Global Attractivity and Periodic Solution of a Discrete Multispecies Cooperation and Competition Predator-Prey System

Abstract

We propose a discrete multispecies cooperation and competition predator-prey systems. For general nonautonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained.

1. Introduction

In this paper, we consider the dynamic behavior of the following non-autonomous discrete 𝑛+𝑚-species cooperation and competition predator-prey systems𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)exp𝑟_1𝑖(𝑘)𝑥1−^𝑖(𝑘)𝑎_𝑖(𝑘)+∑^𝑛_{𝑙=1,𝑙≠𝑖}𝑏_𝑖𝑙(𝑘)𝑥_𝑙(𝑘)−𝑐_𝑖(𝑘)𝑥_𝑖(𝑘)−^𝑚_𝑙=1𝑑_𝑖𝑙(𝑘)𝑦_𝑙(𝑘),𝑦_𝑗(𝑘+1)=𝑦_𝑗(𝑘)exp𝑟_2𝑗(𝑘)+^𝑛_𝑙=1𝑒_𝑗𝑙(𝑘)𝑥_𝑙(𝑘)−^𝑚_𝑙=1𝑝_𝑗𝑙(𝑘)𝑦_𝑙(𝑘), (1.1) where 𝑖=1,2,…𝑛;𝑗=1,2,…,𝑚;𝑥_𝑖(𝑘) is the density of prey species 𝑖 at 𝑘th generation. 𝑦_𝑗(𝑘) is the density of predator species 𝑗 at 𝑘th generation.

Dynamic behaviors of population models governed by difference equations had been studied by a number of papers, see [1–4] and the references cited therein. It has been found that the autonomous discrete systems can demonstrate quite rich and complicated dynamics, see [5, 6]. Recently, more and more scholars paid attention to the non-autonomous discrete population models, since such kind of model could be more appropriate.

May [7] suggested the following set of equations to describe a pair of mutualists: 𝑢̇=𝑟₁𝑢𝑢1−𝑎₁+𝑏₁𝑣−𝑐₁𝑢,𝑣̇=𝑟₂𝑣𝑣1−𝑎₂+𝑏₂𝑢−𝑐₂𝑣, (1.2) where 𝑢,𝑣 are the densities of the species 𝑈,𝑉 at time 𝑡, respectively. 𝑟_𝑖,𝑎_𝑖,𝑏_𝑖,𝑐_𝑖,𝑖=1,2 are positive constants. He showed that system (1.2) has a globally asymptotically stable equilibrium point in the region 𝑢>0,𝑣>0.

Bai et al. [8] argued that the discrete case of cooperative system is more appropriate, and they proposed the following system: 𝑥₁(𝑘+1)=𝑥₁(𝑘)exp𝑟₁(𝑘𝑥)1−¹(𝑘)𝑎₁(𝑘)+𝑏₁(𝑘)𝑥₂(𝑘)−𝑐₁(𝑘)𝑥₁(𝑘,𝑥₂)(𝑘+1)=𝑥₂(𝑘)exp𝑟₂(𝑘𝑥)1−²(𝑘)𝑎₂(𝑘)+𝑏₂(𝑘)𝑥₂(𝑘)−𝑐₂(𝑘)𝑥₁(𝑘). (1.3)

Chen [9] investigated the dynamic behavior of the following discrete 𝑛+𝑚-species Lotka-Volterra competition predator-prey systems𝑥_𝑖(𝑘+1)=𝑥_𝑖(𝑘)exp𝑏_𝑖(𝑘)−^𝑛_𝑙=1𝑎_𝑖𝑙(𝑘)𝑥_𝑙(𝑘)−^𝑚_𝑙=1𝑐_𝑖𝑙(𝑘)𝑦_𝑙(𝑘,𝑦_𝑗)(𝑘+1)=𝑦_𝑗(𝑘)exp𝑟_𝑗(𝑘)+^𝑛_𝑙=1𝑑_𝑗𝑙(𝑘)𝑥_𝑙(𝑘)−^𝑚_𝑙=1𝑒_𝑗𝑙(𝑘)𝑦_𝑙(𝑘), (1.4) he investigated the dynamic behavior of the system (1.4).

The aim of this paper is, by further developing the analysis technique of Huo and Li [10] and Chen [9], to obtain a set of sufficient condition which ensure the permanence and the global stability of the system (1.1); for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system (1.1) are obtained.

We say that system (1.1) is permanent if there are positive constants 𝑀 and 𝑚 such that for each positive solution (𝑥₁(𝑘),…,𝑥_𝑛(𝑘),𝑦₁(𝑘),…,𝑦_𝑚(𝑘)) of system (1.1) satisfies𝑚≤lim_𝑘→+∞inf𝑥_𝑖(𝑘)≤lim_𝑘→+∞sup𝑥_𝑖(𝑘)≤𝑀,𝑚≤lim_𝑘→+∞inf𝑦_𝑖(𝑘)≤lim_𝑘→+∞sup𝑦_𝑖(𝑘)≤𝑀, (1.5) for all 𝑖=1,2,…,𝑛;𝑗=1,2,…,𝑚.

Throughout this paper, we assume that 𝑟_1𝑖(𝑘), 𝑏_𝑖𝑙(𝑘), 𝑎_𝑖(𝑘), 𝑐_𝑖(𝑘), 𝑟_2𝑗(𝑘), 𝑑_𝑖𝑙(𝑘), 𝑒_𝑗𝑙(𝑘), 𝑝_𝑗𝑙(𝑘) are all bounded nonnegative sequence, and use the following notations for any bounded sequence {𝑎(𝑘)}𝑎^𝑢=sup_𝑘∈𝑁𝑎(𝑘),𝑎^𝑙=inf_𝑘∈𝑁𝑎(𝑘). (1.6)

For biological reasons, we only consider solution (𝑥₁(𝑘),…,𝑥_𝑛(𝑘),𝑦₁(𝑘),…,𝑦_𝑚(𝑘)) with𝑥_𝑖(0)>0;𝑖=1,2,…,𝑛,𝑦_𝑗(0)>0,𝑗=1,2,...,𝑚. (1.7)

Then system (1.1) has a positive solution (𝑥₁(𝑘),…,𝑥_𝑛(𝑘),𝑦₁(𝑘),…,𝑦_𝑚(𝑘))^∞_𝑘=0 passing through (𝑥₁(0),…,𝑥_𝑛(0),𝑦₁(0),…,𝑦_𝑚(0)).

The organization of this paper is as follows. In Section 2, we obtain sufficient conditions which guarantee the permanence of the system (1.1). In Section 3, we obtain sufficient conditions which guarantee the global stability of the positive solution of system (1.1). As a consequence, for periodic case, we obtain sufficient conditions which ensure the existence of a globally stable positive solution of system (1.1).

2. Permanence

In this section, we establish permanence results for system (1.1).

Lemma 2.1 (see [11]). Let 𝑘∈𝑁⁺_𝑘0={𝑘₀,𝑘₀+1,…,𝑘₀+𝑙,…},𝑟≥0. For any fixed 𝑘,𝑔(𝑘,𝑟) is a non-decreasing function with respect to 𝑟, and for 𝑘≥𝑘₀, the following inequalities hold: 𝑦()(𝑘),𝑢()((𝑘.𝑘+1≤𝑔𝑘,𝑦𝑘+1≥𝑔𝑘,𝑢)) (2.1) If 𝑦(𝑘₀)≤𝑢(𝑘₀), then 𝑦(𝑘)≤𝑢(𝑘) for all 𝑘≥𝑘₀.

Now let one consider the following single species discrete model: 𝑁(𝑘+1)=𝑁(𝑘)exp{𝑎(𝑘)−𝑏(𝑘)𝑁(𝑘)}, (2.2) where {𝑎(𝑘)} and {𝑏(𝑘)} are strictly positive sequences of real numbers defined for 𝑘∈𝑁={0,1,2,…} and 0<𝑎^𝑙≤𝑎^𝑢,0<𝑏^𝑙≤𝑏^𝑢.

Lemma 2.2 (see [12]). Any solution of system (2.5) with initial condition 𝑁(0)>0 satisfies 𝑚≤lim_𝑘→+∞inf𝑁(𝑘)≤lim_𝑘→+∞sup𝑁(𝑘)≤𝑀, (2.3) where 1𝑀=𝑏^𝑙exp𝑎^𝑢𝑎−1,𝑚=^𝑙𝑏^𝑢exp𝑎^𝑙−𝑏^𝑢𝑀. (2.4)

Proposition 2.3. Assume that −𝑟^𝑢_2𝑗+^𝑛_𝑙=1𝑒^𝑙_𝑗𝑙𝑀_𝑙>0,𝑗=1,2,…,𝑚 (2.5) holds, then lim_𝑘→+∞sup𝑥_𝑖(𝑘)≤𝑀_𝑖,𝑖=1,2,…,𝑛,lim_𝑘→+∞sup𝑦_𝑖(𝑘)≤𝑄_𝑖,𝑖=1,2,...,𝑚, (2.6) where 𝑀_𝑖=1𝑐^𝑙_𝑖𝑟^𝑙_1𝑖exp𝑟^𝑢_1𝑖,𝑄−1_𝑖=1𝑝^𝑙_𝑖𝑖exp^𝑛_𝑙=1𝑒^𝑢_𝑖𝑙𝑀_𝑙−𝑟^𝑙_2𝑖−1. (2.7)

Proof. Let 𝑢(𝑘)=(𝑥₁(𝑘),…,𝑥_𝑛(𝑘),𝑦₁(𝑘),…,𝑦_𝑚(𝑘)) be any positive solution of system (1.1), from the 𝑖th equation of (1.1) we have 𝑥_𝑖(𝑘+1)≤𝑥_𝑖(𝑘)exp𝑟_1𝑖(𝑘)1−𝑐_𝑖(𝑘)𝑥_𝑖(𝑘)=𝑥_𝑖(𝑘)exp𝑟_1𝑖(𝑘)−𝑟_1𝑖(𝑘)𝑐_𝑖(𝑘)𝑥_𝑖(𝑘). (2.8) By applying Lemmas 2.1 and 2.2, it immediately follows that lim_𝑘→+∞sup𝑥_𝑖(𝑘)≤1𝑐^𝑙_𝑖𝑟^𝑙_1𝑖exp𝑟^𝑢_1𝑖−1∶=𝑀_𝑖. (2.9) For any positive constant 𝜀 small enough, it follows from (2.9) that there exists enough large 𝐾₀ such that 𝑥_𝑖(𝑘)≤𝑀_𝑖+𝜀,𝑖=1,2,…,𝑛,∀𝑘≥𝐾₀. (2.10)
From the 𝑛+𝑗th equation of the system (1.1) and (2.10), we can obtain 𝑦_𝑗(𝑘+1)≤𝑦_𝑗(𝑘)exp−𝑟_2𝑗(𝑘)+^𝑛_𝑙=1𝑒_𝑗𝑙(𝑘)𝑀_𝑙+𝜀−𝑝_𝑗𝑗(𝑘)𝑦_𝑗(𝑘). (2.11) Condition (2.5) shows that Lemmas 2.1 and 2.2 could be applied to (2.11), and so, by applying Lemmas 2.1 and 2.2, it immediately follows that lim_𝑘→+∞sup𝑦_𝑗(𝑘)≤1𝑝^𝑙_𝑗𝑗exp^𝑛_𝑙=1𝑒^𝑢_𝑗𝑙𝑀_𝑙+𝜀−𝑟^𝑙_2𝑗−1,1≤𝑗≤𝑚. (2.12) Setting 𝜀→0 in the above inequality leads to lim_𝑘→+∞sup𝑦_𝑗(𝑘)≤1𝑝^𝑙_𝑗𝑗exp^𝑛_𝑙=1𝑒^𝑢_𝑗𝑙𝑀_𝑙−𝑟^𝑙_2𝑗−1∶=𝑄_𝑗,1≤𝑗≤𝑚. (2.13) This completes the proof of Proposition 2.3.

Now we are in the position of stating the permanence of the system (1.1).

Theorem 2.4. In addition to (2.5), assume further that 𝑟^𝑙_1𝑖−^𝑚_𝑙=1𝑑^𝑢_𝑖𝑙𝑄_𝑙>0,𝑖=1,2,…,𝑛,−𝑟^𝑢_2𝑗+^𝑛_𝑙=1𝑒^𝑙_𝑗𝑙𝑚_𝑙−^𝑚_{𝑙=1,𝑙≠𝑗}𝑝^𝑢_𝑗𝑙𝑄_𝑙>0,𝑗=1,2,…𝑚, (2.14) then system (1.1) is permanent, where 𝑚_𝑖=𝑟^𝑙_1𝑖−∑^𝑚_𝑙=1𝑑^𝑢_𝑖𝑙𝑄_𝑙𝑟^𝑢_1𝑖1/𝑎^𝑙_𝑖+𝑐^𝑢_𝑖exp𝑟^𝑙_1𝑖−^𝑚_𝑙=1𝑑^𝑢_𝑖𝑙𝑄_𝑙−𝑟^𝑢_1𝑖1𝑎^𝑙_𝑖+𝑐^𝑢_𝑖𝑀_𝑖,𝑄_𝑖=1𝑝^𝑙_𝑖𝑖exp^𝑛_𝑙=1𝑒^𝑢_𝑖𝑙𝑀_𝑙−𝑟^𝑙_2𝑖−1. (2.15)

Proof. By applying Proposition 2.3, we see that to end the proof of Theorem 2.4, it is enough to show that under the conditions of Theorem 2.4, lim_𝑘→+∞inf𝑥_𝑗(𝑘)≥𝑚_𝑖,1≤𝑖≤𝑛,lim_𝑘→+∞inf𝑦_𝑗(𝑘)≥𝑞_𝑗,1≤𝑗≤𝑚. (2.16) From Proposition 2.3, ∀𝜀>0, there exists a 𝐾₁>0,𝐾₁∈𝑁,∀𝑘>𝐾₁, 𝑥_𝑖(𝑘)≤𝑀_𝑖+𝜀,𝑖=1,2,…,𝑛,𝑦_𝑗(𝑘)≤𝑄_𝑗+𝜀,𝑗=1,2,…,𝑚. (2.17) From the 𝑖th equation of system (1.1) and (2.17), we have 𝑥_𝑖(𝑘+1)≥𝑥_𝑖(𝑘)exp𝑟_1𝑖(𝑘)−𝑟_1𝑖(𝑘)1𝑎^𝑙_𝑖+𝑐^𝑢_𝑖𝑥_𝑖(𝑘)−^𝑚_𝑙=1𝑑_𝑖𝑙(𝑘)𝑄_𝑙+𝜀=𝑥_𝑖(𝑘)exp𝑟_1𝑖(𝑘)−^𝑚_𝑙=1𝑑_𝑖𝑙(𝑘)𝑄_𝑙+𝜀−𝑟_1𝑖(𝑘)1𝑎^𝑙_𝑖+𝑐^𝑢_𝑖𝑥_𝑖(𝑘), (2.18) for all 𝑘>𝐾₁.
By applying Lemmas 2.1 and 2.2 to (2.18), it immediately follows that lim_𝑘→+∞inf𝑥_𝑖(𝑘)≥𝑟^𝑙_1𝑖−∑^𝑚_𝑙=1𝑑^𝑢_𝑖𝑙𝑄_𝑙+𝜀𝑟^𝑢_1𝑖1/𝑎^𝑙_𝑖+𝑐^𝑢_𝑖×exp𝑟^𝑙_1𝑖−^𝑚_𝑙=1𝑑^𝑢_𝑖𝑙𝑄_𝑙+𝜀−𝑟^𝑢_1𝑖1𝑎^𝑙_𝑖+𝑐^𝑢_𝑖𝑀_𝑖. (2.19) Setting 𝜀→0 in (2.19) leads to lim_𝑘→+∞inf𝑥_𝑖(𝑘)≥𝑟^𝑙_1𝑖−∑^𝑚_𝑙=1𝑑^𝑢_𝑖𝑙𝑄_𝑙𝑟^𝑢_1𝑖1/𝑎^𝑙_𝑖+𝑐^𝑢_𝑖×exp𝑟^𝑙_1𝑖−^𝑚_𝑙=1𝑑^𝑢_𝑖𝑙𝑄_𝑙−𝑟^𝑢_1𝑖1𝑎^𝑙_𝑖+𝑐^𝑢_𝑖𝑀_𝑖∶=𝑚_𝑖. (2.20)
Then, for any positive constant 𝜀 small enough, from (2.20) we know that there exists an enough large 𝐾₂>𝐾₁ such that 𝑥_𝑖(𝑘)≥𝑚_𝑖−𝜀,∀𝑘≥𝑘₂. (2.21) Equations (2.17), (2.21) combining with the 𝑛+𝑗th equation of the system (1.1) leads to 𝑦_𝑗(𝑘+1)≥𝑦_𝑗(𝑘)exp−𝑟_2𝑗(𝑘)+^𝑛_𝑙=1𝑒_𝑗𝑙(𝑘)𝑚_𝑙−−𝜀_{𝑚𝑙=1,𝑙≠𝑗}𝑝_𝑗𝑙(𝑘)𝑄_𝑙+𝜀−𝑝_𝑗𝑗(𝑘)𝑦_𝑗(𝑘), (2.22) under the condition (2.14), by applying Lemmas 2.1 and 2.2 to (2.22), it immediately follows that lim_𝑘→+∞inf𝑦_𝑗(𝑘)≥−𝑟^𝑢_2𝑗+∑^𝑛_𝑙=1𝑒^𝑙_𝑗𝑙𝑚_𝑙−𝜀−∑^𝑚_{𝑙=1,𝑙≠𝑗}𝑝^𝑢_𝑗𝑙𝑄_𝑙+𝜀𝑝^𝑢_𝑗𝑗×exp−𝑟^𝑢_2𝑗+^𝑛_𝑙=1𝑒^𝑙_𝑗𝑙𝑚_𝑙−𝜀−^𝑚_{𝑙=1,𝑙≠𝑗}𝑝^𝑢_𝑗𝑙𝑄_𝑙+𝜀−𝑝^𝑢_𝑗𝑗𝑄_𝑗. (2.23) Setting 𝜀→0 in (2.23) leads to lim_𝑘→+∞inf𝑦_𝑗(𝑘)≥−𝑟^𝑢_2𝑗+∑^𝑛_𝑙=1𝑒^𝑙_𝑗𝑙𝑚_𝑙−∑^𝑚_{𝑙=1,𝑙≠𝑗}𝑝^𝑢_𝑗𝑙𝑄_𝑙𝑝^𝑢_𝑗𝑗×exp−𝑟^𝑢_2𝑗+^𝑛_𝑙=1𝑒^𝑙_𝑗𝑙𝑚_𝑙−^𝑚_{𝑙=1,𝑙≠𝑗}𝑝^𝑢_𝑗𝑙𝑄_𝑙−𝑝^𝑢_𝑗𝑗𝑄_𝑗∶=𝑞_𝑗. (2.24) This ends the proof of Theorem 2.4.

It should be noticed that, under the assumption of Theorem 2.4, the set 𝑚₁,𝑀₁×⋯𝑚_𝑛,𝑀_𝑛×𝑞₁,𝑄₁×⋯𝑞_𝑚,𝑄_𝑚 (2.25) is an invariant set of system (1.1).

3. Global Stability

Now we study the stability property of the positive solution of system (1.1).

Theorem 3.1. Assume that 𝜆_𝑖=max1−𝑐^𝑢_𝑖+1𝑎^𝑙_𝑖𝑟^𝑢_1𝑖𝑀_𝑖,1−𝑐^𝑙_𝑖+1𝑎^𝑢_𝑖𝑟^𝑙_1𝑖𝑚_𝑖+𝑟^𝑢_1𝑖𝑀_𝑖𝑎^𝑙_𝑖^2𝑛_{𝑙=1,𝑙≠𝑖}𝑏^𝑢_𝑖𝑙𝑀_𝑙+^𝑚_𝑙=1𝑑^𝑢_𝑖𝑙𝑄_𝑙𝛿<1,_𝑗=max1−𝑝^𝑢_𝑗𝑗𝑄_𝑗,1−𝑝^𝑙_𝑗𝑗𝑞_𝑗+^𝑛_𝑙=1𝑒^𝑢_𝑗𝑙𝑀_𝑙+^𝑛_𝑙=1𝑝^𝑢_𝑗𝑙𝑄_𝑙<1. (3.1) Then for any two positive solution (𝑥₁(𝑘),…,𝑥_𝑛(𝑘),𝑦₁(𝑘),…,𝑦_𝑚(𝑘)) and (𝑥₁(𝑘),…,𝑥_𝑛(𝑘),𝑦₁(𝑘),…,𝑦_𝑚(𝑘)) of system (1.1), one has lim_𝑘→+∞𝑥_𝑖(𝑘)−𝑥_𝑖(𝑘)=0,lim_𝑘→+∞𝑦_𝑗(𝑘)−𝑦_𝑗(𝑘)=0. (3.2)

Proof. Let 𝑥_𝑖(𝑘)=𝑥_𝑖(𝑘)exp𝑢_𝑖(𝑘),𝑦_𝑗(𝑘)=𝑦_𝑗(𝑘)exp𝑣_𝑗(𝑘). (3.3) Then system (1.1) is equivalent to 𝑢_𝑖(𝑘+1)=𝑢_𝑖(𝑘)+𝑟_1𝑖(𝑘)𝑥_𝑖(𝑘)𝑎_𝑖(𝑘)+∑^𝑛_{𝑙=1,𝑙≠𝑖}𝑏_𝑖𝑙(𝑘)𝑥_𝑙(𝑘)−𝑟_1𝑖(𝑘)𝑥_𝑖(𝑘)exp𝑢_𝑖(𝑘)𝑎_𝑖(𝑘)+∑^𝑛_{𝑙=1,𝑙≠𝑖}𝑏_𝑖𝑙(𝑘)𝑥_𝑙(𝑘)exp𝑢_𝑙(𝑘)−𝑟_1𝑖(𝑘)𝑐_𝑖(𝑘)𝑥_𝑖(𝑘)exp𝑢_𝑖(𝑘)−1−^𝑚_𝑙=1𝑑_𝑖𝑙(𝑘)𝑦_𝑙(𝑘)exp𝑣_𝑙(𝑘,𝑣)−1_𝑗(𝑘+1)=𝑣_𝑗(𝑘)−^𝑛_𝑙=1𝑒_𝑗𝑙(𝑘)𝑥_𝑙(𝑘)exp𝑢_𝑙(𝑘−)−1_𝑚𝑙=1𝑝_𝑗𝑙(𝑘)𝑦_𝑙(𝑘)exp𝑣_𝑙(𝑘)−1. (3.4) So, 𝑢_𝑖(𝑘+1)≤1−𝑐_𝑖(𝑘)+1𝑎_𝑖(𝑘)𝑟_1𝑖(𝑘)𝑥_𝑖(𝑘)exp𝜃_𝑖(𝑘)𝑢_𝑖(𝑘)𝑢_𝑖(𝑘+𝑟_1𝑖)(𝑘)𝑥_𝑖(𝑘)𝑎²_𝑖(𝑘)^𝑛_{𝑙=1,𝑙≠𝑖}𝑏_𝑖𝑙(𝑘)𝑥_𝑙(𝑘)exp𝜃_𝑙(𝑘)𝑢_𝑙(𝑘)𝑢_𝑙(𝑘+_𝑚𝑙=1)𝑑_𝑖𝑙(𝑘)𝑦_𝑙(𝑘)exp𝜉_𝑙(𝑘)𝑣_𝑙(𝑘)𝑣_𝑙(𝑘𝑣_𝑖)(𝑘+1)≤1−𝑝_𝑗𝑗(𝑘)𝑦_𝑗(𝑘)exp𝜉_𝑗(𝑘)𝑣_𝑗(𝑘)𝑣_𝑗(𝑘+_𝑛𝑙=1)𝑒_𝑗𝑙(𝑘)𝑥_𝑙(𝑘)exp𝜃_𝑙(𝑘)𝑢_𝑙(𝑘)𝑢_𝑙(𝑘+_𝑚𝑙=1)𝑝_𝑖𝑙(𝑘)𝑦_𝑙(𝑘)exp𝜉_𝑙(𝑘)𝑣_𝑙(𝑘)𝑣_𝑙(𝑘), (3.5) where 𝜃_𝑙(𝑘),𝜉_𝑙(𝑘)∈[0,1], to complete the proof, it suffices to show that lim_𝑘→+∞𝑢_𝑖(𝑘)=0,lim_𝑘→+∞𝑣_𝑗(𝑘)=0. (3.6) In view of (3.1), we can choose 𝜀>0 small enough such that 𝜆^𝜀_𝑖=max1−𝑐^𝑢_𝑖+1𝑎^𝑙_𝑖𝑟^𝑢_1𝑖𝑀_𝑖+𝜀,1−𝑐^𝑙_𝑖+1𝑎^𝑢_𝑖𝑟^𝑙_1𝑖𝑚_𝑖+𝑟−𝜀_𝑢1𝑖𝑀_𝑖+𝜀𝑎^𝑙_𝑖^2𝑛_{𝑙=1,𝑙≠𝑖}𝑏^𝑢_𝑖𝑙𝑀_𝑙+𝜀+^𝑚_𝑙=1𝑑^𝑢_𝑖𝑙𝑄_𝑙𝛿+𝜀<1,_𝜀𝑗=max1−𝑝^𝑢_𝑗𝑗𝑄_𝑗+𝜀,1−𝑝^𝑙_𝑗𝑗𝑞_𝑗−𝜀+^𝑛_𝑙=1𝑒^𝑢_𝑗𝑙𝑀_𝑙++𝜀_𝑛𝑙=1𝑝^𝑢_𝑗𝑙𝑄_𝑙+𝜀<1. (3.7) For the above 𝜀>0, according to Theorem 2.4 in Section 2, there exists a 𝑘^∗∈𝑁 such that 𝑚_𝑖−𝜀≤𝑥_𝑖(𝑘),𝑥_𝑖(𝑘)≤𝑀_𝑖𝑞+𝜀,_𝑗−𝜀≤𝑦_𝑗(𝑘),𝑦_𝑗(𝑘)≤𝑄_𝑗+𝜀, (3.8) for all 𝑘≥𝑘^∗.
Noticing that 𝜃_𝑙(𝑘),𝜉_𝑙(𝑘)∈[0,1] implies that 𝑥_𝑙(𝑘)exp{𝜃_𝑙(𝑘)𝑢_𝑙(𝑘)} lies between 𝑥_𝑙(𝑘) and 𝑥_𝑙(𝑘),𝑦_𝑙(𝑘)exp{𝜉_𝑙(𝑘)𝑣_𝑙(𝑘)} lies between 𝑦_𝑙(𝑘) and 𝑦_𝑙(𝑘). From (3.5), we get 𝑢_𝑖(𝑘+1)≤max1−𝑐^𝑢_𝑖+1𝑎^𝑙_𝑖𝑟^𝑢_1𝑖𝑀_𝑖+𝜀,1−𝑐^𝑙_𝑖+1𝑎^𝑢_𝑖𝑟^𝑙_1𝑖𝑚_𝑖−𝜀𝑢_𝑖(𝑘+𝑟_𝑢1𝑖)𝑀_𝑖+𝜀𝑎^𝑙_𝑖^2𝑛_{𝑙=1,𝑙≠𝑖}𝑏^𝑢_𝑖𝑙𝑀_𝑙+𝜀𝑢_𝑙(𝑘)+^𝑚_𝑙=1𝑑^𝑢_𝑖𝑙𝑄_𝑙+𝜀𝑣_𝑙(𝑘,𝑣_𝑗)(𝑘+1)≤max1−𝑝^𝑢_𝑗𝑗𝑄_𝑗+𝜀,1−𝑝^𝑙_𝑗𝑗𝑞_𝑗−𝜀𝑣_𝑗(𝑘+_𝑛𝑙=1)𝑒^𝑢_𝑗𝑙𝑀_𝑙+𝜀𝑢_𝑙(𝑘)+^𝑛_𝑙=1𝑝^𝑢_𝑗𝑙𝑄_𝑙+𝜀𝑣_𝑙(𝑘). (3.9) Let 𝛾=max{𝜆^𝜀_𝑖,𝛿^𝜀_𝑗}, then 𝛾<1. In view of (3.9), for 𝑘≥𝑘^∗, we get max𝑢_𝑖(𝑘+1),𝑣_𝑗(𝑘+1)≤𝛾max𝑢_𝑖(𝑘),𝑣_𝑗(𝑘). (3.10) This implies max𝑢_𝑖(𝑘),𝑣_𝑗(𝑘)≤𝛾^{𝑘−𝑘∗}max𝑢_𝑖𝑘^∗,𝑣_𝑗𝑘^∗. (3.11) From (3.11), we have lim_𝑘→+∞𝑢_𝑖(𝑘)=0,lim_𝑘→+∞𝑣_𝑗(𝑘)=0. (3.12) This ends the proof of Theorem 3.1.

4. Existence and Stability of Periodic Solution

In this section, we further assume that the coefficients of the system (1.1) satisfies (4.1).

There exists a positive integer 𝜔 such that for 𝑘∈𝑁, 0<𝑟_1𝑖(𝑘+𝜔)=𝑟_1𝑖(𝑘),0<𝑏_𝑖𝑙(𝑘+𝜔)=𝑏_1𝑖(𝑘),0<𝑎_𝑖(𝑘+𝜔)=𝑎_𝑖(𝑘),0<𝑐_𝑖(𝑘+𝜔)=𝑐_𝑖(𝑘),0<𝑟_2𝑗(𝑘+𝜔)=𝑟_2𝑗(𝑘),0<𝑑_𝑖𝑙(𝑘+𝜔)=𝑑_𝑖𝑙(𝑘),0<𝑒_𝑗𝑙(𝑘+𝜔)=𝑒_𝑗𝑙(𝑘),0<𝑝_𝑗𝑙(𝑘+𝜔)=𝑝_𝑗𝑙(𝑘). (4.1)

Our first result concerned with the existence of a positive periodic solution of system (1.1).

Theorem 4.1. Assume that (2.5) and (2.14) hold, then system (1.1) admits at least one positive 𝜔-periodic solution which ones denotes by (𝑥₁(𝑘),…,𝑥_𝑛(𝑘),𝑦₁(𝑘),…,𝑦_𝑚(𝑘)).

Proof. As noted at the end of Section 2, 𝐷^𝑛+𝑚∶=𝑚₁,𝑀₁×⋯𝑚_𝑛,𝑀_𝑛×𝑞₁,𝑄₁×⋯𝑞_𝑚,𝑄_𝑚 (4.2) is an invariant set of system (1.1). Thus, we can define a mapping 𝐹 on 𝐷^𝑛+𝑚 by 𝐹𝑥₁(0),…,𝑥_𝑛(0),𝑦₁(0),…,𝑦_𝑚(0)=𝑥₁(𝜔),…,𝑥_𝑛(𝜔),𝑦₁(𝜔),…,𝑦_𝑚(𝜔), (4.3) for (𝑥₁(0),…,𝑥_𝑛(0),𝑦₁(0),…,𝑦_𝑚(0))∈𝐷^𝑛+𝑚. Obviously, 𝐹 depends continuously on (𝑥₁(0),…,𝑥_𝑛(0),𝑦₁(0),…,𝑦_𝑚(0)). Thus, 𝐹 is continuous and maps the compact set 𝐷^𝑛+𝑚 into itself. Therefore, 𝐹 has a fixed point. It is easy to see that the solution (𝑥₁(𝑘),…,𝑥_𝑛(𝑘),𝑦₁(𝑘),…,𝑦_𝑚(𝑘)) passing through this fixed point is an 𝜔-periodic solution of the system (1.1). This completes the proof of Theorem 4.1.

Theorem 4.2. Assume that (2.5), (2.14), and (3.1) hold, then system (1.1) has a global stable positive 𝜔-periodic solution.

Proof. Under the assumption of Theorem 4.2, it follows from Theorem 4.1 that system (1.1) admits at least one positive 𝜔-periodic solution. Also, Theorem 3.1 ensures the positive solution to be globally stable. This ends the proof of Theorem 4.2.