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International Journal of Engineering and Advanced Research Technology (IJEART) ISSN: 2454-9290, Volume-1, Issue-6, December 2015

Global attractivity and positive almost periodic solution of a multispecies discrete mutualism system with time delays Hui Zhang 

initial condition (1.2), by utilizing an almost periodic functional hull theory and constructing a suitable Lyapunov functional and applying the analysis technique of papers [3, 12, 13].

The remaining part of this paper is organized as follows: In Section 2, we will introduce some definitions and several useful lemmas. In the next section, we establish the permanence of system (1.1). Sufficient conditions for the global attractivity of system (1.1) are showed in Section 4.

Then, in Section 5, we establish sufficient conditions to ensure the existence of a unique strictly positive almost periodic solution, which is globally attractive. The main result is illustrated by an example with a numerical simulation in the last section.

Abstract— In this paper, we consider an almost periodic multispecies discrete Lotka-Volterra mutualism system with time delays. We first obtain the permanence and global attractivity of the system. By means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, sufficient conditions are obtained for the existence of a unique strictly positive almost periodic solution which is globally attractive. An example together with numerical simulation indicates the feasibility of the main results.

Index Terms—Almost periodic solution, Mutualism system, Discrete, Global attractivity

I. INTRODUCTION Recently, investigating the almost periodic solutions of discrete and continuous population dynamics model with time delays has more extensively practical application value(see [1–19] and the references cited therein). In this paper, we are concerned with the following multispecies discrete Lotka-Volterra mutualism system with time delays

II. PRELIMINARIES First, we give the definitions of the terminologies involved.

Definition 2.1([20]) A sequence x: Z→R is called an almost periodic sequence if the ε-translation set of x is a relatively dense set in Z for all ε > 0; that is, for any given ε > 0, there exists an integer l(ε) > 0 such that each interval of length l(ε) contains an integer τ∈E{ε, x} with τ is called an ε-translation number of x(n).

Definition 2.2([21]) Let D be an open subset of Rm, f : Z ×D →Rm. f(n, x) is said to be almost periodic in n uniformly for x ∈ D if for any ε > 0 and any compact set S ⊂ D, there exists a positive integer l=l(ε, S) such that any interval of length l contains an integer τ for which

where {ai(k)}, {bi(k)}, {cij(k)} and {dij(k)} are bounded nonnegative almost periodic sequences such that

τ is called an ε-translation number of f(n,x).

Definition 2.3( [22]) The hull of f, denoted by H(f), is defined by

By the biological meaning, we will focus our discussion on the positive solutions of system (1.1). So it is assumed that the initial conditions of system (1.1) are the form:

for some sequence {τk}, where S is any compact set in D. Definition 2.4 Suppose that X(k) = (x1(k), x2(k), · · · , xn(k)) is any solution of system (1.1). X(k) is said to be a strictly positive solution in Z if for k∈Z and i = 1, 2, · · · , n To the best of our knowledge, this is the first paper to investigate the global stability of positive almost periodic solution of multispecies discrete Lotka-Volterra mutualism system with time delays. The aim of this paper is to obtain sufficient conditions for the existence of a unique globally attractive almost periodic solution of the systems (1.1) with

Now, we state several lemmas which will be useful in proving our main result.

Lemma 2.1( [23]) {x(n)} is an almost periodic sequence if '

and only if for any integer sequence { k i }, there exists a

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Global attractivity and positive almost periodic solution of a multispecies discrete mutualism system with time delays '

subsequence { ki } ⊂ { k i } such that the sequence {x(n+ ki )} converges uniformly for all n∈Z as i →∞. Furthermore, the limit sequence is also an almost periodic sequence.

Lemma 2.2( [24]) Assume that sequence {x(n)} satisfies x(n) > 0 and for n∈N, where a(n) and b(n) are non-negative sequences bounded above and below by positive constants. Then

Substituting (3.7) into system (1.1), for k≥K+ζi, it follows that

Thus, as a direct corollary of Lemma 2.3, according to (3.4) and (3.8), one has where

Lemma 2.3( [24]) Assume that sequence {x(n)} satisfies Letting ε→0, it follows that and x(N0)>0, where a(n) and b(n) are non-negative sequences bounded above and below by positive constants and N0∈N. Then

III. PERMANENCE In this section, we establish the permanence result for system (1.1).

Theorem 3.1 System (1.1) with initial condition (1.2) is permanent, that is, there exist positive constants mi and Mi(i = 1, 2, · · · , n) which are independent of the solutions of system (1.1), such that for any positive solution (x1(k), x2(k), · · · , xn(k)) of system (1.1), one has:

Proof. Let (x1(k), x2(k), · · · , xn(k)) be any positive solution of system (1.1) with initial condition (1.2). From the first equation of system (1.1) it follows that

where

Then, (3.4) and (3.10) show that system (1.1) is permanent.

The proof is completed.

IV. GLOBAL ATTRACTIVITY In this section, by constructing a non-negative Lyapunovlike functional, we will obtain sufficient conditions for global attractivity of positive solutions of system (1.1) with initial condition (1.2). We first introduce a definition and prove a theorem which will be useful to obtain our main result.

Definition 4.1 A solution (x1(k), x2(k), · · · , xn(k)) of system (1.1) with initial condition (1.2) is said to be globally attractive if for any other solution (x∗1(k), x∗2(k), · · · , x∗n(k)) of system (1.1) with initial condition (1.2), we have Lemma 4.1 For any two positive solutions (x1(k), x2(k), · · · , xn(k)) and (x∗1(k), x∗2(k), · · · , x∗n(k)) of system (1.1) with initial condition (1.2), we have for k ≥ 2τ

By using (3.1), one could easily obtain that

Substituting (3.2) into the first equation of system (1.1), it follows that

Thus, as a direct corollary of Lemma 2.2, according to (3.3), one has

For any small positive constant ε > 0, from (3.4) it follows that there exists a positive constants K > 0 such that for all k > K and i = 1, 2, · · · , n, xi(k) ≤ Mi + ε.

(3.5) For k ≥ K+ζi, from (3.5) and system (1.1), we have

where

Proof. For i = 1, 2, · · · , n, we can have from system (1.1)

Thus, by using (3.6) we obtain

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International Journal of Engineering and Advanced Research Technology (IJEART) ISSN: 2454-9290, Volume-1, Issue-6, December 2015

Since where ξi(k) lies between xi(k) and x∗i (k), i = 1, 2, · · · , n, it follows that that is

By Theorem 3.1, there are constants Mi > 0, and a positive integer k0 such that for k > k0, 0 < xi(k), x∗i (k) ≤ Mi, i = 1, 2, · · · , n. Then from (4.8) and (4.9) we can obtain that for k ≥ k0 + 2τ ,

Since

and for k≥2τ

Secondly, let

By a simple calculation, we can obtain Using the Mean Value Theorem, we get

here Ai(s), Bi(s) are defined by (4.2). Then from (4.3)-(4.5), we can easily obtain (4.1). The proof is completed.

Theorem 4.1 Assume that in system (1.1) with initial condition (1.2), there exist positive constants βi(i =1, 2, · · · , n) and η > 0 such that Thirdly, let where

Then for any two positive solutions (x1(k), x2(k), · · · , xn(k)) and (x∗1(k), x∗2(k), · · · , x∗n(k)) of system (1.1) with initial condition (1.2), we have

Then we can derive

Proof. Firstly, let From (4.1), we have that for k ≥ 2τ,

Now we set

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Global attractivity and positive almost periodic solution of a multispecies discrete mutualism system with time delays

Then from (4.8)-(4.13), we have that for k ≥ k0+2τ ,

Now we define a Lyapunov-like discrete functional V (k) by

It is easy to see that V(k0+2τ)<+∞. Calculating the difference of V(k) along the solution of system (1.1) with initial condition (1.2), we have that for k ≥ k0+2τ,

where Ei and Fij are defined by (4.7).

Then we have that

Then we get a hull equation of system (1.1) as follows:

By the almost periodic theory, we can conclude that if system (1.1) satisfies (4.6), then the hull equation (5.1) of system (1.1) also satisfies (4.6).

By Theorem 3.4 in [26], we can easily obtain the lemma as follows.

Lemma 5.1 If each hull equation of system (1.1) has a unique strictly positive solution, then the almost periodic difference system (1.1) has a unique strictly positive almost periodic solution.

Theorem 5.1 If the almost periodic difference system (1.1) satisfies (4.6), then the almost periodic difference system (1.1) admits a unique strictly positive almost periodic solution, which is globally attractive.

Proof. By Lemma 5.1, we only need to prove that each hull equation of system (1.1) has a unique globally attractive almost periodic sequence solution; hence we firstly prove that each hull equation of system (1.1) has at least one strictly positive solution (the existence), and then we prove that each hull equation of system (1.1) has a unique strictly positive solution (the uniqueness).

Now we prove the existence of a strictly positive solution of any hull equation (5.1). By the almost periodicity of {a ∗i (k)}, {b∗i (k)}, {c∗ij(k)} and {d∗ij(k)}, there exists an integer valued sequence {τm } with τm→ ∞ as m → ∞ such that a∗i (k + τm) → a∗i (k), b∗i (k + τm) → b∗i (k), c∗ij(k + τm) → c∗ij(k), d∗ij(k + τm) → d∗ij(k), as m→∞ for k∈Z. Suppose that X(k) = (x1(k), x2(k), · · · , xn(k)) is any solution of hull equation (5.1). By the proof of Lemma 2.2 and 2.3, we have

which implies And also That is

and then

It means that (x1(k), x2(k), · · · , xn(k)) is globally attractive.

This completes the proof of Theorem 4.1.

V. ALMOST PERIODIC SOLUTION In this section, we will study the existence of a globally attractive almost periodic sequence solution of system (1.1) with initial condition (1.2) by means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, and obtain the sufficient conditions.

Let {δm} be any integer valued sequence such that δm →∞ as m→∞. According to Lemma 2.1, taking a subsequence if necessary, we have

Let ε be an arbitrary small positive number. It from (5.2) that there exists a positive integer k0 such that mi−ε≤xi(k)≤ Mi + ε, k ≥ k0, i = 1, 2, · · · , n. Write Xm(k) = X(k + τm) = (x1m(k), x2m(k), · · · , xnm(k)), for all k ≥ k0 + τ − τm,m ∈ Z+. We claim that there exists a sequence {yi(k)}, and a subsequence of {τk}, we still denote by {τk} such that xim(k) → yi(k), uniformly in k on any finite subset B of Z as m→∞, where B = {a1, a2, . . . , ap}, ah∈Z(h = 1, 2, . . . , p) and p is a finite number.

In fact, for any finite subset B⊂Z, when m is large enough, τm + ah − τ>k0, h = 1, 2, . . . , p. So that is, {xi(k+τm)} are uniformly bounded for large enough m.

Now, for a1∈B, we can choose a subsequence {τ(1)m } of {τm} such that {xi(a1+τ(1)m)} uniformly converges on Z+ for m large enough.

Similarly, for a2∈B, we can choose a subsequence {τ(2)m } of {τ(1)m } such that {xi(a2 + τ(2)m )} uniformly converges on Z+ for m large enough.

Repeating this procedure, for ap ∈ B, we can choose a subsequence {τ(p)m} of {τ(p−1)m} such that {xi(ap+τ (p)m)} uniformly converges on Z+ for m large enough.

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International Journal of Engineering and Advanced Research Technology (IJEART) ISSN: 2454-9290, Volume-1, Issue-6, December 2015 Now pick the sequence {τ(p)m } which is a subsequence of {τm}, we still denote it as {τm}, then for all k∈B, we have xi(k+τm)→yi(k) uniformly in k∈B, as m→∞.

By the arbitrary of B, the conclusion is valid.

Combined with

Define Q 

n

  Q , where i 1

i

i

Let ε be an arbitrary small positive number. It follows from (5.5) that there exists a positive integer K > 0 such that gives Therefore, for k<−K, i = 1, 2, · · · , n

We can easily see that Y(k) = (y1(k), y2(k), · · · , yn(k)) is a solution of hull equation (5.1) and mi – ε ≤ yi(k)≤ Mi + ε, i = 1, 2, · · · , n, for k ∈ Z. Since ε is an arbitrary small positive number, it follows that mi ≤ yi(k) ≤ Mi, i = 1, 2, · · · , n, for k ∈ Z, that is Hence each hull equation of almost periodic difference system (1.1) has at least one strictly positive solution.

Now we prove the uniqueness of the strictly positive solution of each hull equation (5.1). Suppose that the hull equation (5.1) has two arbitrary strictly positive solutions (x∗1(k), x∗2(k), · · · , x∗n(k)) and (y∗1(k), y∗2(k),· · · ,y∗n(k)).

Like in the proof of Theorem 4.1, we construct a Lyapunov functional

It follows from (5.3) and above inequalities that

so lim V * ( k )  0 . Note that V∗(k) is a non-increasing k 

function on Z, and then V∗(k)≡0. That is x∗i (k) =y∗i (k), i = 1, 2, · · · , n, for all k∈Z, Therefore, each hull equation of system (1.1) has a unique strictly positive solution.

In view of the above discussion, any hull equation of system (1.1) has a unique strictly positive solution. By Lemma 2.2-2.3 and Theorem 4.1, the almost periodic difference system (1.1) has a unique strictly positive almost periodic solution which is globally attractive. The proof is completed.

where

VI AN EXAMPLE AND NUMERICAL SIMULATION In this section, we give the following example to check the feasibility of our result.

Example Consider the following almost periodic discrete Lotka-Volterra mutualism model with delays: Calculating the difference of V∗(k) along the solution of the hull equation (5.1), like in the discussion of (4.14), one has

From (5.4), we can see that V∗(k) is a non-increasing function on Z. Summing both sides of the above inequalities from k to 0, we have

Note that V∗(k) is bounded. Hence we have

which implies that

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Global attractivity and positive almost periodic solution of a multispecies discrete mutualism system with time delays

By simple computation, we derive

Then

Also it is easy to see that the condition (4.6) is verified.

Therefore, system (6.1) has a unique strictly positive almost periodic solution which is globally attractive. Our numerical simulations support our results (see Figs.1-3).

FIGURE1: Dynamic behavior of x1(k) of system (6.1) with the initial conditions (x1(k), x2(k), x3(k)) = (0.015, 0.019, 0.011) and (0.034, 0.022, 0.028), k = 1, 2, 3, 4 for k ∈ [1, 100] and k ∈ [500, 550], respectively.

FIGURE2: Dynamic behavior of x2(k) of system (6.1) with the initial conditions (x1(k), x2(k), x3(k)) = (0.015, 0.019, 0.011) and (0.034, 0.022, 0.028), k = 1, 2, 3, 4 for k ∈ [1, 100] and k ∈ [500, 550], respectively.

FIGURE3: Dynamic behavior of x3(k) of system (6.1) with the initial conditions (x1(k), x2(k), x3(k)) = (0.015, 0.019, 0.011) and (0.034, 0.022, 0.028), k = 1, 2, 3, 4 for k ∈ [1, 100] and k ∈ [500, 550], respectively.

ACKNOWLEDGEMENT The authors declare that there is no conflict of interests regarding the publication of this paper, and there are no financial interest conflicts between the authors and the commercial identity.

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[2] Yongzhi Liao, Tianwei Zhang, Almost Periodic Solutions of a Discrete Mutualism Model with Variable Delays, Discrete Dynamics in Nature and Society, Volume 2012, Article ID 742102, 27 pages.

[3] Hui Zhang, Yingqi Li, Bin Jing, Global attractivity and almost periodic solution of a discrete mutualism model with delays, Mathematical Methods in the Applied Sciences, 2013, DOI: 10.1002/mma.3039.

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Hui Zhang is a lecturer of Xi’an Research Institute of High-tech Hongqing Town. His major is almost periodicity of continuous and discrete dynamic system.

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