Permanence and almost periodic solution of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales

In this paper, we consider the almost periodic dynamics of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales. By establishing some dynamic inequalities on time scales, a permanence result for the model is obtained. Furthermore, by means of the almost periodic functional hull theory on time scales and Lyapunov functional, some criteria are obtained for the existence, uniqueness and global attractivity of almost periodic solutions of the model. Our results complement and extend some scientific work in recent years. Finally, an example is given to illustrate the main results.


Introduction
Recently, there are many scholars concerning with the dynamics of the mutualism model.Topics such as permanence, global attractivity, and periodicity of mutualism systems governed by differential equations were extensively investigated (see [1][2][3][4][5][6][7][8][9][10]).For example, in [10], the author studied the existence of positive periodic solutions of the periodic mutualism model: where r i , K i , α i ∈ C(R, R + ), α i > K i , i = 1, 2, τ i , σ i ∈ C(R, R + ), i = 1, 2, r i , K i , α i , τ i , σ i (i = 1, 2) are functions of period ω > 0. However, in applications, if the various constituent components of the temporally nonuniform environment is with incommensurable periods, then one has to consider the environment to be almost periodic since there is no a priori reason to expect the existence of periodic solutions.Hence, if we consider the effects of the environmental factors, almost periodicity is sometimes more realistic and more general than periodicity.In recent years, the almost periodic solution of the models in biological populations has been studied extensively (see [11][12][13][14][15][16][17][18] and the references cited therein).In addition, some recent attention was on the permanence and global stability of discrete mutualism system, and many excellent results have been derived (see [19][20][21][22][23][24]). In [24], the authors considered the following discrete multispecies Lotka-Volterra mutualism system: x j (k) where x i (k) stand for the densities of species x i at the kth generation, a i (k) represent the natural growth rates of species x i at the kth generation, b i (k) are the intraspecific effects of the kth generation of species x i on own population, c ij (k) measure the interspecific mutualism effects of the kth generation of species x j on species x i (i, j = 1, 2, . . ., n, i = j), and d ij (≥ 1) are positive control constants.By means of the theory of difference inequality and Lyapunov function, sufficient conditions are established for the existence and uniformly asymptotic stability of unique positive almost periodic solution to system (1.2).Furthermore, so many processes, both natural and manmade, in biology, medicine, chemistry, physics, engineering, economics, etc. involve time delays.Time delays occur so often so if we ignore them, we ignore reality.Generally, the meaning of time delay is that some time elapses between causes and their effects (for instance, in population dynamics, individuals always need some time to mature, or in medicine, infectious diseases have incubation periods).Specially, in the real world, the delays in differential equations of biological phenomena are usually time-varying.Thus, it is worthwhile continuing to study the existence and stability of a unique almost periodic solution of the multispecies Lotka-Volterra mutualism system with time varying delays.
Since permanence is one of the most important topics on the study of population dynamics, one of the most interesting questions in mathematical biology concerns the survival of species in ecological models.Biologically, when a system of interacting species is persistent in a suitable sense, it means that all the species survive in the long term.It is reasonable to ask for conditions under which the system is permanent.
Also, as we known, the study of dynamical systems on time scales is now an active area of research.The theory of times scales has received a lot of attention which was introduced by Stefan Hilger in his Ph.D. thesis in 1988, providing a rich theory that unifies and extends continuous and discrete analysis [25].In fact, both continuous and discrete systems are very important in implementation and applications.But it is troublesome to study the dynamics for continuous and discrete systems respectively.Therefore, it is significate to study that on time scales which can unify the continuous and discrete situations.Motivated by the above reasons, in this paper, we are concerned with the following multispecies Lotka-Volterra mutualism system with time varying delays on time scales: where T is an almost periodic time scale.
3) is reduced to the following system: ) which is a generalization of (1.1).If T = Z, then system (1.3) is reduced to the following system: let τ i (k) = 0, δ j (k) = 0, then system (1.3) is reduced to system (1.2).
By the biological meaning, we will focus our discussion on the positive solutions of system (1.3).So, it is assumed that the initial condition of system (1.3) is the form where For convenience, we denote Throughout this paper, we assume that To the best of our knowledge, there is no paper published on the permanence, the existence and uniqueness of globally attractive almost periodic solutions to systems (1.4) and (1.5).The main purpose of this paper is by establishing some dynamic inequalities on time scales to discuss the permanence of system (1.3) and by using the almost periodic functional hull theory on time scales to establish criteria for the existence and uniqueness of globally attractive almost periodic solutions of system (1.3).
The paper is organized as follows.In Section 2, we introduce some basic definitions, necessary lemmas and establishing some dynamic inequalities on time scales which will be used in later sections.In Section 3, we discuss the permanence of system (1.3).In Section 4, we consider the global attractivity of almost periodic solutions of system (1.3) by means of Lyapunov functional.In Section 5, some sufficient conditions are obtained for the existence of positive almost periodic solutions of system (1.3) by use of the almost periodic functional hull theory on time scales.The main result in Sections 4 and 5 are illustrated by giving an example in Section 6.

Preliminaries
In this section, we shall recall some basic definitions, lemmas which are used in what follows.
A time scale T is an arbitrary nonempty closed subset of the real numbers, the forward and backward jump operators σ, ρ : T → T and the forward graininess µ : T → R + are defined, respectively, by σ(t) := inf{s ∈ T : s > t}, ρ(t) := sup{s ∈ T : s < t} and µ(t) = σ(t) − t.
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 f be right-dense continuous, if F ∆ (t) = f (t), then we define the delta integral by s r f (t)∆t = F (s) − F (r), r, s ∈ T.
Lemma 2.1.[25] Assume f, g : T −→ R are delta differentiable at t ∈ T, then The set of all regressive and rd-continuous functions p : T → R will be denoted by R = R(T) = R(T, R).We define the set If r ∈ R, then the generalized exponential function e r is defined by for all s, t ∈ T, with the cylinder transformation Let p, q : T → R be two regressive functions, we define p ⊕ q = p + q + µpq, ⊖p = − p 1 + µp , p ⊖ q = p ⊕ (⊖q) = p − q 1 + µq .
Then the generalized exponential function has the following properties.
Lemma 2.2.[25] Assume that p, q : T → R are two regressive functions, then (i) e 0 (t, s) ≡ 1 and e p (t, t) ≡ 1; (ii) e p (σ(t), s) = (1 + µ(t)p(t))e p (t, s); (iii) e p (t, s) = 1/e p (s, t) = e ⊖p (s, t); (iv) e p (t, s)e p (s, r) = e p (t, r); (v) e p (t, s)e q (t, s) = e p⊕q (t, s); (vi) e p (t, s)/e q (t, s) = e p⊖q (t, s); (vi) e σ p (t,s) .Lemma 2.3.[26] Let f : T → R be a continuously increasing function and f (t) > 0 for t ∈ T, then is a relatively dense set in T for all ǫ > 0 and for each compact subset S of D; that is, for any given ǫ > 0 and for each compact subset S of D, there exists a constant l(ǫ, S) > 0 such that each interval of length l(ǫ, S) contains a τ (ǫ, S) ∈ E{ǫ, f, S} such that τ is called the ǫ-translation number of f and l(ǫ, S) is called the inclusion length of E{ǫ, f, S}.
We will introduce the translation operator T , T α f (t, x) = g(t, x), which means that g(t, x) = lim n→+∞ f (t + α n , x) and is written only when the limit exists.The mode of convergence, for example, pointwise, uniform, and so forth, will be specified at each use of the symbol.
[27] A function f (t) is almost periodic if and only if for any sequence {α ′ n } ⊂ Π there exists a subsequence {α n } ⊂ {α ′ n } such that f (t + α n ) converges uniformly on t ∈ T as n → ∞.Furthermore, the limit function is also almost periodic.
Consider the following equation and the corresponding hull equation where f : Similar to the proof of Theorem 3.2 in [28], one can easily get the following.
2) has a unique solution, then these solutions are almost periodic.
Definition 2.4.Suppose that ϕ(t) is any solution of (2.1) on T. ϕ(t) is said to be a strictly positive solution on T if for t ∈ T, Lemma 2.9.If each of the hull equations of system (2.1) has a unique strictly positive solution, then system (2.1) has a unique strictly positive almost periodic solution.
Proof.Suppose that ϕ(t) is a strictly positive solution of system (2.2).Since f is almost periodic in t uniformly for x ∈ S, by Lemma 2.6, for any sequences α ′ , β ′ ⊂ Π, there exist common subsequences which is the common hull equation of system (2.1), with respect to α and β, respectively.Therefore, we have T α+β ϕ(t) = T α T β ϕ(t), then by Lemma 2.6, ϕ(t) is an almost periodic solution of (2.1).Since α ⊂ α ′ ⊂ Π and lim x) exists uniformly in t ∈ T for x ∈ S. For the sequence α ⊂ α ′ , we conclude that T α ϕ(t) = ψ(t) exists uniformly in t ∈ T. According to the uniqueness of the solution and T α ψ(t) = ψ(t), one obtains that ϕ(t) = ψ(t).The proof is completed.Lemma 2.10.[25] Assume that a ∈ R and Proof.The proof of (i).It is obviously that there exists a unique positive root of the equation x(ax − b) − d = 0. Suppose that lim sup t→+∞ x(t) = +∞.Then there exists a subsequence Thus, we have Considering the following inequality Integrating inequality (2.4) from t * 0 to t, we have In view of (2.3) and (2.5), we obtain For every θ ∈ T, if µ(θ) = 0, then Hence, for every θ ∈ T, we have It follows from (2.6) and (2.7) that Especially, if d = 0, then x = b a , we can easily know that Hence lim sup k→+∞ x(t k ) < +∞.This contradicts the assumption.
We claim Otherwise, there exists ε such that x(t) > M + ε for any t ∈ T. So we can choose {t k } ∞ k=1 ⊂ T such that By a similar process as above, we can derive that which is a contradiction.Hence, our claim holds.The proof of (ii).Suppose that lim inf t→+∞ x(t) = 0. Then there exists a subsequence For any positive constant ε small enough, it follows from lim sup t→+∞ x(t) ≤ N that there exists large enough T 1 such that Considering the following inequality For t > t * 0 ≥ t 0 , we have From (2.8) and (2.9), we obtain (2.10) Hence, for every θ ∈ T, we have The proof of Lemma 2.11 is completed.
Similar to the proof of Lemma 2.11, we can easily obtain the following results: Lemma 2.12.Assume that x(t) > 0 on T, b ≥ 0, a, d > 0, t − τ (t) ∈ T, where τ (t) : T → R + is a rd-continuous function and τ = sup Proof.The proof of (i).Suppose that lim sup t→+∞ x(t) = +∞.Then there exists a subsequence Thus, we have Considering the following inequality Integrating inequality (2.13) from t * 0 to t, we have In view of (2.12) and (2.14), we obtain For every θ ∈ T, if µ(θ) = 0, then Hence, for every θ ∈ T, we have The proof of (ii).Suppose that lim inf t→+∞ x(t) = 0. Then there exists a subsequence For any positive constant ε small enough, it follows from lim sup t→+∞ x(t) ≤ Ñ that there exists large enough T 2 such that Considering the following inequality For t > t * 0 ≥ t 0 , we have From (2.17) and (2.18), we obtain For every θ ∈ T, if µ(θ) = 0, then Hence, for every θ ∈ T, we have By use of (2. 19) and (2.20), we obtain The proof of Lemma 2.12 is completed.

Permanence
In this section, we will give our main results about the permanence of system (1.3).For convenience, we introduce the following notations: where μ = sup t∈T {µ(t)}.
Lemma 3.1.Assume that (H 1 ) − (H 3 ) hold.Let x(t) = (x 1 (t), x 2 (t)), . . ., x n (t)) be any solution of system (1.3) with initial condition (1.6), then Proof.Let x(t) = (x 1 (t), x 2 (t)), . . ., x n (t)) be any solution of system (1.3) with initial condition (1.6).From (1.3) it follows that Let N i (t) = e x i (t) , obviously N i (t) > 0, the above inequality yields that In view of Lemma 2.4, we have then By applying Lemma 2.14, there exists a constant T 0 such that On the other hand, from (1.3) it follows that t) , obviously N i (t) > 0, then the above inequality yields that ).In view of Lemma 2.3, we have then By applying Lemma 2.12 and a l i exp The proof is complete.

Global attractivity
In this section, we will study the global attractivity of system (1.3).
where x m i , x M i defined in Lemma 3.1 and μ = sup t∈T {µ(t)}.

Consider a Lyapunov function
In view of (4.3) − (4.7), we can obtain From (4.8), we get By use of (4.8) and (4.9), we have . n.This completes the proof.

Almost periodic solutions
In this section, we investigate the existence and uniqueness of almost periodic solutions of system (1.3) by use of the almost periodic functional hull theory on time scales.
Let {s p } ⊂ Π be any sequence such that s p → +∞ as p → +∞.According to Lemma 2.9, taking a subsequence if necessary, we have for t ∈ T, i, j = 1, 2, . . ., n, i = j.Then, we get the hull equations of system (1.3) as follows: (5.1)By use of the almost periodic theory on time scales and Lemma 2.7, it is easy to obtain the following lemma.Proof.By Lemma 2.9, in order to prove the existence of a unique strictly positive almost periodic solution of system (1.3), we only need to prove that each hull equations of system (1.3) has a unique strictly positive solution.
We can easily see that y(t) = (y 1 (t), y 2 (t), . . ., y n (t)) is a solution of system (5.1) and x m i −ǫ ≤ y i (t) ≤ x M i + ǫ for t ∈ T, i = 1, 2, . . ., n.Since ǫ is an arbitrary small positive number, it follows that x m i ≤ y i (t) ≤ x M i for t ∈ T, i = 1, 2, . . ., n, which implies that each of the hull equations (5.1) has at least one strictly positive solution.Now, we prove the uniqueness of the strictly positive solution of each of the hull equations (5.1).Suppose that the hull equations (5.1)Hence, for i, j = 1, 2, . . .n with i = j, one has which imply that So, lim t→−∞ V * (t) = 0.
Note that V * (t) is a nonincreasing nonnegative function on T, and that V * (t) = 0.That is x * i (t) = y * i (t), t ∈ T, i = 1, 2, . . ., n.Therefore, each of the hull equations (5.1) has a unique strictly positive solution.In view of the previous discussion, any of the hull equations (5.1) has a unique strictly positive solution.By Lemma 2.9, system (1.3) has a unique strictly positive almost periodic solution.The proof is completed.

An example
Consider the following multispecies Lotka-Volterra mutualism system with time delays on almost periodic time scale T: x ∆ i (t) = a i (t) − b i (t)e x i (t−τ i (t)) + 2 j=1,j =i c ij (t) e x j (t−δ j (t)) d ij + e x j (t−δ j (t)) , i = 1, 2, t ∈ T. (6.1)

Definition 2 . 1 .
[27] A time scale T is called an almost periodic time scale ifΠ = τ ∈ R : t ± τ ∈ T, ∀t ∈ T = {0}.Throughout this paper, E n denotes R n or C n , D denotes an open set in E n or D = E n , and S denotes an arbitrary compact subset of D. Definition 2.2.[27] Let T be an almost periodic time scale.A function