Permanence and Global Stability of Positive Periodic Solutions of a Discrete Competitive System

Abstract

We consider the dynamic behaviors of a discrete competitive system. A good understanding of the permanence, existence, and global stability of positive periodic solutions is gained. Numerical simulations are also presented to substantiate the analytical results.

1. Introduction

In biomathematics, one of the most challenging aspects of mathematical biology is competition modeling. Although the mathematical idea is simple [1], this type of modeling is so difficult to carry out in any generality since there are so many ways for a population to compete; many classical competitive models have been established to describe the relationships between species and the outer environment, and the connections between different species. Lotka-Volterra competitive model of two species communities is probably the best known model of mathematical ecology. In 1934, Gause [2] found out the competitive exclusion theory, which states that two species that compete for the exact same resources cannot stably coexist. One of the two competitors will always have an ever so slight advantage over the other that leads to extinction of the second competitor in the long run. Since then the competitive model has increasingly won attention as an important and fundament model in biomathematics. The dynamic relationship between species and their competitors has long been one of the dominant theses in both ecology and mathematical ecology. As a consequence, many excellent results concerned with permanence, extinction, stability and hopf bifurcations, and existence and global stability of positive periodic solutions of Lotka-Volterra competitive system are obtained (see [3–16]).

Although much progress has been seen for Lotka-Volterra competitive systems, such systems are not well studied in the sense that most results are continuous time versions related. Many authors [17–21] have argued that the discrete-time models governed by difference equations are more appropriate than the continuous ones when populations have a short life expectancy, nonoverlapping generations in the real word. Discrete-time models can provide efficient computational models of continuous models for numerical simulations. So it is reasonable to study discrete-time competitive systems governed by difference equations.

In this paper, we will consider the dynamic behavior of a discrete-time competitive system. Let us first introduce its continuous time version which is motivated in [22] 𝑥̇₁(𝑡)=𝑥₁(𝑡)[𝑟₁(𝑡)−𝑎₁(𝑡)𝑥₁𝑐(𝑡)−²(𝑡)𝑥₂(𝑡)1+𝑥₂𝑥̇(𝑡)],₂(𝑡)=𝑥₂(𝑡)[𝑟₂(𝑡)−𝑎₂(𝑡)𝑥₂𝑐(𝑡)−¹(𝑡)𝑥₁(𝑡)1+𝑥₁(𝑡)], (1.1) where 𝑥₁(𝑡),𝑥₂(𝑡) are the population densities of two competing species; 𝑟₁(𝑡),𝑟₂(𝑡) are the intrinsic growth rates of species; 𝑎₁(𝑡),𝑎₂(𝑡) are the rates of intraspecific competition of the first and second species, respectively; 𝑐₁(𝑡),𝑐₂(𝑡) are the rates of interspecific competition of the first and second species, respectively. All the coefficients above are continuous and bounded above and below by positive constants.

Following the same idea and method in [21], one can easily derive the discrete analogue of system (1.1), which takes the form of 𝑥₁(𝑛+1)=𝑥₁(𝑛)exp[𝑟₁(𝑛)−𝑎₁(𝑛)𝑥₁𝑐(𝑛)−²(𝑛)𝑥₂(𝑛)1+𝑥₂𝑥(𝑛)],₂(𝑛+1)=𝑥₂(𝑛)exp[𝑟₂(𝑛)−𝑎₂(𝑛)𝑥₂𝑐(𝑛)−¹(𝑛)𝑥₁(𝑛)1+𝑥₁(𝑛)],𝑛=0,1,2…. (1.2) The exponential form of system (1.2) is more biologically reasonable than that directly derived by replacing the differential by difference in system (1.1) because this exponential form can assure 𝑥_𝑖(𝑛+1)>0 if 𝑥_𝑖(0)>0(𝑖=1,2). Here 𝑥_𝑖(𝑛) represent the densities of species 𝑥_𝑖 at the 𝑛th generation, 𝑟_𝑖(𝑛) are the intrinsic growth rates of species 𝑥_𝑖 at the 𝑛th generation, 𝑎_𝑖(𝑛) measure the intraspecific effects of the 𝑛th generation of species 𝑥_𝑖 on own population, and 𝑐_𝑖(𝑛) stand for the interspecific effects of the 𝑛th generation of species 𝑥_𝑖 on species 𝑥_𝑗(𝑖,𝑗=1,2;𝑖≠𝑗).

It is well known that, compared to the continuous time systems, the discrete-time ones are more difficult to deal with. The principle aim of this paper is to explore the permanence, existence, and global stability of positive periodic solutions of system (1.2). To the best of our knowledge, no work has been done for system (1.2).

For the sake of simplicity and convenience in the following discussion, the notations below will be used through this paper: 𝑓^𝑈=sup_𝑛∈ℕ𝑓(𝑛),𝑓^𝐿=inf_𝑛∈ℕ𝑓(𝑛), (1.3) where {𝑓(𝑛)} is a bounded sequence and ℕ is the set of nonnegative integer numbers.

For biological reasons, in system (1.2) we only consider the solution {𝑥₁(𝑛),𝑥₂(𝑛)} with the initial value {𝑥₁(0),𝑥₂(0)}>0.

The organization of this paper is as follows. In the next section, we establish the permanence of system (1.2). In Section 3, we obtain sufficient conditions which ensure the existence and global stability of positive periodic solutions of system (1.2). Numerical simulations are present to illustrate the feasibility of our main results in final section.

2. Permanence

In this section, we will establish sufficient conditions for the permanence of system (1.2).

Definition 2.1. System (1.2) is said to be permanent if there exist positive constants 𝑚_𝑖 and 𝑀_𝑖 such that each positive solution {𝑥₁(𝑛),𝑥₂(𝑛)} of system (1.2) satisfies 𝑚_𝑖≤liminf_𝑛→+∞𝑥_𝑖(𝑛)≤limsup_𝑛→+∞𝑥_𝑖(𝑛)≤𝑀_𝑖,𝑖=1,2. (2.1)

Proposition 2.2. Any positive solution {𝑥₁(𝑛),𝑥₂(𝑛)} of system (1.2) satisfies limsup_𝑛→+∞𝑥_𝑖(𝑛)≤𝑀=_𝑖^def[exp(𝑟^𝑈_𝑖−1)]𝑎^𝐿_𝑖,𝑖=1,2. (2.2)

Proof. To prove Proposition 2.2, we consider Case 1 and Case 2.
Case 1. Assume that there exists an 𝑛₀∈ℕ such that 𝑥₁(𝑛₀+1)≥𝑥₁(𝑛₀), from the first equation of system (1.2), it follows that 𝑟₁(𝑛₀)−𝑎₁(𝑛₀)𝑥₁(𝑛₀𝑐)−²(𝑛₀)𝑥₂(𝑛₀)1+𝑥₂(𝑛₀)≥0, (2.3) which implies, 𝑥₁(𝑛₀𝑟)≤¹(𝑛₀)𝑎₁(𝑛₀)≤𝑟^𝑈₁𝑎^𝐿₁. (2.4) Then 𝑥₁(𝑛₀+1)=𝑥₁(𝑛₀)exp[𝑟₁(𝑛₀)−𝑎₁(𝑛₀)𝑥₁(𝑛₀𝑐)−²(𝑛₀)𝑥₂(𝑛₀)1+𝑥₂(𝑛₀)]≤𝑥₁(𝑛₀)exp[𝑟^𝑈₁−𝑎^𝐿₁𝑥₁(𝑛₀≤)][exp(𝑟^𝑈₁−1)]^𝐿_1d𝑎_ef=𝑀₁, (2.5) where we use the fact that max_𝑥∈ℝ[𝑥exp(𝑎−𝑏𝑥)]=[exp(𝑎−1)]/𝑏for𝑎,𝑏>0, and ℝ is the set of all real numbers. Hence 𝑥₁(𝑛₀)≤𝑀₁.
We claim that 𝑥₁(𝑛)≤𝑀₁ for all 𝑛≥𝑛₀. By way of contradiction, assume that there exists a 𝑝₀>𝑛₀ such that 𝑥₁(𝑝₀)>𝑀₁, then 𝑝₀≥𝑛₀+2. Let 𝑝₀=min{𝑝₀∶𝑝₀≥𝑛₀+2,𝑥₁(𝑝₀)>𝑀₁}, (2.6) that is, 𝑥₁(𝑝₀)>𝑀₁ and 𝑝₀≥𝑛₀+2, then 𝑥₁(𝑝₀)>𝑀₁≥𝑥₁(𝑝₀−1). It is easy to obtain that 𝑥₁(𝑝₀)≤𝑀₁ from the above argument, which is a contradiction. Therefore, 𝑥₁(𝑛)≤𝑀₁ for all 𝑛≥𝑛₀, then limsup_𝑛→+∞𝑥₁(𝑛)≤𝑀₁. This proves the claim.
Case 2. Suppose that 𝑥₁(𝑛+1)<𝑥₁(𝑛) for all 𝑛∈ℕ. In particular, lim_𝑛→+∞𝑥₁(𝑛) exists, denoted by 𝑥₁, we will prove 𝑥₁≤𝑟^𝑈₁/𝑎^𝐿₁ by way of contradiction as follows. Assume that 𝑥₁>𝑟^𝑈₁/𝑎^𝐿₁, taking limit in the first equation of system (1.2), which leads to lim_𝑛→+∞[𝑟₁(𝑛)−𝑎₁(𝑛)𝑥₁𝑐(𝑛)−²(𝑛)𝑥₂(𝑛)1+𝑥₂(𝑛)]=0, (2.7) however, lim_𝑛→+∞[𝑟₁(𝑛)−𝑎₁(𝑛)𝑥₁𝑐(𝑛)−²(𝑛)𝑥₂(𝑛)1+𝑥₂](𝑛)≤lim_𝑛→+∞[𝑟₁(𝑛)−𝑎₁(𝑛)𝑥₁(𝑛)]≤𝑟^𝑈₁−𝑎^𝐿₁𝑥₁<0, (2.8) which is a contradiction. This proves the claim. By the fact that min_𝑥∈ℝ⁺{[exp(𝑥−1)]/𝑥}=1, we obtain that 𝑥₁≤𝑟^𝑈₁/𝑎^𝐿₁≤𝑟^𝑈₁/𝑎^𝐿₁⋅exp(𝑟^𝑈₁−1)/𝑟=^𝑈₁^def𝑀₁. Therefore, limsup_𝑛→+∞𝑥₁(𝑛)≤𝑀₁=[exp(𝑟^𝑈₁−1)]𝑎^𝐿₁. (2.9) Analogously, limsup_𝑛→+∞𝑥₂(𝑛)≤𝑀₂=[exp(𝑟^𝑈₂−1)]𝑎^𝐿₂. (2.10) This completes the proof of Proposition 2.2.

Proposition 2.3. Suppose that system (1.2) satisfies the following assumptions: 𝑟^𝐿₁>𝑐^𝑈₂,𝑟^𝐿₂>𝑐^𝑈₁. (2.11) Then any positive solution {𝑥₁(𝑛),𝑥₂(𝑛)} of system (1.2) satisfies liminf_𝑛→+∞𝑥_𝑖(𝑛)≥𝑚=_𝑖^def𝑟^𝐿_𝑖−𝑐^𝑈_𝑗𝑎^𝑈_𝑖exp(𝑟^𝐿_𝑖−𝑎^𝑈_𝑖𝑀_𝑖−𝑐^𝑈_𝑗),𝑖≠𝑗;𝑖,𝑗=1,2. (2.12)

Proof. By Proposition 2.2, since limsup_𝑛→+∞𝑥₁(𝑛)≤𝑀₁ for each 𝜀>0, then there exists an 𝑛^∗∈ℕ such that 𝑥₁(𝑛)≤𝑀₁+𝜀 for 𝑛≥𝑛^∗. In order to prove Proposition 2.3, there are two cases to be considered as follows.
Case 1. Assume that there exists an 𝑛₀≥𝑛^∗ such that 𝑥₁(𝑛₀+1)≤𝑥₁(𝑛₀), by the first equation of system (1.2), it derives that 𝑥₁(𝑛₀+1)=𝑥₁(𝑛₀)exp[𝑟₁(𝑛₀)−𝑎₁(𝑛₀)𝑥₁(𝑛₀𝑐)−²(𝑛₀)𝑥₂(𝑛₀)1+𝑥₂(𝑛₀)]≥𝑥₁(𝑛₀)exp[𝑟^𝐿₁−𝑎^𝑈₁𝑥(𝑛₀)−𝑐^𝑈₂], (2.13) therefore, 𝑟^𝐿₁−𝑎^𝑈₁𝑥(𝑛₀)−𝑐^𝑈₂≤0. (2.14) It follows from the inequality (2.11) that 𝑥₁(𝑛₀)≥(𝑟^𝐿₁−𝑐^𝑈₂)𝑎^𝑈₁>0. (2.15) By (2.13) and (2.15) we have 𝑥₁(𝑛₀+1)=𝑥₁(𝑛₀)exp[𝑟₁(𝑛₀)−𝑎₁(𝑛₀)𝑥₁(𝑛₀𝑐)−²(𝑛₀)𝑥₂(𝑛₀)1+𝑥₂(𝑛₀)]≥𝑟^𝐿₁−𝑐^𝑈₂𝑎^𝑈₁exp[𝑟^𝐿₁−𝑎^𝑈₁(𝑀₁+𝜀)−𝑐^𝑈₂]>0. (2.16) Hence 𝑥₁(𝑛₀)≥𝑥^𝜀₁, where 𝑥=^𝜀₁^def((𝑟^𝐿₁−𝑐^𝑈₂)/𝑎^𝑈₁)exp[𝑟^𝐿₁−𝑎^𝑈₁(𝑀₁+𝜀)−𝑐^𝑈₂].
In the following we will prove 𝑥₁(𝑛)≥𝑥^𝜀₁ for all 𝑛≥𝑛₀. By way of contradiction, assume that there exists a 𝑝₀>𝑛₀ such that 𝑥₁(𝑝₀)<𝑥^𝜀₁, then 𝑝₀≥𝑛₀+2. Let 𝑝₀=min{𝑝₀∶𝑝₀≥𝑛₀+2,𝑥₁(𝑝₀)<𝑥^𝜀₁}, (2.17) that is, 𝑥₁(𝑝₀)<𝑥^𝜀₁ and 𝑝₀≥𝑛₀+2, then 𝑥₁(𝑝₀)<𝑥^𝜀₁≤𝑥₁(𝑝₀−1), the above argument produces that 𝑥₁(𝑝₀)≥𝑥^𝜀₁, which is a contradiction. Therefore, 𝑥₁(𝑛)≥𝑥^𝜀₁forall𝑛≥𝑛₀, since 𝜀 can be sufficiently small, it gives that 𝑥₁𝑟(𝑛)≥^𝐿1−𝑐^𝑈₂𝑎^𝑈₁exp(𝑟^𝐿₁−𝑎^𝑈₁𝑀₁−𝑐^𝑈₂)=^def𝑚₁>0, (2.18) then liminf_𝑛→+∞𝑥₁(𝑛)≥𝑚₁. This proves the claim.
Case 2. Assume that 𝑥₁(𝑛+1)>𝑥₁(𝑛) for a sufficiently large 𝑛≥𝑛^∗. In this case, lim_𝑛→+∞𝑥₁(𝑛) exists, denoted by 𝑥₁. For the sake of proving 𝑥₁≥(𝑟^𝐿₁−𝑐^𝑈₂)/𝑎^𝑈₁, by way of contradiction, assume that 𝑥₁<(𝑟^𝐿₁−𝑐^𝑈₂)/𝑎^𝑈₁ , taking limit in the first equation of system (1.2), it follows that lim_𝑛→+∞[𝑟₁(𝑛)−𝑎₁(𝑛)𝑥₁𝑐(𝑛)−²(𝑛)𝑥₂(𝑛)1+𝑥₂(𝑛)]=0, (2.19) however, lim_𝑛→+∞[𝑟₁(𝑛)−𝑎₁(𝑛)𝑥₁𝑐(𝑛)−²(𝑛)𝑥₂(𝑛)1+𝑥₂(𝑛)]≥lim_𝑛→+∞[𝑟₁(𝑛)−𝑎₁(𝑛)𝑥₁(𝑛)−𝑐₂(𝑛)]≥𝑟^𝐿₁−𝑎^𝑈₁𝑥₁−𝑐^𝑈₂>0, (2.20) which is a contradiction. It implies that 𝑥₁≥(𝑟^𝐿₁−𝑐^𝑈₂)/𝑎^𝑈₁. By the fact that min_𝑥∈ℝ⁺{[exp(𝑥−1)]/𝑥}=1, we obtain that 𝑀₁=[exp(𝑟^𝑈₁−1)]/𝑎^𝐿₁≥𝑟^𝑈₁/𝑎^𝐿₁≥𝑟^𝐿₁/𝑎^𝑈₁. Thus 𝑥₁>(𝑟^𝐿₁−𝑐^𝑈₁)/𝑎^𝑈₁≥𝑥^𝜀₁. Therefore, liminf_𝑛→+∞𝑥₁(𝑛)≥𝑥^𝜀₁. Since 𝜀 can be sufficiently small, we have liminf_𝑛→+∞𝑥₁(𝑛)≥𝑚=₁^def𝑟^𝐿₁−𝑐^𝑈₂𝑎^𝑈₁exp(𝑟^𝐿₁−𝑎^𝑈₁𝑀₁−𝑐^𝑈₂)>0. (2.21) Analogously, by the second inequality in (2.11), we can obtain that liminf_𝑛→+∞𝑥₂(𝑛)≥𝑚=₂^def𝑟^𝐿₂−𝑐^𝑈₁𝑎^𝑈₂exp(𝑟^𝐿₂−𝑎^𝑈₂𝑀₂−𝑐^𝑈₁)>0. (2.22) This completes the proof of Proposition 2.3.

Now, we are in a position to state Theorem 2.4 whose proof is a direct consequence of Propositions 2.2 and 2.3.

Theorem 2.4. If the inequalities in (2.11) hold, then system (1.2) is permanent.

3. Existence and Global Stability of Positive Periodic Solutions

In this section, we suppose system (1.2) is a periodic system, and then we investigate the existence and global stability of positive periodic solutions of such system. To do this, assume that all the coefficients of system (1.2) are 𝜔-periodic, in other words, 𝑟_𝑖(𝑛+𝜔)=𝑟_𝑖(𝑛),𝑎_𝑖(𝑛+𝜔)=𝑎_𝑖(𝑛),𝑐_𝑖(𝑛+𝜔)=𝑐_𝑖(𝑛),𝑖=1,2. (3.1)

Theorem 3.1. If the inequalities in (2.11) hold, then system (1.2) has at least one strictly positive 𝜔-periodic solution, denoted by {𝑥^∗₁(𝑛),𝑥^∗₂(𝑛)}.

Proof. We know that 𝐾=[𝑚₁,𝑀₁]×[𝑚₂,𝑀₂] is an invariant set of system (1.2) from Propositions 2.2 and 2.3. Define the continuous mapping 𝐹 on 𝐾𝐹{𝑥^∗₁(0),𝑥^∗₂(0)}={𝑥^∗₁(𝜔),𝑥^∗₂(𝜔)},for{𝑥^∗₁(0),𝑥^∗₂(0)}∈𝐾. (3.2) Obviously, 𝐹 depends continuously on {𝑥^∗₁(0),𝑥^∗₂(0)}, then 𝐹 is continuous and maps the compact set 𝐾=[𝑚₁,𝑀₁]×[𝑚₂,𝑀₂] into itself. Therefore, 𝐹 has a fixed point {𝑥^∗₁,𝑥^∗₂}. It is easy to see that the solution {𝑥^∗₁(𝑛),𝑥^∗₂(𝑛)} which passes through {𝑥^∗₁,𝑥^∗₂} is an 𝜔-periodic solution of system (1.2). The proof is complete.

Next, we derive sufficient conditions which guarantee that the positive periodic solution of system (1.2) is globally stable. We first give the definition of global stability.

Definition 3.2. A positive periodic solution {𝑥^∗₁(𝑛),𝑥^∗₂(𝑛)} of system (1.2) is globally stable if each other solution {𝑥₁(𝑛),𝑥₂(𝑛)} of system (1.2) with positive initial value defined for all 𝑛>0 satisfies lim_𝑛→+∞|𝑥₁(𝑛)−𝑥^∗₁(𝑛)|=0,lim_𝑛→+∞|𝑥₂(𝑛)−𝑥^∗₂(𝑛)|=0. (3.3)

Now, we give the main result in this section.

Theorem 3.3. In addition to (2.11), assume further that the following assumptions 𝜆=₁^defmax{|1−𝑎^𝐿₁𝑚₁|,|1−𝑎^𝑈₁𝑀₁|}+𝑐^𝑈₂𝜆<1,_2def=max{|1−𝑎^𝐿₂𝑚₂|,|1−𝑎^𝑈₂𝑀₂|}+𝑐^𝑈₁<1 (3.4)(3.5) hold. Then the positive periodic solution of system (1.2) is globally stable.

Proof. Let {𝑥^∗₁(𝑛),𝑥^∗₂(𝑛)} be a positive periodic solution of system (1.2). We make the change of variables 𝑥₁(𝑛)=𝑥^∗₁(𝑛)exp𝑢(𝑛) and 𝑥₂(𝑛)=𝑥^∗₂(𝑛)exp𝑣(𝑛), then system (1.2) is rewritten as 𝑢(𝑛+1)−𝑢(𝑛)=𝑎₁(𝑛)𝑥^∗₁(𝑛)[1−exp𝑢(𝑛)]+𝑐₂1(𝑛){1+𝑥^∗₂−1(𝑛)exp[𝑣(𝑛)]1+𝑥^∗₂(𝑛)},𝑣(𝑛+1)−𝑣(𝑛)=𝑎₂(𝑛)𝑥^∗₂(𝑛)[1−exp𝑣(𝑛)]+𝑐₁1(𝑛){1+𝑥^∗₁−1(𝑛)exp[𝑢(𝑛)]1+𝑥^∗₁(𝑛)}. (3.6) By the mean-value theorem, it derives that 𝑢(𝑛+1)=𝑢(𝑛){1−𝑎₁(𝑛)𝑥^∗₁(𝑛)exp[𝜃₁𝑢(𝑛)]}−𝑐₂𝑥(𝑛)𝑣(𝑛)^∗2(𝑛)exp[𝜃₂𝑣(𝑛)]{1+𝑥^∗₂(𝑛)exp[𝜃₂𝑣(𝑛)]}²,𝑣(𝑛+1)=𝑣(𝑛){1−𝑎₂(𝑛)𝑥^∗₂(𝑛)exp[𝜃₃𝑣(𝑛)]}−𝑐₁𝑥(𝑛)𝑢(𝑛)^∗1(𝑛)exp[𝜃₄𝑢(𝑛)]{1+𝑥^∗₁(𝑛)exp[𝜃₄𝑢(𝑛)]}², (3.7) where the constants 𝜃₁,𝜃₂,𝜃₃,𝜃₄∈(0,1).
Now, by (3.4) and (3.5), we choose the constant 𝜀 sufficiently small such that 𝜆^𝜀₁=max{|1−𝑎^𝐿₁(𝑚₁−𝜀)|,|1−𝑎^𝑈₁(𝑀₁+𝜀)|}+𝑐^𝑈₂𝜆<1,_𝜀2=max{|1−𝑎^𝐿₂(𝑚₂−𝜀)|,|1−𝑎^𝑈₂(𝑀₂+𝜀)|}+𝑐^𝑈₁<1. (3.8) In view of Propositions 2.2 and 2.3, there exists an 𝑛₀∈𝑁 such that 𝑛≥𝑛₀, we have 0<𝑚₁−𝜀≤𝑥^∗₁(𝑛)≤𝑀₁+𝜀,0<𝑚₁−𝜀≤𝑥₁(𝑛)≤𝑀₁+𝜀,0<𝑚₂−𝜀≤𝑥^∗₂(𝑛)≤𝑀₂+𝜀,0<𝑚₂−𝜀≤𝑥₂(𝑛)≤𝑀₂+𝜀. (3.9) Since 𝜃₁,𝜃₂,𝜃₃,𝜃₄∈(0,1), both 𝑥^∗₁(𝑛)exp[𝜃₁𝑢(𝑛)] and 𝑥^∗₁(𝑛)exp[𝜃₄𝑢(𝑛)] are between 𝑥₁(𝑛) and 𝑥^∗₁(𝑛). Meanwhile, both 𝑥^∗₂(𝑛)exp[𝜃₂𝑣(𝑛)] and 𝑥^∗₂(𝑛)exp[𝜃₃𝑣(𝑛)] are between 𝑥₂(𝑛) and 𝑥^∗₂(𝑛). From the first equation of system (3.7), it follows that |𝑢(𝑛+1)|≤|𝑢(𝑛)||1−𝑎₁(𝑛)𝑥^∗₁(𝑛)exp[𝜃₁𝑢(𝑛)]|+|𝑐₂𝑥(𝑛)𝑣(𝑛)^∗2(𝑛)exp[𝜃₂𝑣(𝑛)]1+𝑥^∗₂(𝑛)exp[𝜃₂|𝑣(𝑛)]≤max{|1−𝑎^𝐿₁(𝑚₁−𝜀)|,|1−𝑎^𝑈₁(𝑀₁+𝜀)|}|𝑢(𝑛)|+max{|𝑐₂(𝑛)𝑣(𝑛)|}≤max{|1−𝑎^𝐿₁(𝑚₁−𝜀)|,|1−𝑎^𝑈₁(𝑀₁+𝜀)|}|𝑢(𝑛)|+𝑐^𝑈₂|𝑣(𝑛)|≤𝜆^𝜀₁max{|𝑢(𝑛)|,|𝑣(𝑛)|}. (3.10) Similarly, it follows from (3.5) that |𝑣(𝑛+1)|≤𝜆^𝜀₂max{|𝑢(𝑛)|,|𝑣(𝑛)|}. (3.11) Denote 𝜆^𝜀=max{𝜆^𝜀₁,𝜆^𝜀₂}, then 𝜆^𝜀<1. Therefore, when 𝑛≥𝑛₀, max{|𝑢(𝑛+1)|,|𝑣(𝑛+1)|}≤𝜆^𝜀max{|𝑢(𝑛)|,|𝑣(𝑛)|}≤(𝜆^𝜀)^𝑛−𝑛0max{|𝑢(𝑛₀)|,|𝑣(𝑛₀)|}, (3.12) as a consequence, lim_𝑛→+∞|𝑥₁(𝑛)−𝑥^∗₁(𝑛)|=0,lim_𝑛→+∞|𝑥₂(𝑛)−𝑥^∗₂(𝑛)|=0. By using Definition 3.2, it follows that the positive periodic solution {𝑥^∗₁(𝑛),𝑥^∗₂(𝑛)} of system (1.2) is globally stable. This completes the proof.

Remark 3.4. Theorem 3.3 shows that {𝑥^∗₁(𝑛),𝑥^∗₂(𝑛)} is the global attractor of all positive solutions of system (1.2), then {𝑥^∗₁(𝑛),𝑥^∗₂(𝑛)} is the unique positive periodic solution of system (1.2).

4. Example and Numerical Simulation

In this paper, we have investigated the permanence and global stability of positive periodic solutions of a discrete competitive system. Each species is not isolated from its living environment, but competes with the other for the same resource. Sufficient conditions which guarantee the permanence, existence and global stability of positive periodic solutions are established, respectively. The theoretical results are confirmed by the following examples and their numerical results.

To verify the sufficient conditions for permanence of system (1.2), we assume that 𝑟₁(𝑛)=0.46+0.01sin𝜋𝑛, 𝑟₂(𝑛)=0.88−0.02sin𝜋𝑛, 𝑎₁(𝑛)=0.75+0.01cos𝜋𝑛, 𝑎₂(𝑛)=0.52+0.01sin𝜋𝑛, 𝑐₁(𝑛)=0.28+0.02sin𝜋𝑛, 𝑐₂(𝑛)=0.13−0.01sin𝜋𝑛, and the initial condition (𝑥₁(0),𝑥₂(0))=(0.25,0.01). Clearly, (2.11) in Theorem 2.4 are satisfied, and hence system (1.2) is permanent (see Figure 1).

(a)

(b)

Figure 1 

Permanence of system (1.2) with initial value (0.25,0.01). (a) Time series of 𝑥1 for 𝑛∈[0,50]. (b) Time series of 𝑥2 for 𝑛∈[0,50].

Now, we further verify the sufficient conditions for the existence and global stability of positive periodic solutions of periodic system (1.2). Let us assume that all the coefficients of system (1.2) are periodic and listed in Table 1.

Table 1 

The coefficient values when 𝐧 is under different conditions.

Besides, we choose the positive periodic solution with initial values (0.09,0.01), denoted by (𝑥^∗₁,𝑥^∗₂), and the positive solution with initial value (0.08,0.02) denoted by (𝑥₁,𝑥₂). By Theorem 3.3, a simple calculation shows that the assumptions in (3.4) and (3.5) hold. So from Theorem 3.3 we know that periodic system (1.2) has a positive 2-periodic solution which is globally stable. From Figures 2(a), 2(b), and 2(c), we see that 𝑥₁ with 𝑥₁(0)=0.08 will tend to 𝑥^∗₁ with 𝑥^∗₁(0)=0.09. Similarly, from Figures 3(a), 3(b), and 3(c), we see that 𝑥₂ with 𝑥₂(0)=0.02 will tend to 𝑥^∗₂ with 𝑥^∗₂(0)=0.01. Furthermore, Figures 4(a) and 4(b) show the phase portrait of periodic system (1.2) with (𝑥^∗₁(0),𝑥^∗₂(0))=(0.09,0.01) for 𝑛∈[0,50] and (𝑥₁(0),𝑥₂(0))=(0.08,0.02) for 𝑛∈[0,50], respectively. From Figure 4(c), we can see that periodic system (1.2) has a positive 2-periodic solution which is globally stable.

(a)

(b)

(c)

Figure 2 

(a) Time series of 𝑥∗1 with 𝑥∗1(0)=0.09 for 𝑛∈[0,50]. (b) Time series of 𝑥1 with 𝑥1(0)=0.08 for 𝑛∈[0,50]. (c) Time series of 𝑥∗1 and 𝑥1 for 𝑛∈[0,50] in the same coordinate system.

(a)

(b)

(c)

Figure 3 

(a) Time series of 𝑥∗2 with 𝑥∗2(0)=0.01 for 𝑛∈[0,50]. (b) Time series of 𝑥2 with 𝑥2(0)=0.02 for 𝑛∈[0,50]. (c) Time series of 𝑥∗2 and 𝑥2 for 𝑛∈[0,50] in the same coordinate system.

(a)

(b)

(c)

Figure 4 

(a) Phase portrait of 𝑥∗1 and 𝑥∗2 with initial value (0.09,0.01) for 𝑛∈[0,50]. (b) Phase portrait of 𝑥1 and 𝑥2 with initial value (0.08,0.02) for 𝑛∈[0,50]. (c) Phase portraits of 𝑥∗1,𝑥∗2 and 𝑥1,𝑥2 for 𝑛∈[30,50] in the same coordinate system.

5. Acknowledgement

The work is supported by the Innovation Term of Educational Department of Hubei Province in China (T200804), the National Science Foundation of Hubei Province in China (2008CDB068), and the Innovation Project of Hubei Institute for Nationalities for postgraduate students. We would like to thank the Editor Professor A. Vecchio and the referee for careful reading of the original manuscript and valuable comments and suggestions that greatly improved the presentation of this work.