is a contraction mapping on B * .Since sup which implies that T is a contraction.By the fixed point theorem in Banach space, T has a unique fixed point ϕ * ∈ B * such that T ϕ * = ϕ * .In view of (4.5), we see that ϕ * is a solution of (1.2).Therefore, (1.2) has a positive almost periodic solution in the region B * .This completes the proof.
If there exist positive constants λ with ⊖λ ∈ R + and M > 1 such that such that for an arbitrary solution Then the solution x * (t) is said to be exponentially stable.Theorem 4.2.Assume that (H 1 ), (H 3 )-(H 5 ) hold.Then the positive almost periodic solution x * (t) in the region B * of (1.2) is unique and exponentially stable.

An example
In this section, we present an example to illustrate the feasibility of our results obtained in previous sections.

Conclusion
In this paper, we proposed a new concept of almost periodic time scales, two new definitions of almost periodic functions on time scales and investigated some basic properties of them, which can unify the continuous and the discrete cases effectively.As an application, we obtain some sufficient conditions for the existence and exponential stability of positive almost periodic solutions for a class of Nicholson's blowflies models on time scales.Our methods and results of this paper may be used to study almost periodicity of general dynamic equations on time scales.Besides, based on our this new concept of almost periodic time scales, one can further study the problems of pseudo almost periodic functions, pseudo almost automorphic functions and pseudo almost periodic set-valued functions on times as well as the problems of pseudo almost periodic, pseudo almost automorphic and pseudo almost periodic set-valued dynamic systems on times and so on.

Remark 3 . 1 . 4 = 2 . 3 . 1 .
Obviously, if T is an almost periodic time scale under Definition 3.1, then inf T = −∞ and sup T = +∞.If T is an almost periodic time scale under Definition 2.2, then T is also an almost periodic time scale under Definition 3.1 and in this case, T = T. Example 3.1.Let T = Z ∪ { 1 4 }.For every τ ∈ Z, we have T τ = Z and T1 {0}.Hence Π = Z and T = τ ∈Π T τ = Z = ∅.So, T is an almost periodic time scale under Definition 3.1 but it is not an almost periodic time scale under Definition 2.Lemma If T is an almost periodic time scales under Definition 3.1, then T is an almost periodic time scale under Definition 2.2.

Definition 3 . 4 .
Let T be an almost periodic time scale under sense of Definition 3.1.A function

Remark 3 . 3 .
It is clear that if a function is an almost periodic function under Definition 3.2, then it is also an almost periodic function under Definition 3.4.

Remark 4 . 5 .
When T = R or T = Z, our results of this section are also new.If we take T = R, A 1 = 1, A 2 = e, then Lemma 4.1, Theorem 4.1 and Theorem 4.1 improve Lemma 2.4, Theorem 3.1 and Theorem 3.2 in [14], respectively.
* (t) implies that the uniqueness of the positive almost periodic solution.The proof is complete.