Global Dynamics of Discrete Competitive Models with Large Intrinsic Growth Rates

Abstract

The global dynamics of discrete competitive model of Lotka-Volterra type with two species is considered. Earlier works have shown that the unique positive equilibrium is globally attractive under the assumption that the intrinsic growth rates of the two competitive species are less than 1+ln 2, and further the unique positive equilibrium is globally asymptotically stable under the stronger condition that the intrinsic growth rates of the two competitive species are less than or equal to 1. We prove that the system can also be globally asymptotically stable when the intrinsic growth rates of the two competitive species are greater than 1 and globally attractive when the intrinsic growth rates of the two competitive species are greater than 1 + ln 2.

1. Introduction

In this paper, we further consider the global dynamics of discrete Lotka-Volterra model

𝑥₁(𝑛+1)=𝑥₁(𝑛)exp(𝑟₁−𝑎₁₁𝑥₁(𝑛)−𝑎₁₂𝑥₂𝑥(𝑛)),₂(𝑛+1)=𝑥₂(𝑛)exp(𝑟₂−𝑎₂₁𝑥₁(𝑛)−𝑎₂₂𝑥₂(𝑛)), (1.1) with positive initial conditions 𝑥₁(0),𝑥₂(0)>0. Here 𝑥_𝑖(𝑛)(𝑖=1,2) is the density of population 𝑖 at 𝑛th generation, 𝑟_𝑖(𝑖=1,2) is the intrinsic growth rate of population 𝑖. 𝑎_𝑖𝑗(𝑖,𝑗=1,2) represents the intensity of intraspecific competition or interspecific action of the two species. It is assumed that 𝑟_𝑖 and 𝑎_𝑖𝑗(𝑖,𝑗=1,2) are positive constants throughout this paper.

The discrete Lotka-Volterra models have wide applications in applied mathematics. They were first established in biomathematical background and then have proved to be a rich source in analysis for the dynamical systems in different research fields such as physics, chemistry, and economy [1].

Model (1.1) was first introduced in May [2] then was investigated by many authors [3–14]. The difference system (1.1) is autonomous, some of the works mentioned above are the nonautonomous case of (1.1). Many results about the global dynamics of (1.1) such as permanence, global attractivity, global asymptotical stability have been obtained. For example, it is shown in [10] that (1.1) can be globally asymptotically stable when 𝑟_𝑖≤1(𝑖=1,2). And from [3] we know that (1.1) can be globally attractive under the assumption that 𝑟_𝑖<1+ln2(𝑖=1,2).

It is well known that for the single-species Logistic model 𝑥(𝑛+1)=𝑥(𝑛)exp[𝑟−𝑎𝑥(𝑛)], (1.2) the positive equilibrium 𝑥^∗=𝑟/𝑎 is globally asymptotically stable if and only if 𝑟≤2 and there exists periodic cycles when 𝑟>2. When 𝑟≥3.13, (1.2) exhibits chaotic behavior (e.g., see [15]). That is, the global dynamics of (1.2) is very complex when the intrinsic growth rate 𝑟 is large. It is clear that (1.1) is a coupling of two equations described by (1.2). And it is proved in [16] that (1.1) also exhibits chaotic behavior when 𝑟_𝑖=𝑟≥3.13(𝑖=1,2). Therefore, questions can be proposed naturally: how to investigate the global dynamics of (1.1) when 1+ln2<𝑟_𝑖<3.13(𝑖=1,2)? Can model (1.1) be also globally asymptotically stable when 𝑟_𝑖>1(𝑖=1,2)? Can model (1.1) be globally attractive when 𝑟_𝑖>1+ln2(𝑖=1,2)?

Our aim of this paper is to obtain some global dynamics of (1.1) when the intrinsic growth rate 𝑟_𝑖(𝑖=1,2) is large (𝑟_𝑖≥1,𝑖=1,2) and give answers to the above questions. First we obtain permanent result of (1.1), then global attractivity of (1.1) is obtained through geometrical properties of (1.1). Last, we obtain the global asymptotical stability of (1.1) by applying a theorem in [10]. After these theoretical results for (1.1) obtained, we give numerical examples to confirm these theoretical results and show that our theoretical results imply that (1.1) can be globally attractive when 𝑟_𝑖>1+ln2(𝑖=1,2) and (1.1) can also be globally asymptotically stable when 𝑟_𝑖>1(𝑖=1,2).

The paper is organized as follows. We give some preliminaries in Section 2. In Section 3, we discuss permanence, global attractivity, and global asymptotical stability of (1.1) theoretically. Numerical examples are given in Section 4 to show the feasibility of the assumptions of the main results and on the other hand, to show that our main results can be applied to larger intrinsic growth rates than earlier works. Brief conclusion is given in Section 5.

2. Preliminaries

A pair of sequences of real positive numbers {𝑥₁(𝑛),𝑥₂(𝑛)}^∞_𝑛=1 that satisfies (1.1) is a positive solution of (1.1). It is clear that the solutions of system (1.1) with initial values 𝑥₁(0)>0,𝑥₂(0)>0 are positive ones, which is accordant with the biological background of (1.1). That is, we only need to investigate the dynamics of system (1.1) in the plane domain 𝐺={(𝑥,𝑦)∣𝑥≥0,𝑦≥0}. (2.1)

If a solution of (1.1) is a pair of real constants {𝑥₁,𝑥₂}, then it is an equilibrium of (1.1).

Lemma 2.1. Assume that 𝐷≜𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁≠0, (2.2) then system (1.1) has four equilibria.

Proof. Solving the following scalar equation system: 𝑥₁=𝑥₁exp(𝑟₁−𝑎₁₁𝑥₁−𝑎₁₂𝑥₂),𝑥₂=𝑥₂exp(𝑟₂−𝑎₂₁𝑥₁−𝑎₂₂𝑥₂). (2.3) We obtain that the four equilibria of system (1.1) are 𝑟(0,0),(¹𝑎₁₁𝑟,0),(0,²𝑎₂₂),(𝑥^∗₁,𝑥^∗₂), (2.4) respectively. Here and the following, we denote 𝑥^∗₁≜𝐷₁𝐷,𝑥^∗₂≜𝐷₂𝐷,𝐷₁≜𝑟₁𝑎₂₂−𝑟₂𝑎₁₂,𝐷₂≜𝑟₂𝑎₁₁−𝑟₁𝑎₂₁. (2.5)

The equilibria (0,0),(𝑟₁/𝑎₁₁,0) and (0,𝑟₂/𝑎₂₂) are the so-called “boundary equilibrium.” If we further assume that 𝐷₁>0,𝐷₂>0, (2.6) which implies that 𝐷>0, then (𝑥^∗₁,𝑥^∗₂) is the unique positive equilibrium of (1.1).

Lemma 2.2. Denote 𝑓(𝑥,𝑦)=𝑥exp(𝑟₁−𝑎₁₁𝑥−𝑎₁₂𝑦), 𝑟₁≥1, then the maximum 𝑀₁ of 𝑓(𝑥,𝑦) in the domain 𝐺₁≜{(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑟₁−𝑎₁₁𝑥−𝑎₁₂𝑦≤0,𝑟₂−𝑎₂₁𝑥−𝑎₂₂𝑦≤0} (2.7) is (1)if𝑎₂₂≤𝐷₁𝑜𝑟𝑎₂₂/𝐷≥𝑟₁/𝑎₂₁,then𝑀₁=max{(𝑟₂/𝑎₂₁)exp(𝑟₁−𝑎₁₁(𝑟₂/𝑎₂₁)),𝐷₁/𝐷},(2)if𝐷₁/𝐷<𝑎₂₂/𝐷<𝑟₁/𝑎₂₁,then𝑀₁=(𝑎₂₂/𝐷)exp((𝐷₁/𝑎₂₂)−1).Denote 𝑔(𝑥,𝑦)=𝑦exp(𝑟₂−𝑎₂₁𝑥−𝑎₂₂𝑦), 𝑟₂≥1, then the maximum 𝑀₂ of 𝑔(𝑥,𝑦) in domain 𝐺₁ is(1)if𝑎₁₁≤𝐷₂𝑜𝑟𝑎₁₁/𝐷≥𝑟₂/𝑎₁₂,then𝑀₂=max{(𝑟₁/𝑎₁₂)exp(𝑟₂−𝑎₂₂(𝑟₁/𝑎₁₂)),𝐷₂/𝐷},(2)if𝐷₂/𝐷<𝑎₁₁/𝐷<𝑟₂/𝑎₁₂,then𝑀₂=(𝑎₁₁/𝐷)exp((𝐷₂/𝑎₁₁)−1).

Proof. For any fixed 𝑥(or𝑦), let 𝑦→+∞(or𝑥→+∞) we get 𝑓(𝑥,𝑦)→0. Note that lim_{𝑥→+∞,𝑦→+∞}𝑓(𝑥,𝑦)=0, (2.8) therefore, the maximum of 𝑓(𝑥,𝑦) in domain 𝐺₁ exists. Direct computation gives 𝑀₁, we omit the details. Similarly, 𝑀₂ exists and its value can be obtained directly.

Lemma 2.3. (1) If 𝐹: 𝑅⁺=[0,+∞)→𝑅  is monotonously increasing, then for each positive sequence {𝑦(𝑛)}^∞_𝑛=1, limsup_𝑛→∞𝐹(𝑦(𝑛))=𝐹(limsup_𝑛→∞𝑦(𝑛)),liminf_𝑛→∞𝐹(𝑦(𝑛))=𝐹(liminf_𝑛→∞𝑦(𝑛)). (2.9) If 𝐹: 𝑅⁺=[0,+∞)→𝑅is monotonously decreasing, then for each positive sequence {𝑦(𝑛)}^∞_𝑛=1, limsup_𝑛→∞𝐹(𝑦(𝑛))=𝐹(liminf_𝑛→∞𝑦(𝑛)),liminf_𝑛→∞𝐹(𝑦(𝑛))=𝐹(limsup_𝑛→∞𝑦(𝑛)). (2.10)
(2) For any positive sequences {𝑦(𝑛)}^∞_𝑛=1,{𝑧(𝑛)}^∞_𝑛=1 one has limsup_𝑛→∞[𝑦(𝑛)𝑧(𝑛)]≤limsup_𝑛→∞𝑦(𝑛)limsup_𝑛→∞𝑧(𝑛),liminf_𝑛→∞[𝑦(𝑛)𝑧(𝑛)]≥liminf_𝑛→∞𝑦(𝑛)liminf_𝑛→∞𝑧(𝑛). (2.11)

Proof. One can refer to [17] for the proof of this lemma.

Next we give some definitions that will be used in this paper.

Definition 2.4. System (1.1) is permanent if there exist positive constants 𝑚 and 𝑀 such that 𝑚≤liminf_𝑛→∞𝑥_𝑖(𝑛)≤limsup_𝑛→∞𝑥_𝑖(𝑛)≤𝑀,𝑖=1,2. (2.12)

Definition 2.5. System (1.1) is strongly persistent if each positive solution of (1.1) satisfies liminf_𝑛→∞𝑥_𝑖(𝑛)>0,𝑖=1,2. (2.13)

Definition 2.6. The solution {𝑥₁(𝑛),𝑥₂(𝑛)} of system (1.1) with initial values 𝑥₁(0)>0, 𝑥₂(0)>0 is said to be stable if for any 𝜀>0, there is a 𝛿>0 such that if max{|𝑥₁(0)−𝑥₁(0)|,|𝑥₂(0)−𝑥₂(0)|}<𝛿, we have |𝑥₁(𝑛)−𝑥₁(𝑛)|+|𝑥₂(𝑛)−𝑥₂(𝑛)|<𝜀 for all positive integers 𝑛, where {𝑥₁(𝑛),𝑥₂(𝑛)} is the solution of (1.1) with initial values 𝑥₁(0)>0,𝑥₂(0)>0.

Definition 2.7. Suppose that {𝑥^∗₁,𝑥^∗₂} is the positive equilibrium solution of (1.1). If for each positive solution {𝑥₁(𝑛),𝑥₂(𝑛)} of system (1.1), we have max{|𝑥^∗₁−𝑥₁(𝑛)|,|𝑥^∗₂−𝑥₂(𝑛)|}→0 as 𝑛→∞, we say (1.1) is globally attractive or the equilibrium {𝑥^∗₁,𝑥^∗₂} of (1.1) is globally  attractive.

Definition 2.8. The positive equilibrium solution {𝑥^∗₁,𝑥^∗₂} of (1.1) or system (1.1) is said to be globally asymptotically stable if this equilibrium is stable and globally attractive.
The following lemma can be found in [10].

Lemma 2.9. Consider the following difference system: 𝑥_𝑖(𝑛+1)=𝑥_𝑖(𝑛)exp(𝑟_𝑖(𝑛)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑛)𝑥_𝑗(𝑛)),𝑖=1,2,…,𝑙. (2.14) Assume that (i)there exist positive constant 𝜈 and positive constants 𝑐_𝑖 such that 𝑐_𝑖𝑎_𝑖𝑖(𝑛)>^𝑙_{𝑗=1,𝑗≠𝑖}𝑐_𝑗|𝑎_𝑗𝑖(𝑛)|+𝜈,𝑖=1,2,…,𝑙, (2.15) for all large 𝑛;(ii)system (2.14) is strongly persistent;(iii)for any positive solution{𝑥₁(𝑛),𝑥₂(𝑛),…,𝑥_𝑙(𝑘)} of system (2.14), 𝑎_𝑖𝑖(𝑛)𝑥_𝑖(𝑛)≤1,𝑖=1,2,…,𝑙, (2.16) for all large𝑛.
Then system (2.14) is globally asymptotically stable.

3. Main Results

In this section, we will obtain the permanence, global attractivity, and global asymptotical stability of system (1.1) when 𝑟_𝑖≥1(𝑖=1,2).

Lemma 3.1. For every positive solution {x₁(n),x₂(n)} of system (1.1) with initial values x₁(0)>0,x₂(0)>0, one has limsup_n→∞x₁(n)≤s₁,limsup_n→∞x₂(n)≤s₂, (3.1) where s₁=1a₁₁exp(r₁−1),s₂=1a₂₂exp(r₂−1). (3.2)

Proof. Note that 𝑥₁(𝑛+1)=𝑥₁(𝑛)exp(𝑟₁−𝑎₁₁𝑥₁(𝑛)−𝑎₁₂𝑥₂(𝑛))=𝑥₁(𝑛)exp(𝑟₁−𝑎₁₁𝑥₁(𝑛))exp(−𝑎₁₂𝑥₂(𝑛)),exp(−𝑎₁₂𝑥₂(𝑛))≤1 (3.3) for all 𝑛, therefore 𝑥₁(𝑛+1)≤𝑥₁(𝑛)exp(𝑟₁−𝑎₁₁𝑥₁1(𝑛))≤𝑎₁₁exp(𝑟₁−1). (3.4) Here we used max_𝑥≥01𝑥exp(𝑟−𝑎𝑥)=𝑎exp(𝑟−1) (3.5) for 𝑎>0. Then limsup_𝑛→∞𝑥₁1(𝑛)≤𝑎₁₁exp(𝑟₁−1). (3.6) The proof of limsup_𝑛→∞𝑥₂1(𝑛)≤𝑎₂₂exp(𝑟₂−1) (3.7) is similar.

Lemma 3.2. Assume that {𝑥₁(𝑛),𝑥₂(𝑛)} is the solution of (1.1) with initial values 𝑥₁(0)>0,𝑥₂(0)>0  and 1𝑎₁₁exp(𝑟₁𝑟−1)<²𝑎₂₁,1𝑎₂₂exp(𝑟₂𝑟−1)<¹𝑎₁₂, (3.8) then liminf_𝑛→∞𝑥₁(𝑛)≥𝑡₁>0,liminf_𝑛→∞𝑥₂(𝑛)≥𝑡₂>0, (3.9) where 𝑡₁=𝑟₁𝑎₁₁𝑎(1−¹²𝑟₁𝑠₂)exp(𝑟₁−𝑎₁₂𝑠₂−𝑎₁₁𝑟₁𝑠₁𝑡),₂=𝑟₂𝑎₂₂𝑎(1−²¹𝑟₂𝑠₁)exp(𝑟₂−𝑎₂₁𝑠₁−𝑎₂₂𝑟₂𝑠₂), (3.10) and 𝑠₁,𝑠₂ are the same as in Lemma 3.1.

Proof. The proof of this lemma is similar to that of [3, Proposition 2].

Note that 𝑡₁>0,𝑡₂>0, therefore, system (1.1) is permanent from Lemmas 3.1 and 3.2 under the assumption (3.8).

Theorem 3.3. Assume that (3.8) is satisfied then system (1.1) with initial values 𝑥₁(0)>0,𝑥₂(0)>0  is permanent.

Theorem 3.4. Assume that (2.6), and (3.8) hold. The coefficients of (1.1) satisfy 𝑟_𝑖≥1(𝑖=1,2)  and (1)𝑎₂₂≤𝐷₁𝑜𝑟𝑎₂₂/𝐷≥𝑟₁/𝑎₂₁,(2)𝑎₁₁≤𝐷₂𝑜𝑟𝑎₁₁/𝐷≥𝑟₂/𝑎₁₂.Further, assume that 𝑀₁≤𝐷₁𝐷≠1𝑎₁₁,𝑀₂≤𝐷₂𝐷≠1𝑎₂₂, (3.11) where 𝑀₁  and 𝑀₂  are defined in Lemma 2.2. Then the unique positive equilibrium (𝑥^∗₁,𝑥^∗₂) of (1.1) is globally attractive.

Proof. If we denote 𝑙₁=liminf_𝑛→∞𝑥₁(𝑛),𝑙₂=liminf_𝑛→∞𝑥₂𝐿(𝑛),₁=limsup_𝑛→∞𝑥₁(𝑛),𝐿₂=limsup_𝑛→∞𝑥₂(𝑛) (3.12) for any positive solution {𝑥₁(𝑛),𝑥₂(𝑛)} of system (1.1) with initial conditions 𝑥₁(0)>0,𝑥₂(0)>0, we have 0<𝑙₁≤𝐿₁<+∞,0<𝑙₂≤𝐿₂<+∞ (3.13) from Theorem 3.3 and Definition 2.4. Moreover, 𝑙₁≥𝑙₁exp(𝑟₁−𝑎₁₁𝐿₁−𝑎₁₂𝐿₂𝑙),₂≥𝑙₂exp(𝑟₂−𝑎₂₁𝐿₁−𝑎₂₂𝐿₂𝐿),₁≤𝐿₁exp(𝑟₁−𝑎₁₁𝑙₁−𝑎₁₂𝑙₂𝐿),₂≤𝐿₂exp(𝑟₂−𝑎₂₁𝑙₁−𝑎₂₂𝑙₂) (3.14)(3.15)(3.16)(3.17) from (2) of Lemma 2.3.
Note (3.13), the inequalities (3.14)–(3.17) can be written as follows: 𝑟₁−𝑎₁₁𝑙₁−𝑎₁₂𝑙₂𝑟≥0,₂−𝑎₂₁𝑙₁−𝑎₂₂𝑙₂𝑟≥0,₁−𝑎₁₁𝐿₁−𝑎₁₂𝐿₂𝑟≤0,₂−𝑎₂₁𝐿₁−𝑎₂₂𝐿₂≤0. (3.18)(3.19)(3.20)(3.21)
From (3.18)–(3.21), it is clear that (𝑙₁,𝑙₂) lies in the domain 𝐺₂={(𝑥,𝑦)∣𝑥≥0,𝑦≥0,𝑟₁−𝑎₁₁𝑥−𝑎₁₂𝑦≥0,𝑟₂−𝑎₂₁𝑥−𝑎₂₂𝑦≥0}, (3.22) while (𝐿₁,𝐿₂) lies in the domain  𝐺₁  (see (2.7). Therefore, from (3.11) and Lemma 2.2, the maximum of 𝑓(𝑥,𝑦)=𝑥exp(𝑟₁−𝑎₁₁𝑥−𝑎₁₂𝑦) in domain 𝐺₁ is 𝐷₁/𝐷, the maximum of 𝑔(𝑥,𝑦)=𝑦exp(𝑟₂−𝑎₂₁𝑥−𝑎₂₂𝑦) in domain 𝐺₁ is 𝐷₂/𝐷. Then 𝐿₁≤𝐷₁𝐷,𝐿₂≤𝐷₂𝐷. (3.23) But in domain 𝐺₁, only the point (𝑥^∗₁,𝑥^∗₂)=(𝐷₁/𝐷,𝐷₂/𝐷) satisfies these two inequalities, then 𝐿₁=𝐷₁𝐷=𝑥^∗₁,𝐿₂=𝐷₂𝐷=𝑥^∗₂. (3.24)
At this point, we claim that 𝑙₁=𝐷₁𝐷=𝑥^∗₁,𝑙₂=𝐷₂𝐷=𝑥^∗₂. (3.25) Note (3.11), we must consider the following four cases to prove claim (3.25):Case (i):𝐷₁/𝐷<1/𝑎₁₁,𝐷₂/𝐷<1/𝑎₂₂,Case (ii):𝐷₁/𝐷<1/𝑎₁₁,𝐷₂/𝐷>1/𝑎₂₂,Case (iii):𝐷₁/𝐷>1/𝑎₁₁,𝐷₂/𝐷<1/𝑎₂₂,Case (iv):𝐷₁/𝐷>1/𝑎₁₁,𝐷₂/𝐷>1/𝑎₂₂.
It is easy to verify that the function ℎ(𝑥)=𝑥exp(𝑟−𝑎𝑥),𝑎>0 is monotonously increasing when 0<𝑥<1/𝑎 and monotonously decreasing when 𝑥>1/𝑎. With this fact and Lemma 2.3, the proof of the claim is given as below.
Case (i)
We rearrange the two equations of (1.1) as 𝑥₁(𝑛+1)=𝑥₁(𝑛)exp(𝑟₁−𝑎₁₁𝑥₁(𝑛))exp(−𝑎₁₂𝑥₂𝑥(𝑛)),₂(𝑛+1)=𝑥₂(𝑛)exp(𝑟₂−𝑎₂₂𝑥₂(𝑛))exp(−𝑎₂₁𝑥₁(𝑛)). (3.26) Note that 𝐿₁=𝑥^∗₁=𝐷₁𝐷<1𝑎₁₁,𝐿₂=𝑥^∗₂=𝐷₂𝐷<1𝑎₂₂, (3.27) we have 𝑥₁(𝑛)<1/𝑎₁₁,𝑥₂(𝑛)<1/𝑎₂₂ for 𝑛 sufficiently large. Then 𝑙₁≥𝑙₁exp(𝑟₁−𝑎₁₁𝑙₁−𝑎₁₂𝐿₂𝑙),₂≥𝑙₂exp(𝑟₂−𝑎₂₁𝐿₁−𝑎₂₂𝑙₂). (3.28) That is 𝑟₁−𝑎₁₁𝑙₁−𝑎₁₂𝐿₂𝑟≤0,₂−𝑎₂₁𝐿₁−𝑎₂₂𝑙₂≤0. (3.29)(3.30) The inequalities (3.24), (3.29) together with (3.30) imply that 𝑙₁≥𝐷₁𝐷=𝑥^∗₁,𝑙₂≥𝐷₂𝐷=𝑥^∗₂. (3.31) From (3.13) and (3.24), we get 𝑙₁=𝐷₁𝐷=𝑥^∗₁,𝑙₂=𝐷₂𝐷=𝑥^∗₂. (3.32)Case (ii)
Similarly, we have 𝑙₁≥𝑙₁exp(𝑟₁−𝑎₁₁𝑙₁−𝑎₁₂𝐿₂𝑙),₂≥𝐿₂exp(𝑟₂−𝑎₂₁𝐿₁−𝑎₂₂𝐿₂). (3.33)(3.34) From (3.13), (3.24), and (3.33), we get 𝑙₁=𝐷₁/𝐷=𝑥^∗₁. And from (3.13), (3.24), and (3.34), 𝑙₂=𝐿₂=𝐷₂/𝐷=𝑥^∗₂ follows.
The proof of Case (iii) is similar to that of Case (ii).Case (iv)
We have 𝑙₁≥𝐿₁exp(𝑟₁−𝑎₁₁𝐿₁−𝑎₁₂𝐿₂𝑙),₂≥𝐿₂exp(𝑟₂−𝑎₂₁𝐿₁−𝑎₂₂𝐿₂). (3.35) Therefore, 𝑙₁=𝐷₁𝐷=𝑥^∗₁,𝑙₂=𝐷₂𝐷=𝑥^∗₂ (3.36) are consequent from (3.13), (3.24), and (3.35).
The proof of claim (3.25) is completed. Note (3.24) and (3.25), lim_𝑛→∞𝑥₁(𝑛)=𝑥^∗₁,lim_𝑛→∞𝑥₂(𝑛)=𝑥^∗₂ (3.37) for any positive solution {𝑥₁(𝑛),𝑥₂(𝑛)} of system (1.1). That is, (1.1) is globally attractive according to Definition 2.7.

Theorem 3.5. Assume that the assumptions of Theorem 3.4 are satisfied, moreover, 𝐷₁𝐷<1𝑎₁₁,𝐷₂𝐷<1𝑎₂₂, (3.38) then the unique positive equilibrium of system (1.1) is globally asymptotically stable.

Proof. From Theorem 3.3, system (1.1) is strongly persistent. That is, condition (ii) of Lemma 2.9 is satisfied.
𝐷_𝑖>0,𝑖,𝑗=1,2 implies that 𝑟₁𝑎₂₂−𝑟₂𝑎₁₂>0,𝑟₂𝑎₁₁−𝑟₁𝑎₂₁>0. Set 𝑐₁=𝑟₂,𝑐₂=𝑟₁, it is clear that 𝑐₁𝑎₁₁>𝑐₂𝑎₂₁,𝑐₂𝑎₂₂>𝑐₁𝑎₁₂. Thus, condition (i) of Lemma 2.9 is satisfied.
Let {𝑥₁(𝑛),𝑥₂(𝑛)} be any positive solution of system (1.1). We show below that 𝑎₁₁𝑥₁(𝑛+1)≤1,𝑎₂₂𝑥₂(𝑛+1)≤1 (3.39) for all large 𝑛. By Theorem 3.4, we know that (𝑥^∗₁,𝑥^∗₂)=(𝐷₁/𝐷,𝐷₂/𝐷) is globally attractive. That is lim_𝑛→∞𝑥₁𝐷(𝑛)=¹𝐷,lim_𝑛→∞𝑥₂𝐷(𝑛)=²𝐷. (3.40) From (3.38) we first select 𝜀>0, such that 𝐷₁𝐷1+𝜀<𝑎₁₁,𝐷₂𝐷1+𝜀<𝑎₂₂. (3.41) Further from (3.40), we know that there exists 𝑁₁ and 𝑁₂, such that 𝑥₁𝐷(𝑛+1)<¹𝐷+𝜀,for𝑛≥𝑁₁,𝑥₂𝐷(𝑛+1)<²𝐷+𝜀,for𝑛≥𝑁₂, (3.42) respectively. Then denote 𝑁=max{𝑁₁,𝑁₂}, we get 𝑥₁𝐷(𝑛+1)<¹𝐷1+𝜀<𝑎₁₁,𝑥₂𝐷(𝑛+1)<²𝐷1+𝜀<𝑎₂₂ (3.43) for 𝑛≥𝑁 from (3.41). That is, (3.39) is true for all sufficiently large 𝑛. Therefore, condition (iii) of Lemma 2.9 is satisfied. The proof is completed by applying Lemma 2.9.

Theorem 3.6. Assume that (2.6), and (3.8) hold, the coefficients of (1.1) satisfy 𝑟_𝑖≥1(𝑖=1,2) and 𝐷₁𝐷<𝑎₂₂𝐷<𝑟₁𝑎₂₁,𝐷₂𝐷<𝑎₁₁𝐷<𝑟₂𝑎₁₂,𝑎₂₂𝐷𝐷exp(¹𝑎₂₂1−1)<𝑎₁₁,𝑎₁₁𝐷𝐷exp(²𝑎₁₁1−1)<𝑎₂₂, (3.44)(3.45) then the positive equilibrium of system (1.1) is globally asymptotically stable.

Proof. From the proof of Theorem 3.4, we know that (𝐿₁,𝐿₂) lies in domain 𝐺₁. Therefore, we obtain limsup_𝑛→∞𝑥₁𝑎(𝑛)≤²²𝐷𝐷exp(¹𝑎₂₂1−1)<𝑎₁₁,limsup_𝑛→∞𝑥₂𝑎(𝑛)≤¹¹𝐷𝐷exp(²𝑎₁₁1−1)<𝑎₂₂ (3.46) from Lemma 2.2. That is, condition (iii) of Lemma 2.9 is satisfied. Conditions (i) and (ii) of Lemma 2.9 are also satisfied. Then the positive equilibrium of system (1.1) is globally asymptotically stable by applying Lemma 2.9.

4. Numerical Examples

In this section, we give two numerical examples to show the feasibility of the assumptions of the results. The first example also shows that system (1.1) can be globally attractive when the intrinsic growth rates of the two species are greater than 1+ln2≐1.6931, and this result can be obtained by Theorem 3.4.

Example 4.1. Consider the following case of system (1.1): 𝑟₁=1.95,𝑟₂=1.8,𝑎₁₁=0.5,𝑎₁₂=0.1,𝑎₂₂=0.5,𝑎₂₁=0.09, (4.1) then 𝐷₁=0.7950,𝐷₂𝐷=0.7245,₁𝐷=𝑥^∗₁𝐷=3.2988,²𝐷=𝑥^∗₂1=3.0062,𝑎₁₁exp(𝑟₁1−1)=5.1714,𝑎₂₂exp(𝑟₂𝑟−1)=4.4511,₂𝑎₂₁𝑟=20.0000,¹𝑎₁₂1=19.5000,𝑎₁₁=1𝑎₂₂𝑟=2.0000,₂𝑎₂₁exp(𝑟₁−𝑎₁₁𝑟₂𝑎₂₁𝑟)=0.0064,¹𝑎₁₂exp(𝑟₂−𝑎₂₂𝑟₁𝑎₁₂)=0.0069. (4.2)
We see that the conditions of Theorem 3.4 are satisfied. Therefore, the positive equilibrium of system (1.1) is globally attractive (see Figure 1). But this result cannot be obtained by [3, Theorem 3] when consider the autonomous case of this theorem(the model studied in [3] is nonautonomous). In fact, the condition of [3, Theorem 3] must satisfy exp(𝑟_𝑖−1)−1<1(𝑖=1,2) when 𝑟_𝑖>1, that is, 𝑟_𝑖<1+ln2≐1.6931(𝑖=1,2). In Example 4.1, 𝑟_𝑖>1+ln2(𝑖=1,2).
The following example shows that system (1.1) can be globally asymptotically stable when the intrinsic growth rates of the two species are greater than 1, and this result can be obtained by Theorem 3.6.

Figure 1 

Solutions of system (1.1) with initial values (𝑥1(0),𝑥2(0))=(10,9), (1,10), (0.01,0.09), (9,0.1), and 𝑟1=1.95,𝑟2=1.8.

Example 4.2. Consider the following case of system (1.1): 𝑟₁=1.1,𝑟₂=1.2,𝑎₁₁=0.05,𝑎₁₂=0.02,𝑎₂₂=0.06,𝑎₂₁=0.015, (4.3) then 𝐷₁=0.0420,𝐷₂𝑥=0.0435,_∗1=15.5556,𝑥^∗₂1=16.1111,𝑎₁₁exp(𝑟₁1−1)=22.1034,𝑎₂₂exp(𝑟₂𝑟−1)=20.3567,₂𝑎₂₁𝑟=80.0000,¹𝑎₁₂𝑟=55.0000,₂𝑎₂₁exp(𝑟₁−𝑎₁₁𝑟₂𝑎₂₁𝑟)=4.4019,¹𝑎₁₂exp(𝑟₂−𝑎₂₂𝑟₁𝑎₁₂𝑎)=6.7351,₂₂𝐷𝐷exp(¹𝑎₂₂𝑎−1)=16.4626,¹¹𝐷𝐷exp(²𝑎₁₁1−1)=16.2610,𝑎₁₁1=20.0000,𝑎₂₂=16.6667. (4.4)
It is clear that the conditions of Theorem 3.6 are satisfied. Thus by Theorem 3.6 the positive equilibrium of system (1.1) is globally asymptotically stable (see Figure 2).
Example 4.2 shows that our results improve [12, Theorem 3] by providing estimates for the smallness of  𝑟₁,𝑟₂. The work in [10, Theorem 2] states that if 𝐷₁>0,𝐷₂>0,𝑟_𝑖≤1(𝑖=1,2), then the positive equilibrium (𝑥^∗₁,𝑥^∗₂) is globally asymptotically stable. Thus the global asymptotical stability of system (1.1) in the case of Example 4.2 cannot be obtained by [10, Theorem 2] because of 𝑟_𝑖>1(𝑖=1,2).

Figure 2 

Solutions of system (1.1) with initial values(𝑥1(0),𝑥2(0))=(50,45),(1,47),(0.1,0.09),(48,0.1), and 𝑟1=1.1,𝑟2=1.2.

5. Conclusion

In this paper, we further discuss the global dynamics of a discrete autonomous competitive model of Lotka-Volterra type. Sufficient conditions are obtained to guarantee the permanence, global attractivity, and global asymptotical stability of the system. These conditions are expressed by the coefficients of the model and can be easily verified. Numerical examples are also given to show the feasibility of the conditions.

Earlier works have shown that the system of this type can be globally attractive when the intrinsic growth rates of the two species are less than 1+ln2 ([3], for single-species system see [18]). It is shown in [10] that the system can be globally asymptotically stable when the intrinsic growth rates of the two species are less than 1. In [16], it is shown that the system can exhibit chaotic behavior when the intrinsic growth rates of the two species are equal and greater than 3.13. But the global dynamics of the system is not clear enough when the intrinsic growth rates of the two species are greater than 1 and less than 3.13. We obtain that the system can also be globally asymptotically stable when the intrinsic growth rates of the two competitive species are greater than 1 and globally attractive when the intrinsic growth rates of the two competitive species are greater than 1+ln2.

For the global stability of the system, the following condition in Theorem 3.5: 𝐷₁𝐷<1𝑎₁₁,𝐷₂𝐷<1𝑎₂₂, (5.1) can be reduced to the following by direct computation: 𝑟₁𝑎<1+¹²𝑎₂₂,𝑟₂𝑎<1+²¹𝑎₁₁. (5.2) And the above inequalities imply that 𝑟_𝑖(𝑖=1,2) can be greater than 1 while the system is globally asymptotically stable.

Acknowledgments

The authors would like to thank the Editor Professor Yong Zhou and the referees for their valuable comments and suggestions. This work was supported by the Foundation of Jiangsu Polytechnic University (ZMF09020020).