Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations

Abstract Overf the last few years, by utilizing Mawhin’s continuation theorem of coincidence degree theory and Lyapunov functional, many scholars have been concerned with the global asymptotical stability of positive periodic solutions for the non-linear ecosystems. In the real world, almost periodicity is usually more realistic and more general than periodicity, but there are scarcely any papers concerning the issue of the global asymptotical stability of positive almost periodic solutions of non-linear ecosystems. In this paper we consider a kind of delayed two-species competitive model with stage structure. By means of Mawhin’s continuation theorem of coincidence degree theory, some sufficient conditions are obtained for the existence of at least one positive almost periodic solutions for the above model with nonnegative coefficients. Furthermore, the global asymptotical stability of positive almost periodic solution of the model is also studied. The work of this paper extends and improves some results in recent years. An example and simulations are employed to illustrate the main results of this paper.


Introduction
In [1], Zeng et al. proposed the following two-species competitive model with stage structure: (1.1) where x and x are immature and mature population densities of one species, respectively, and x represents the population density of another species. The competition is between x and x . By means of the xed point theory and Lyapunv functional, Zeng et al. studied the existence and uniqueness of globally attractive positive T-periodic solution of system (1.1).
In real world, almost periodicity is more realistic and more general than periodicity. Therefore, more and more scholars have focused on the study of almost periodic dynamics of non-linear ecosystem [2][3][4][5][6]. Moreover, population models with delays have attracted much attention in recent years. Time delays represent an additional level of complexity that can be incorporated in a more detailed analysis of a particular system. So, this article is to consider the following delayed two-species almost periodic competitive model with stage structure: where a i , b i , β j , c, τ, δ and σ are all nonnegative almost periodic functions, i = , , , j = , .
Let R, Z and N + denote the sets of real numbers, integers and positive integers, respectively, C(X, Y) and C (X, Y) be the space of continuous functions and continuously di erential functions which map X into Y, respectively. Especially, C(X) := C(X, X), C (X) := C (X, X). In relation to a continuous bounded function f , we use the following notations: Throughout this paper, we always make the following assumption for system (1.2): (H )ā > ,ā >b ,ā > ,b > andb > . The main purpose of this article is to study the existence and global asymptotic stability of positive almost periodic solution of system (1.2) by using the coincidence degree theory and Lyapunov functional. Finally, a example and some simulations are also given to illustrate the main results.

Preliminaries
De nition 2.1. ([ , ]) x ∈ C(R, R n ) is called almost periodic, if for any ϵ > , it is possible to nd a real number l = l(ϵ) > , for any interval with length l(ϵ), there exists a number τ = τ(ϵ) in this interval such that where · is arbitrary norm of R n . τ is called to the ϵ-almost period of x, T(x, ϵ) denotes the set of ϵ-almost periods for x and l(ϵ) is called to the length of the inclusion interval for T(x, ϵ). The collection of those functions is denoted by AP(R, R n ). Let AP(R) := AP(R, R).

Lemma 2.1. ([ ])
If x ∈ AP(R) is di erentiable, then for ∀ϵ > , it follows: where a is an arbitrary real constant and T > is a constant independent of a. The method to be used in this paper involves the applications of the continuation theorem of coincidence degree.
Let X and Y be real Banach spaces, L : DomL ⊆ X → Y be a linear mapping and N : X → Y be a continuous mapping. The mapping L is called a Fredholm mapping of index zero if ImL is closed in Y and dimKerL = codimImL < +∞. If L is a Fredholm mapping of index zero and there exist continuous projectors P : X → X and Q : Y → Y such that ImP = KerL, KerQ = ImL = Im(I − Q). It follows that L P = L| DomL∩KerP : (I − P)X → ImL is invertible and its inverse is denoted by K P . If Ω is an open bounded subset of X, the mapping N will be called L-compact onΩ if QN(Ω) is bounded and K P (I − Q)N :Ω → X is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ → KerL.

Almost periodic solution
Let where ω is de ned as that in ( . ). Proof. Under the invariant transformation (x , x , x ) T = (e y , e y , e y ) T , system (1.2) reduces to It is easy to see that if system ( . ) has one almost periodic solution (y , y , y ) T , then (x , x , x ) T = (e y , e y , e y ) T is a positive almost periodic solution of system (1.2). Therefore, to complete the proof it su ces to show that system (3.0) has one almost periodic solution.
Take where DomL = {z = (y , y , y ) T ∈ X : y , y , y ∈ C (R)} and With these notations, system (3.0) can be written in the form It is not di cult to verify that KerL = V , ImL = V is closed in Y and dim KerL = = codim ImL. Therefore, L is a Fredholm mapping of index zero (see Lemma 2.12 in [6]). Now de ne two projectors P : Then P and Q are continuous projectors such that ImP = KerL and ImL = KerQ = Im(I − Q). Furthermore, through an easy computation we nd that the inverse K P : ImL → KerP ∩ DomL of L P has the form Then QN : X → Y and K P (I − Q)N : X → X read Lemma 2.13 in [6]).
In order to apply Lemma 2.6, we need to search for an appropriate open-bounded subset Ω.
Finally, we will show that condition (c) of Lemma 2.6 is satis ed. Let us consider the homotopy From the above discussion it is easy to verify that H(ι, z) ≠ on ∂Ω ∩ KerL, ∀ι ∈ [ , ]. Further, Φz = has a solution: (y + , y + , where J is the identity mapping since ImQ = KerL. Obviously, all the conditions of Lemma 2.6 are satis ed. Therefore, system (3.0) has at least one almost periodic solution, that is, system (1.2) has at least one positive almost periodic solution. This completes the proof. Obviously, (H ) in Theorem 3.1 is weaker than (F ). Further, by Theorem 3.1, it is easy to obtain the existence of positive almost periodic of system (1.1) without (F ). So the work of this paper extends and improves the result in [2].
which yields that Letting ϵ → in the above inequality, we get So we can obtain from the rst equation of system (3.31) that Letting ϵ → in the above inequality, we get Similar to the arguments as that in (3.32)-(3.33), it follows from the third equation of system (3.31) that By the similar discussions as that in (3.17), (3.21) and (3.26), we could nd g − i such that Let Ω = {z ∈ X : z X < C }. From the proof in Theorem 3.1, it is easy to verify that Ω satis es conditions (a)-(c) of Lemma 2.6. Obviously, all the conditions of Lemma 2.6 are satis ed. Therefore, system (3.0) has one almost periodic solution, that is, system (1.2) has at least one positive almost periodic solution. This completes the proof.
Then system ( . ) has a unique positive almost periodic solution, which is globally asymptotically stable.
Proof. By Theorem 3.1, we know that system (1.2) has at least one positive almost periodic solution (x * , x * , x * ) T . Suppose that (x , x , x ) T is another positive solution of system (1.2). From (H ), there must exist < θ < a − such that De ne where By calculating the upper right derivative of V i (i = , , , , , ) along the positive solution of system (1.2), it follows that and Together with (4.1)-(4.6), it follows that Therefore, V is non-increasing. Integrating the last inequality from to t leads to Thus, the almost periodic solution of system (1.2) is globally exponentially stable. Next, we show that there is only one positive almost periodic solution of system (1.2). For any two positive almost periodic solutions (x , x , x ) T and (x ,x ,x ) T of system (1.2), we claim that x i (t) ≡x i (t), ∀t ∈ R, i = , , . If not, without loss of generality, there must be at least one t ∈ R such that x (t ) ≠x (t ), i.e., |x (t ) −x (t )| := l > . The global asymptotical stability implies that there exists t > t such that By the almost periodicity of x andx , there must exist l > and τ ∈ [t − t , t − t + l ] such that So we can easily know from (4.7)-(4.8) that which is a contradiction. Then x (t) ≡x (t), ∀t ∈ R. Similarly, we can prove x i (t) ≡x i (t), ∀t ∈ R, i = , . Therefore, the almost periodic solution of system (1.2) is unique. This completes the proof.
Together with Theorem 3.2, we can easily show that

Example and simulations
Example 5.1. Considering the following delayed two-species competitive model with stage structure and di erent periods: Then system ( . ) has a unique positive almost periodic solution, which is globally asymptotically stable.

Conclusions
The stage-structured models have been studied extensively, and many important phenomena have been observed in recent years. In this paper we study an almost periodic nonautonomous delayed two-species competitive model with stage structure, and this motivation comes from a nonautonomous delayed two- species competitive model. We obtain easily veri able su cient criteria for the existence and globally asymptotic stability of positive almost periodic solutions of the above model. In order to obtain a more accurate description of the ecological system perturbed by human exploitation activities such as planting and harvesting and so on, we need to consider the impulsive di erential equations. In this paper, we only studied system (1.2) without impulses. Whether system (1.2) with impulses can be discussed in the same methods or not is still an open problem. We will continue to study this problem in the future.