INTERMITTENT DISPERSAL POPULATION MODEL WITH ALMOST PERIOD PARAMETERS AND DISPERSAL DELAYS

. We establish a class of intermittent bidirectional dispersal population models with almost periodic parameters and dispersal delays between two patches. The form of dispersal discussed in this paper is diﬀerent from both continuous and impulsive dispersals, in which the dispersal behavior occurs either in a sustained manner or instantaneously; instead, it is a synthesis of these types. Dynamical properties such as permanence, existence, uniqueness, and globally asymptotic stability of almost periodic solutions are investigated by using Liapunov-Razumikhin type technique, using the comparison theorem, constructing a suitable Lyapunov functional, using almost periodic functional hull theory and analysis approach, etc. Finally, numerical simulations are pre-sented and discussed to illustrate our analytic results, by which we ﬁnd that intermittent dispersal systems are more complicated than continuous or impulsive dispersal systems.

Takeuchi et al. [40] considered a non-autonomous single-species model with dispersal time delay in patchy environment as follows: where τ is a positive constant, which represents the time for the species to disperse between two patches. The authors established criteria on permanence and extinction for the above system (1), in which bidirectional dispersal behavior of the modelled populations is occurring at every point in time and is occurring simultaneously between any two patches; i.e., it is continuous bidirectional dispersal.
Zhang et al. [51] considered the following autonomous single species model with logistic growth and dissymmetric impulsive bi-directional dispersal: where [τ k , τ k+1 ) denotes the time period during which the population occupies one of the two patches, with diffusion between patches occurring only at every impulsive time τ k+1 (k = 0, 1, 2, ...), and τ k < τ k+1 with lim k→∞ τ k = +∞, ∆x i (t) = x i (t + ) − x i (t − ), x i (τ + k ) = lim t→τ + k x i (t) represents the population density of species x after the k-th impulsive dispersal in the i-th patch, x i (τ − k ) = lim t→τ − k x i (t) = x i (τ k ) (i = 1, 2) the population density of species x before the k-th impulsive dispersal in the i-th patch, b i the immigration rate of species x from j-th patch to i-th patch, a i the emigration rate of species x from i-th patch to j-th patch (i, j = 1, 2, j = i). Criteria on the permanence, extinction, existence, uniqueness and global attractivity of positive periodic solutions for system (2) were obtained.
As a result, it is more realistic, important and general to consider almost periodic parameters than periodic parameters for a population system.
Meng et al. [30] investigated the following general non-autonomous delayed almost periodic single-species dispersal system in multiple patches: Conditions on the permanence, global asymptotic stability, existence and uniqueness of almost periodic solutions were determined.
Here we will restrict our meaning of 'migration' as the movement of individuals of a population back and forth between two spatial units (although this can be extended to multiple spatial units). As we know, the actual movement behavior of a migratory species between patches can have many facets, usually neither purely continuous nor purely impulsive in time. As a result, the whole dynamics of a meta-population can neither be fully studied in any single investigation, nor analyzed with a single type of meta-population model. Some of the environmental conditions in the landscape matrix between habitat patches may permit normal movement patterns between patches to occur only during certain intervals of time within seasons or within life-cycles, instead of allowing movement at all times, or at consistent times of the year. Environmental conditions (e.g., weather, occasional landscape impediments) may not allow animals to complete a migration over a short time period. For example, salmon can make amazing upstream journeys over hundreds of kilometers to reproduce, in which strong currents and rapids are hindrances that can prolong the journey (see Nislow et al. [33], Winemiller and Jepsen [47]). As another example, during their spring migration, because of certain types of food becoming available as snow melts, caribou may stop to take food during migration. In these types of cases, movement back and forth between two patches only occurs during certain time intervals that may vary, without covering whole seasons or life cycles. In fact, migration is a far from uniform process. As noted by Dingle [11]; within a population "Individuals differ not only in whether or not they migrate; when they do migrate, they vary in the distances travelled or the routes taken." This is due to behavioral and physiological variation among individuals within a population, which affects the factors that trigger their choice of whether and when to migrate; such as seasonality, hunger, mating opportunities, etc. In other words, the movement can be intermittent, neither continuous at all times, nor impulsive at fixed time. Therefore, it is not reasonable to characterize the bidirectional movements of migration either with only impulsive diffusion models or only with continuous diffusion models. Instead, some sort of integration of the two types is required.
Motivated by the above considerations, in this paper we propose the following non-autonomous almost periodic single species model with intermittent dispersal This system is composed of two patches. When t ∈ [τ 2k , τ 2k+1 ), the species x inhabits patch i (i = 1, 2) and does not disperse. When t → τ 2k+1 , the intrinsic relationship of species x in each of the two patches will change, as the channels between the two patches are open, which permits species x disperse bi-directionally from one patch to another. The parameter d i represents survival rate during the switching from stage 1 (without dispersal movement) to stage 2 (dispersal movement). Then the dispersal movement between the two patches will continue during the time interval t ∈ [τ 2k+1 , τ 2k+2 ), when t → τ 2k+2 , the gate of dispersing for species x between the two patches will close, and the species x will stop dispersing and occupy patch i, where D i is the survival rate of switching from stage 2 to stage 1, a i (t) presents the intrinsic growth rate of population in the ith patch over the time interval [τ 2k , τ 2k+1 ),ã i (t) is the intrinsic growth rate of population in the ith patch over the time interval [τ 2k+1 , τ 2k+2 ), b i (t) is the density-dependent coefficient of population in the ith patch over the time interval [τ 2k , τ 2k+1 ),b i (t) is the density-dependent coefficient of population in the ith patch over the time interval [τ 2k+1 , τ 2k+2 ), D ij (t) is the dispersal rate from the ith patch to the jth patch during time interval [τ 2k+1 , τ 2k+2 ), τ i are positive constants, represents the time for the population to disperse from patch j to i (i = j, i, j = 1, 2). Our main purpose in this paper is to analyze the dynamical properties of intermittent dispersal, such as permanence, existence, uniqueness and global stability of almost periodic solutions. Furthermore, we try to explore and determine the differences in dynamics among continuous dispersal, impulse dispersal and intermittent dispersal.
This paper is organized as follows: In Section 2, some preliminaries, assumptions and useful lemmas are presented. In Section 3.1, we obtain the sufficient conditions on existence, uniqueness of positive almost periodic solutions and global asymptotic stability of the auxiliary system for system (4). In Section 3.2, criteria on permanence, global asymptotic stability, existence and uniqueness of positive almost periodic solutions of system (4) are established. Finally, a brief discussion and numerical simulations are illustrated in section 4.

2.
Preliminaries. Let R denote the set of real numbers, R 2 the 2-dimensional Euclidean linear space equipped with the norm In this paper, we assume that all solutions of system (4) satisfy the following initial conditions Where φ = (φ 1 , φ 2 ) ∈ BP C + . By the fundamental theory of impulsive functional differential equation ([4, 15, 24]), system (4) has a unique solution x(t, φ) = x 1 (t, φ), x 2 (t, φ) satisfying the initial conditions (5). Obviously, the solution x(t, φ) is positive in its maximal interval of existence.
B} is said to be uniformly almost periodic if for arbitrary ε > 0, there exists a relatively dense set in R of ε− almost periods common for all of the sequences, here is said to be almost periodic if the following conditions hold (a) The set of sequences {τ i k } is uniformly almost periodic, k, i ∈ Z. (b) For any ε > 0 there exists a real number δ > 0 such that if the points t ′ and t ′′ belong to the same interval of continuity of ϕ(t) and satisfying | t ′ − t ′′ |< δ, then | ϕ(t ′ ) − ϕ(t ′′ ) |< ε.
The hull function H(f (t)) of f (t) is the set of real function g(t) such that for any g(t) ∈ H(f (t)), there exists a sequence {t n } satisfying lim n→+∞ f (t + t n ) = g(t) uniformly on R.
x 2 (t)) of system (4) with initial conditions (5) eventually enters and remains in the region Ω.
is any solution of system (4), x(t) is said to be a strictly positive solution if for t ∈ R and i = 1, 2 such that Definition 2.6. ( [23]) System (4) is said to be globally asymptotically stable if for any two positive solutions x(t) and y(t) of system (4) with initial conditions (5) such that Throughout this paper, we shall use the following notations.
Notation. If f (t), t ∈ R, is an almost periodic function, we define We set x(t))}. In this paper, for system (4) we always suppose that the following assumptions hold for each i, j = 1, 2, and i = j. ( (H 5 ) There are constants η i > 0 (i = 1, 2), such that for all t ≥ 0, Now, we introduce several lemmas which will be useful in the proofs of the main results. We consider the following vector impulsive differential equation where , is a non-decreasing function, we have the following comparison theorem for system (6).
The following lemma will be used in the proof of the global asymptotic stability for the almost periodic system (4).
For the convenience to present our results, we rewrite system (4) as follows: Remark 1. The intrinsic rules of species x in system (4) are divided into 2 parts, two regarding the inhabiting process in patch 1 and patch 2, i.e., equations 1 and 2, and two regarding the migration process between patch 1 and patch 2, i.e., equations 5 and 6, which makes it inconvenient to investigate the whole dynamic properties of system (4). Therefore, by rewriting the system (4) into (8), it is more convenient to present system (4) and investigate it.
Lemma 2.10. Assume that system (8) satisfies (H 1 )-(H 6 ), if each of hull equations (7) has a unique strictly positive solution, then system (8) has a unique strictly positive almost periodic solution.
is an almost periodic function with respect to t ∈ R, obviously, {h k } is an almost periodic sequence with respect to k ∈ Z. Let φ(t) be a strictly positive solution of (7) for t on R. There exist sequences of real values α ′ and β ′ , which have common subsequences are solutions of the following common hull equations of system (7) , thus according to Lemma 2, [37], φ(t) is an almost periodic solution of system (8). The proof is complete.
(iii) There exists some constant H > 0 such that for any solution x(t) of (4) (4) is uniformly ultimately bounded.

Results.
3.1. Auxiliary theorems. In this subsection, we first consider the following auxiliary system (9), and establish a series of criteria on the existence, uniqueness and globally asymptotical stability of positive almost periodic solutions for system (9), which will be used in the next subsection.
We consider the following auxiliary system where About system (9), we have the following result.
Proof. For any a ik (t) ∈ H(a ik (t)), b ik (t) ∈ H(b ik (t)) and τ * k ∈ H(τ k ), we have the following hull equations of Eq. (9): By conditions (H 3 ) − (H 5 ) and Lemma 2.9, we obtain that there are positive constants δ and ω such that for all t ∈ R, and i = 1, 2. Hence, there are positive constants, for all t ≥ −m.
If the inequality (14) is not true, then from For t 4 , there are the following two cases.
For case 1, t 3 is an impulsive time. Hence, there is an integer k > 0 such that , we can obtain that Choosing an integer p ≥ 0 such that t 4 ∈ [t 3 + pω, t 3 + (p + 1)ω), then integrating Eq.(10) from t 3 to t 4 , by (11) and (12), we have which is a contradiction. Thus we finally have for all t ≥ −m, and i = 1, 2.
For t = τ * k+1 , calculating the Dini upper right derivative of V * (t), we have From this, for any t ≤ t 0 , where t 0 is any initial time, we further obtain that Since t0 −∞ bξdt = ∞ and V * (t) is a non-increasing and nonnegative bounded function on R, then V * (t 0 ) = 0. That is,x * i (t 0 ) =ȳ * i (t 0 ) for i = 1, 2, by the uniqueness of solutions of the initial value problem of Eq.(10). Hence we havex * i (t) =ȳ * i (t) for all t ∈ R, and i = 1, 2.
Proof. For two arbitrary positive solutions of system (9) x(t) = (x 1 (t), x 2 (t)) and y(t) = (y 1 (t), y 2 (t)), according to Theorem 3.1, we obtain that there are positive constants r and R such that for all t ≥ −m and i = 1, 2. Choose a Liapunov function For t = τ k+1 , calculating the Dini upper right derivative of V (t), we have for all t ≥ −m and i = 1, 2, where constant α ≥ 0 satisfying c i b l ik ≥ α, for all t ≥ −m and i = 1, 2. From this, for any t ≥ −m we further obtain for i = 1, 2. By using (15) and Lemma 2.8, we have that | x i (t) − y i (t) | (i=1,2) are bounded and uniformly continuous, and by using the Barbalat' Lemma (see [18]  This completes the proof of Theorem 3.2.

Main results.
In this subsection, we study the permanence, uniqueness and global asymptotic stability of almost periodic solutions for system (4). We first discuss the permanence of system (4). Proof. We let which on substituting into (4) becomes for k ∈ R + , i, j = 1, 2, and i = j. Suppose x(t) = (x 1 (t), x 2 (t)) is any positive solution for system (16) with initial conditions (5). We first prove that the system (16)  Define a Liapunov function x i (t).
By Lemma 2.11, we let W 1 (s) = s, W 2 (s) = 2s. For t = τ k , calculating the derivative of V (t, x) along solution of (16), we have Let P (s) = M 0 qs, where for s > 0 satisfying q > 1, M 0 > 0. We choose a sufficiently large H, when x(t) ≥ H and P (V (t, x(t))) > V (s, x(s)) for s ∈ [t − τ, t], we have Then we obtain a positive constant µ such that Finally,

LONG ZHANG, GAO XU AND ZHIDONG TENG
Therefore, by Lemma 2.11, we can see that there must exist positive constants T and M such that x i (t) ≤ M (i = 1, 2) for all t ≥ T . Next, we will prove that there is a constant m > 0 such that any positive solution of system (16) satisfies From the almost periodic system (16), and assumption (H 1 ), we have Consider the following auxiliary system According to hypotheses (H 1 ) − (H 6 ), Theorem 3.1, we can see that system (17) has a unique positive almost periodic solution u(t) = (u 1 (t), u 2 (t)) satisfying the initial condition (5), which is globally asymptotically stable. Therefore, there are positive constants m and T 1 > T such that u i (t) ≥ m (i = 1, 2) for all t ≥ T 1 . Using Lemma 2.7, we obtain that for arbitrary positive solution x(t) = (x 1 (t), x 2 (t)) of system (16), This completes the proof of Theorem 3.3.

Remark 2.
Takeuchi et al. [40] showed that a nonautonomous single-species model with dispersal time delays in patchy environment is permanent if each "food-poor" patch is connected to at least one "food-rich" patch; population is permanent in "food-rich" patches in the sense that partial permanence ensures permanence. In this paper, the dispersal (or migration) movement of the population can't happen all the time, i.e., when t ∈ [τ 2k , τ 2k+1 ), d ijk = 0, species x inhabits in two patches respectively and there exists no movement of dispersal between two patches; when t ∈ [τ 2k+1 , τ 2k+2 ), d ijk > 0, dispersal (or migration) between two patches occurs. Therefore, an interesting open question is that whether the results in [40] are still true for system (4). Now, we further discuss the global asymptotic stability of system (4) and introduce the following assumption.

LONG ZHANG, GAO XU AND ZHIDONG TENG
We state In fact, there are two cases as follows: From (18), (19) and (H 7 ), we obtain for all t ≥T ≥ T . Integrating both sides of (20) on interval [T , t], we have for i = 1, 2. By using Lemma 2.8 and (21), we have 2) are bounded and uniformly continuous. By using the Barbalat's Lemma (see [18]  In the following section, we discuss the existence and uniqueness of almost periodic solutions of system (4).
Proof. By Lemma 2.10, we only need to prove that each hull equations of almost periodic system (4) has a unique strictly positive solution. We first prove the existence of strictly positive solution of any hull equations (22). Let {t ′ m } be an arbitrary sequence of real numbers, then there exists a subsequence {t n }, , uniformly for all t ∈ R and the sequences {τ k − t n } (k ∈ Z), are convergent to the sequence {τ * k } uniformly with respect to k ∈ Z as n → ∞. Suppose x(t) = (x 1 (t), x 2 (t)) is any positive solution of (22). By the proof of Theorem 3.3, we have for all t ≥ T 1 and i = 1, 2.
Obviously, (x 1 (t + t n ), x 2 (t + t n )) satisfy the following auxiliary system: From (23), there exists a positive constant K which does not depend on n such that |ẋ i (t + t n ) |≤ K, for all t ≥ −τ − t n , and i = 1, 2. Then, for any positive integer p, the sequence {(x 1 (t + t n ), x 2 (t + t n ) : n ≥ p} is uniformly bounded and equi-continuous on interval [−τ − t n , ∞]. Using the Ascolia-Arzela lemma, we obtain that there exists a subsequence {t k } ⊂ {t n } such that the sequence {x 1 (t + t k ), x 2 (t + t k )} converges uniformly in t on any compact set of R as k → ∞. Let x * i (t) be the limit function of {x i (t + t k )}, obviously, (x * 1 (t), x * 2 (t)) defined on R that satisfies Eq. (22) and m ≤ x * i (t) ≤ M, for all t ≥ T 1 , and i = 1, 2. Hence, (x * 1 (t), x * 2 (t)) is a strictly positive solution of Eq. (22).
Next, we prove the uniqueness of a strictly positive solution of Eq. (22). Suppose (y * 1 (t), y * 2 (t)) is also a strictly positive solution of Eq. (22). Then we construct a Liapunov function V * (t) on R as follows: By the similar discussion in the proof of Theorem 3.4, for t = τ k+1 , we have is continuous for all t ∈ R. The rest of the proof of Theorem 3.5 is exactly the same as that of Theorems 3.1, 3.2, 3.4, so we omit it here.
Therefore, any hull equations of system (4) have a unique strictly positive solution. By Lemma 2.10 and Theorem 3.4, system (4) has a unique strictly positive almost periodic solution which is global asymptotically stable. This completes the proof of Theorem 3.5.
Remark 3. In this paper, we only discussed the case of two patches. However, in the real ecosystem, many migratory populations can move among many habitats during their whole migration cycle to inhabit, restore, mature, breed, etc. Therefore, it is more proper to consider the multi-patches cases for a long-distance migration populations. Thereby, we introduce the following dispersal model with n patches.
For system (24) we assume that the following assumptions hold for each i, j = 1, 2, ..., n, and i = j.
(B 1 ) The bounded almost periodic functions a i (t),ã i (t), b i (t),b i (t), and D ij (t) are continuous for all t ∈ R, and b l i ≥ 0,b l i ≥ 0, 0 < d i , D i ≤ 1. (B 2 ) The set of sequences {τ i k }, k, i ∈ Z, is uniformly almost periodic, and inf k∈Z | τ k+1 − τ k |> 0.
where η i , ω are positive constants.

Corollary 2. System
Corollary 3. Assume that system (24) satisfies (B 1 )-(B 7 ), then it has a unique strictly positive almost periodic solution which is global asymptotically stable.
Remark 4. Based on the work of this paper, it is more realistic and interest to consider an open problem: how to establish the necessary and sufficient conditions on the extinction, permanence, existence, uniqueness and global asymptotical stability of positive almost periodic solutions for system (4). Moreover, we only discuss the finite delayed impulsive almost periodic population dynamical systems. An important and interesting open problem is whether the results obtained in this paper can be similarly extended to the case of infinite delays, we leave this for a future work.
4. Numerical simulations and discussion. In this paper, we have proposed a single-species almost periodic bidirectional dispersal system with impulses and time delays. Sufficient criteria on the permanence, existence, uniqueness and globally asymptotical stability of almost periodic solution are established. In order to validate our theoretical results for system (4), we perform some numerical simulations by using the values of parameters in Table 1. Due to the system (4) being an almost periodic system, we will show the numerical simulation on the following ten intervals in Table 2.    ln D 2 ≈ 6.8915 > 0, all the conditions required in Theorem 3.3 are satisfied. By numerical simulations, we obtain that system (4) is permanent (see Fig.1(a), Fig.1(b)).
Furthermore, we show the effect of time delays on the populations, here we take τ 1 , τ 2 in five different cases and keep other parameters unchanged. The details are given in Table 3.   The populations x 1 and x 2 are both permanent and globally asymptotically stable (see Fig.3(a)- Fig.3(d) and Fig.2(a)). However, comparing Fig.3(a)- Fig.3(d) and Fig.2(a), the values of delays vary within half a ω-period. We can see that the longer duration of the time delays, the more beneficial to the survival of the population x in the second patch. No matter what the values of τ 1 or τ 2 are, they both have great impact on the balance of populations in two habitats. We realize that system (4) with time delays is more complicated than that without delays.
Moreover, we shall consider the following nonautonomous continuous dispersal single-species almost periodic system with time delays and ignore the influence of impulse on the population x of system (4): we use the values of parameters in Table 1 and Table 2. The population x has a uniquely positive almost periodic solution which is globally stable with or without delays (See Fig. 4(a), 4(b)). In order to illustrate the difference of the results between our mathematical models (4) and (25), we use the values of parameters in Table 4.   We change some parameter values i.e.,ã 1 (t),ã 2 (t),b 1 (t),b 2 (t), D 12 (t) and D 21 (t), and keep the other parameters unchanged, we can see some significant phenomena happen, the details are given in Table 5.  (4) and (25). This shows that intermittent bidirectional dispersal between two patches is beneficial to population x in system (4) but harmful to population x in continuous dispersal system (25), although both with the same parameters. Which means intermittent dispersal is more reasonable and a better choice for migratory populations in the real ecosystems than those with continuous or impulse dispersal movements.
In a word, systems with intermittent dispersal, impulse perturbations and time delays are more complicated than continuous dispersal systems, but are very consistent with real ecosystems, which help us better and more deeply read the ecosystems.