Global attractivity and positive almost periodic solution of a multispecies discrete mutualism system with time delays Hui Zhang 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 {a i (k)}, {b i (k)}, {c ij (k)} and {d ij (k)} are bounded nonnegative almost periodic sequences such that 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: 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 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. 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 R m. 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 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 for some sequence {τ k }, S is any compact set in D. Definition 2.4 Suppose that X(k) = (x 1 (k), x 2 (k),, x n (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 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 }, there exists a ' i 57 www.ijeart.com
Global attractivity and positive almost periodic solution of a multispecies discrete mutualism system with time delays subsequence { k i } { k ' i } such that the sequence {x(n+ k i )} 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, 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+ζ i, it follows that Thus, as a direct corollary of Lemma 2.3, according to (3.4) and (3.8), one has Lemma 2.3( [24]) Assume that sequence {x(n)} satisfies Letting ε 0, it follows that and x(n 0 )>0, a(n) and b(n) are non-negative sequences bounded above and below by positive constants and N 0 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 M i (i = 1, 2,, n) which are independent of the solutions of system (1.1), such that for any positive solution (x 1 (k), x 2 (k),, x n (k)) of system (1.1), one has: Proof. Let (x 1 (k), x 2 (k),, x n (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 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 (x 1 (k), x 2 (k),, x n (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 (x 1 (k), x 2 (k),, x n (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, x i (k) M i + ε. (3.5) For k K+ζ i, from (3.5) and system (1.1), we have Proof. For i = 1, 2,, n, we can have from system (1.1) Thus, by using (3.6) we obtain 58 www.ijeart.com
Since ξ i (k) lies between x i (k) and x i (k), i = 1, 2,, n, it follows that that is Since By Theorem 3.1, there are constants M i > 0, and a positive integer k 0 such that for k > k 0, 0 < x i (k), x i (k) M i, i = 1, 2,, n. Then from (4.8) and (4.9) we can obtain that for k k 0 + 2τ, and for k 2τ Secondly, let Using the Mean Value Theorem, we get By a simple calculation, we can obtain here A i (s), B i (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 β i (i =1, 2,, n) and η > 0 such that Thirdly, let Then for any two positive solutions (x 1 (k), x 2 (k),, x n (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 59 www.ijeart.com
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 k 0 +2τ, Then we get a hull equation of system (1.1) as follows: Now we define a Lyapunov-like discrete functional V (k) by It is easy to see that V(k 0 +2τ)<+. Calculating the difference of V(k) along the solution of system (1.1) with initial condition (1.2), we have that for k k 0 +2τ, E i and F ij are defined by (4.7). Then we have that which implies That is and then It means that (x 1 (k), x 2 (k),, x n (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 {δ m } be any integer valued sequence such that δ m as m. According to Lemma 2.1, taking a subsequence if necessary, we have 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 {τ m } with τ m as m such that a i (k + τ m ) a i (k), b i (k + τ m ) b i (k), c ij(k + τ m ) c ij(k), d ij(k + τ m ) d ij(k), as m for k Z. Suppose that X(k) = (x 1 (k), x 2 (k),, x n (k)) is any solution of hull equation (5.1). By the proof of Lemma 2.2 and 2.3, we have And also Let ε be an arbitrary small positive number. It from (5.2) that there exists a positive integer k 0 such that m i ε x i (k) M i + ε, k k 0, i = 1, 2,, n. Write X m (k) = X(k + τ m ) = (x 1m (k), x 2m (k),, x nm (k)), for all k k 0 + τ τ m,m Z +. We claim that there exists a sequence {y i (k)}, and a subsequence of {τ k }, we still denote by {τ k } such that x im (k) y i (k), uniformly in k on any finite subset B of Z as m, B = {a 1, a 2,..., a p }, a h Z(h = 1, 2,..., p) and p is a finite number. In fact, for any finite subset B Z, when m is large enough, τ m + a h τ>k 0, h = 1, 2,..., p. So that is, {x i (k+τ m )} are uniformly bounded for large enough m. Now, for a 1 B, we can choose a subsequence {τ (1) m } of {τ m } such that {x i (a 1 +τ (1) m)} uniformly converges on Z + for m large enough. Similarly, for a 2 B, we can choose a subsequence {τ (2) m } of {τ (1) m } such that {x i (a 2 + τ (2) m )} uniformly converges on Z + for m large enough. Repeating this procedure, for a p B, we can choose a subsequence {τ (p) m} of {τ (p 1) m} such that {x i (a p +τ (p) m)} uniformly converges on Z + for m large enough. 60 www.ijeart.com
Now pick the sequence {τ (p) m } which is a subsequence of {τ m }, we still denote it as {τ m }, then for all k B, we have x i (k+τ m ) y i (k) uniformly in k B, as m. By the arbitrary of B, the conclusion is valid. Combined with n Define Q iqi, i 1 gives Let ε be an arbitrary small positive number. It follows from (5.5) that there exists a positive integer K > 0 such that Therefore, for k< K, i = 1, 2,, n We can easily see that Y(k) = (y 1 (k), y 2 (k),, y n (k)) is a solution of hull equation (5.1) and m i ε y i (k) M i + ε, i = 1, 2,, n, for k Z. Since ε is an arbitrary small positive number, it follows that m i y i (k) M i, 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. 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 61 www.ijeart.com
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). FIGURE2: Dynamic behavior of x 2 (k) of system (6.1) with the initial conditions (x 1 (k), x 2 (k), x 3 (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. FIGURE1: Dynamic behavior of x 1 (k) of system (6.1) with the initial conditions (x 1 (k), x 2 (k), x 3 (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 x 3 (k) of system (6.1) with the initial conditions (x 1 (k), x 2 (k), x 3 (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. REFERENCES [1] Hui Zhang, Yingqi Li, Bin Jing, Weizhou Zhao, Global stability of almost periodic solution of multispecies mutualism system with time delays and impulsive effects, Applied Mathematics and Computation, 232(2014)1138-1150. [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. 62 www.ijeart.com
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