Global attractivity of a discrete Lotka-Volterra competition system with infinite delays and feedback controls

Abstract

In this paper, we propose a discrete Lotka-Volterra competition system with infinite delays and feedback controls. Sufficient conditions which ensure the global attractivity of the system are obtained. An example together with its numerical simulation shows the feasibility of the main results.

1 Introduction

For the last decades, the ecological competition systems governed by differential equations of Lotka-Volterra type have been investigated extensively. Many interesting results concerned with the global existence and attractivity of periodic solution, persistence and extinction of the population, etc. have been obtained; we refer to [1–4] and the references therein. Already, many authors [5–21] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Particularly, the persistence, permanence, extinction, local and global stability and the existence of positive periodic solutions, etc., for discrete competitive systems are studied in [5, 9, 12, 13, 15–18, 21]. Chen and Zhou [9] discussed the following discrete Lotka-Volterra competition system:

$$\{\begin{array}{c}{x}_1(n+1)=x_1(n)exp[r_1(n)(1-\frac{x_1(n)}{K_1(n)}-{\mu }_1(n)x_2(n))],\hfill \\ x_2(n+1)=x_2(n)exp[r_2(n)(1-\frac{x_2(n)}{K_2(n)}-{\mu }_2(n)x_1(n))].\hfill \end{array}$$

(1.1)

They obtained sufficient conditions which guarantee the persistence of system (1.1). Also, for the periodic case, they obtained sufficient conditions for the existence of a globally stable periodic solution.

Chen [12] studied the following nonautonomous two-species discrete competitive systems with deviating arguments:

$$\{\begin{array}{c}{x}_1(n+1)=x_1(n)exp[r_1(n)(1-\frac{x_1(n)}{K_1(n)}-{\mu }_1(n){\sum }_{s=-\mathrm{\infty }}^n{H}_1(n-s)x_2(s))],\hfill \\ x_2(n+1)=x_2(n)exp[r_2(n)(1-\frac{x_2(n)}{K_2(n)}-{\mu }_2(n){\sum }_{s=-\mathrm{\infty }}^n{H}_2(n-s)x_1(s))].\hfill \end{array}$$

(1.2)

They obtained sufficient conditions for the permanence of system (1.2).

On the other hand, feedback control is the basic mechanism by which systems, whether mechanical, electrical or biological, maintain their equilibrium or homeostasis. In the higher life forms, the conditions under which life can continue are quite narrow. A change in body temperature of half a degree is generally a sign of illness. The homeostasis of the body is maintained through the use of feedback control [22]. A primary contribution of C.R. Darwin during the last century was the theory that feedback over long time periods is responsible for the evolution of species. In 1931 Volterra [23] explained the balance between two populations of fish in a closed pond using the theory of feedback. Later, a series of mathematical models have been established to describe the dynamics of feedback control systems; see [14, 16–20, 23–25] and the references therein.

The purpose of this paper is to study the global attractivity of the following discrete Lotka-Volterra competition system with infinite delays and feedback controls:

$$\{\begin{array}{c}{x}_1(n+1)=x_1(n)exp[r_1-a_{11}{x}_1(n)-a_{12}{\sum }_{s=-\mathrm{\infty }}^n{H}_1(n-s)x_2(s)\hfill \\ \phantom{x_1(n+1)=}-c_1{\sum }_{s=-\mathrm{\infty }}^n{H}_2(n-s){\mu }_1(s)],\hfill \\ x_2(n+1)=x_2(n)exp[r_2-a_{22}{x}_2(n)-a_{21}{\sum }_{s=-\mathrm{\infty }}^n{H}_3(n-s)x_1(s)\hfill \\ \phantom{x_2(n+1)=}-c_2{\sum }_{s=-\mathrm{\infty }}^n{H}_4(n-s){\mu }_2(s)],\hfill \\ {\mu }_1(n+1)=(1-a_1){\mu }_1(n)+b_1{\sum }_{s=-\mathrm{\infty }}^n{H}_5(n-s)x_1(s),\hfill \\ {\mu }_2(n+1)=(1-a_2){\mu }_2(n)+b_2{\sum }_{s=-\mathrm{\infty }}^n{H}_6(n-s)x_2(s),\hfill \end{array}$$

(1.3)

where $0<a_i<1$, $r_i,b_i,c_i,a_{ij}\in (0,\mathrm{\infty })$, $i=1,2$, $j=1,2$, $x_i(n)$ ($i=1,2$) are the density of the i species at time n and ${\mu }_i(n)$ ($i=1,2$) are the control variables at time n. $H_i(n)$ ($i=1,2,\dots ,6$) are bounded nonnegative sequences such that ${\sum }_{n=0}^{\mathrm{\infty }}{H}_i(n)=1$.

By the biological meaning, we focus our discussion on the positive solutions of (1.3). So, it is assumed that the initial conditions of (1.3) are of the form

$$x_i(s)={\mathrm{\Phi }}_i(s)\ge 0,{\mathrm{\Phi }}_i(0)>0,{\mu }_i(s)={\mathrm{\Psi }}_i(s)\ge 0,{\mathrm{\Psi }}_i(0)>0,i=1,2,$$

(1.4)

where $s=\dots ,-n,-n+1,\dots ,-1,0$. One can easily show that the solutions of (1.3) with (1.4) remain positive for all $n\in Z_{+}$, where $Z_{+}=\{0,1,2,\dots \}$.

Further, assume

$$(r_1{a}_{21}-r_2{d}_1)(r_2{a}_{12}-r_1{d}_2)>0,$$

(1.5)

where $d_1=\frac{1}{a_1}(a_1{a}_{11}+b_1{c}_1)$, $d_2=\frac{1}{a_2}(a_2{a}_{22}+b_2{c}_2)$. Then system (1.3) has a unique positive equilibrium $(x_1^{\ast },x_2^{\ast },{\mu }_1^{\ast },{\mu }_2^{\ast })$ with

$$x_1^{\ast }=\frac{r_2{a}_{12}-r_1{d}_2}{a_{12}{a}_{21}-d_1{d}_2},x_2^{\ast }=\frac{r_1{a}_{21}-r_2{d}_1}{a_{12}{a}_{21}-d_1{d}_2},{\mu }_1^{\ast }=\frac{a_1}{b_1}{x}_1^{\ast },{\mu }_2^{\ast }=\frac{a_2}{b_2}{x}_2^{\ast }.$$

The aim of this paper is, by developing the analysis technique of Chen [12], Liao and Yu [14], Chen and Teng [15], to obtain a set of sufficient conditions for the global attractivity of system (1.3). The paper is organized as follows. In Section 2, as preliminaries, some useful lemmas are given. In Section 3, we study the global attractivity of positive equilibrium of system (1.3). In Section 4, the numerical simulations on the global attractivity of equilibrium are given.

2 Preliminaries

In this section, we introduce some auxiliary lemmas which will be useful in the following.

Lemma 1 (see [15])

Let the function $f(u)=uexp(\alpha -\beta u)$, where α and β are positive constants. Then $f(u)$ is nondecreasing on $u\in (0,\frac{1}{\beta }]$.

Lemma 2 (see [15])

Assume that the sequence $\{u(n)\}$ satisfies

$$u(n+1)=u(n)exp(\alpha -\beta u(n)),n=1,2,\dots ,$$

where α and β are positive constants and $u(0)>0$. We have

• (i) if $\alpha <2$, then ${lim}_{n\to \mathrm{\infty }}u(n)=\frac{\alpha }{\beta }$.

• (ii) if $\alpha \le 1$, then $u(n)\le \frac{1}{\beta }$ for all $n=2,3,\dots$ .

Lemma 3 (see [8])

Suppose that functions $f,g:Z_{+}\times [0,\mathrm{\infty })\to [0,\mathrm{\infty })$ satisfy $f(n,x)\le g(n,x)$ ($f(n,x)\ge g(n,x)$) for $n\in Z_{+}$ and $x\in [0,\mathrm{\infty })$ and $g(n,x)$ is nondecreasing with respect to $x>0$. If sequences $\{x(n)\}$ and $\{u(n)\}$ are the nonnegative solutions of the following difference equations:

$$x(n+1)=f(n,x(n)),u(n+1)=g(n,u(n)),n=0,1,2,\dots ,$$

respectively, and $x(0)\le u(0)$ ($x(0)\ge u(0)$), then for all $n\ge 0$, we have

$$x(n)\le u(n)(x(n)\ge u(n)).$$

Lemma 4 (see [12])

Let $x:Z\to R$ be a nonnegative bounded sequence, and let $H:Z_{+}\to R$ be a nonnegative sequence such that ${\sum }_{n=0}^{\mathrm{\infty }}H(n)=1$, where $Z=\{0,\pm 1,\pm 2,\dots \}$, $R=(-\mathrm{\infty },\mathrm{\infty })$. Then

$$\underset{n\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}x(n)\le \underset{n\to +\mathrm{\infty }}{lim\hspace{0.17em}inf}\sum _{s=-\mathrm{\infty }}^{n}H(n-s)x(s)\le \underset{n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}\sum _{s=-\mathrm{\infty }}^{n}H(n-s)x(s)\le \underset{n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}x(n).$$

We further consider the following discrete linear equation:

$$u(n+1)=(1-{\gamma }_1)u(n)+{\gamma }_2\sum _{s=-\mathrm{\infty }}^{n}H(n-s)x(s),n\in Z_{+},$$

(2.1)

where $0<{\gamma }_1<1$, ${\gamma }_2\in (0,\mathrm{\infty })$. $H(n)$ is a nonnegative sequence defined on $Z_{+}$ such that ${\sum }_{n=0}^{\mathrm{\infty }}H(n)=1$ and $x(n)$ is a nonnegative bounded sequence defined on Z with

$$x_{\ast }\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}x(n)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}x(n)\le x^{\ast },$$

where $x_{\ast }$, $x^{\ast }$ are nonnegative constants.

Lemma 5 Any solution of system (2.1) with $u(0)>0$ satisfies

$$\frac{{\gamma }_2}{{\gamma }_1}{x}_{\ast }\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}u(n)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}u(n)\le \frac{{\gamma }_2}{{\gamma }_1}{x}^{\ast }.$$

Proof From Lemma 4,

$$\underset{n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}\sum _{s=-\mathrm{\infty }}^{n}H(n-s)x(s)\le \underset{n\to +\mathrm{\infty }}{lim\hspace{0.17em}sup}x(n)\le x^{\ast }.$$

Hence, for each $\epsilon >0$, there exists an enough large integer $n_0$ such that for $n\ge n_0$,

$$\sum _{s=-\mathrm{\infty }}^{n}H(n-s)x(n)\le x^{\ast }+\epsilon .$$

By system (2.1), we can obtain

$$\begin{array}{rcl}u(n)& =& u(n_0){(1-{\gamma }_1)}^{n-n_0}+{\gamma }_2\sum _{i=n_0}^{n-1}[{(1-{\gamma }_1)}^{n-i-1}\sum _{s=-\mathrm{\infty }}^{i}H(i-s)x(s)]\\ \le & u_0{(1-{\gamma }_1)}^{n-n_0}+{\gamma }_2(x^{\ast }+\epsilon )\sum _{i=n_0}^{n-1}{(1-{\gamma }_1)}^{n-i-1}\\ =& u_0{(1-{\gamma }_1)}^{n-n_0}+{\gamma }_2(x^{\ast }+\epsilon )\frac{1-{(1-{\gamma }_1)}^{n-n_0}}{{\gamma }_1}\to \frac{{\gamma }_2(x^{\ast }+\epsilon )}{{\gamma }_1},n\to \mathrm{\infty }.\end{array}$$

Thus,

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}u(n)\le \frac{{\gamma }_2(x^{\ast }+\epsilon )}{{\gamma }_1}.$$

By the arbitrariness of ε, we can obtain

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}u(n)\le \frac{{\gamma }_2}{{\gamma }_1}{x}^{\ast }.$$

We can prove ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}u(n)\ge \frac{{\gamma }_2}{{\gamma }_1}{x}_{\ast }$ in a similar way. Thus, we complete the proof. □

Lemma 6 Assume ${lim}_{n\to \mathrm{\infty }}x(n)=\overline{x}$. For every solution $u(n)$ of equation (2.1), we have

$$\underset{n\to \mathrm{\infty }}{lim}u(n)=\frac{{\gamma }_2}{{\gamma }_1}\overline{x}.$$

By Lemma 5, the proof of Lemma 6 is obtained easily. Hence, we omit it here.

3 Global attractivity

In this section, we derive sufficient conditions which guarantee that the positive equilibrium of system (1.3) is globally attractive. The technique of proofs is to use an iteration scheme.

Theorem 1 Assume

$$(r_1{a}_{21}-r_2{d}_1)(r_2{a}_{12}-r_1{d}_2)>0$$

and

$$\frac{r_2{a}_{12}}{a_{22}}+\frac{b_1{c}_1{r}_1}{a_1{a}_{11}}<r_1\le 1,\frac{r_1{a}_{21}}{a_{11}}+\frac{b_2{c}_2{r}_2}{a_2{a}_{22}}<r_2\le 1.$$

Then equilibrium $(x_1^{\ast },x_2^{\ast },{\mu }_1^{\ast },{\mu }_2^{\ast })$ of system (1.3) with (1.4) is globally attractive.

Proof Let $(x_1(n),x_2(n),{\mu }_1(n),{\mu }_2(n))$ be any solution of system (1.3) with (1.4). Denote

$$U_i=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{x}_i(n),V_i=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{x}_i(n),i=1,2,$$

and

$$P_i=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\mu }_i(n),Q_i=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\mu }_i(n),i=1,2.$$

We now claim that $U_i=V_i=x_i^{\ast }$, $P_i=Q_i={\mu }_i^{\ast }$, $i=1,2$.

From the first equation of system (1.3), we obtain

$$x_1(n+1)\le x_1(n)exp[r_1-a_{11}{x}_1(n)],n=0,1,2,\dots .$$

Consider the auxiliary equation

$$p(n+1)=p(n)exp[r_1-a_{11}p(n)].$$

(3.1)

From $r_1\le 1$, by the conclusion (ii) of Lemma 2, we have that $p(n)\le \frac{1}{a_{11}}$ for all $n\ge 2$, where $p(n)$ is any solution of equation (3.1) with initial value $p(0)>0$. From Lemma 1, we have $f(p)=pexp(r_1-a_{11}p)$ is nondecreasing for $p\in (0,\frac{1}{a_{11}}]$.

Hence, from Lemma 3, we obtain $x_1(n)\le p(n)$ for all $n\ge 2$, where $p(n)$ is the solution of equation (3.1) with $p(2)=x_1(2)$. Further, combining it with the conclusion (i) of Lemma 2, we obtain

$$U_1=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{x}_1(n)\le \underset{n\to \mathrm{\infty }}{lim}p(n)=\frac{r_1}{a_{11}}:=M_1^1.$$

From the second equation of system (1.3), we obtain

$$x_2(n+1)\le x_2(n)exp[r_2-a_{22}{x}_2(n)],n=0,1,2,\dots .$$

By a similar argument as that above, we have

$$U_2=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{x}_2(n)\le \frac{r_2}{a_{22}}:=M_1^2.$$

By Lemma 4 and Lemma 5, we obtain

$$P_i=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\mu }_i(n)\le \frac{b_i}{a_i}{M}_1^i,i=1,2.$$

Then, for any constant $\epsilon >0$ sufficiently small, there is an integer $n_1>2$ such that if $n\ge n_1$, then

$$x_i(n)\le M_1^i+\epsilon ,{\mu }_i(n)\le \frac{b_i}{a_i}{M}_1^i+\epsilon ,i=1,2.$$

Further, from Lemma 4 and the first equation of system (1.3), we have

$$x_1(n+1)\ge x_1(n)exp[r_1-a_{11}{x}_1(n)-a_{12}(M_1^2+\epsilon )-c_1(\frac{b_1}{a_1}{M}_1^1+\epsilon )],n\ge n_1.$$

Consider the auxiliary equation

$$p(n+1)=p(n)exp[r_1-a_{11}p(n)-a_{12}(M_1^2+\epsilon )-c_1(\frac{b_1}{a_1}{M}_1^1+\epsilon )],n\ge n_1.$$

(3.2)

From $\frac{r_2{a}_{12}}{a_{22}}+\frac{b_1{c}_1{r}_1}{a_1{a}_{11}}<r_1\le 1$ and the arbitrariness of $\epsilon >0$, we have

$$0<r_1-a_{12}(M_1^2+\epsilon )-c_1(\frac{b_1}{a_1}{M}_1^1+\epsilon )<1.$$

By the conclusion (ii) of Lemma 2, we have that $p(n)\le \frac{1}{a_{11}}$ for all $n\ge n_1$, where $p(n)$ is any solution of equation (3.2) with initial value $p(n_1)>0$. From Lemma 1, we have

$$f(p)=exp[r_1-a_{11}p-a_{12}(M_1^2+\epsilon )-c_1(\frac{b_1}{a_1}{M}_1^1+\epsilon )]$$

is nondecreasing for $p\in (0,\frac{1}{a_{11}}]$.

Hence, from Lemma 3, we have $x_1(n)\ge p(n)$ for all $n\ge n_1$, where $p(n)$ is the solution of equation (3.2) with $p(n_1)=x_1(n_1)$. Combining it with the conclusion (i) of Lemma 2, we obtain

$$V_1=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{x}_1(n)\ge \underset{n\to \mathrm{\infty }}{lim}p(n)=\frac{1}{a_{11}}[r_1-a_{12}(M_1^2+\epsilon )-c_1(\frac{b_1}{a_1}{M}_1^1+\epsilon )].$$

From the arbitrariness of $\epsilon >0$, we conclude $V_1\ge m_1^1$, where

$$m_1^1=\frac{1}{a_{11}}[r_1-a_{12}{M}_1^2-c_1\frac{b_1}{a_1}{M}_1^1].$$

From Lemma 4 and the second equation of system (1.3), we further have

$$x_2(n+1)\ge x_2(n)exp[r_2-a_{22}{x}_2(n)-a_{21}(M_1^1+\epsilon )-c_2(\frac{b_2}{a_2}{M}_1^2+\epsilon )],n\ge n_1.$$

By a similar argument as that above, we can obtain

$$V_2=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{x}_2(n)\ge \frac{1}{a_{22}}[r_2-a_{21}{M}_1^2-c_1\frac{b_1}{a_1}{M}_1^1]:=m_1^2.$$

By Lemma 4 and Lemma 5, we further obtain

$$Q_i=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\mu }_i(n)\ge \frac{b_i}{a_i}{m}_1^i,i=1,2.$$

Hence, for $\epsilon >0$ sufficiently small, there is an $n_2>n_1$ such that if $n\ge n_2$, then

$$x_i(n)\ge m_1^i-\epsilon ,{\mu }_i(n)\ge \frac{b_i}{a_i}{m}_1^i-\epsilon ,i=1,2.$$

From Lemma 4 and the first equation of system (1.3), we further have

$$x_1(n+1)\le x_1(n)exp[r_1-a_{11}{x}_1(n)-a_{12}(m_1^2-\epsilon )-c_1(\frac{b_1}{a_1}{m}_1^1-\epsilon )],n\ge n_2.$$

Consider the auxiliary equation

$$p(n+1)\le p(n)exp[r_1-a_{11}p(n)-a_{12}(m_1^2-\epsilon )-c_1(\frac{b_1}{a_1}{m}_1^1-\epsilon )],n\ge n_2.$$

From $\frac{r_2{a}_{12}}{a_{22}}+\frac{b_1{c}_1{r}_1}{a_1{a}_{11}}<r_1\le 1$ and the arbitrariness of $\epsilon >0$, we have

$$0<r_1-a_{12}(m_1^2-\epsilon )-c_1(\frac{b_1}{a_1}{m}_1^1-\epsilon )<1.$$

Similarly to the above discussion, we can obtain

$$U_1=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{x}_1(n)\le \underset{n\to \mathrm{\infty }}{lim}p(n)=\frac{1}{a_{11}}[r_1-a_{12}(m_1^2-\epsilon )-c_1(\frac{b_1}{a_1}{m}_1^1-\epsilon )].$$

From the arbitrariness of $\epsilon >0$, we conclude $U_1\le M_2^1$, where

$$M_2^1=\frac{1}{a_{11}}[r_1-a_{12}{m}_1^2-c_1\frac{b_1}{a_1}{m}_1^1].$$

From Lemma 4 and the second equation of system (1.3), we further have

$$x_2(n+1)\le x_2(n)exp[r_2-a_{22}{x}_2(n)-a_{21}(m_1^1-\epsilon )-c_2(\frac{b_2}{a_2}{m}_1^2-\epsilon )],n\ge n_2.$$

By a similar argument as that above, we can obtain

$$U_2=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{x}_2(n)\le \frac{1}{a_{22}}[r_2-a_{21}{m}_1^1-c_2\frac{b_2}{a_2}{m}_1^2]:=M_2^2.$$

By Lemma 4 and Lemma 5, we obtain

$$P_i=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\mu }_i(n)\le \frac{b_i}{a_i}{M}_2^i,i=1,2.$$

Hence, for $\epsilon >0$ sufficiently small, there is an $n_3$ such that if $n\ge n_3$,

$$x_i(n)\le M_2^i+\epsilon ,{\mu }_i(n)\le \frac{b_i}{a_i}{M}_2^i+\epsilon ,i=1,2.$$

From Lemma 4 and the first equation of system (1.3), we have

$$x_1(n+1)\ge x_1(n)exp[r_1-a_{11}{x}_1(n)-a_{12}(M_2^2+\epsilon )-c_1(\frac{b_1}{a_1}{M}_2^1+\epsilon )],n\ge n_3.$$

Consider the auxiliary equation

$$p(n+1)=p(n)exp[r_1-a_{11}p(n)-a_{12}(M_2^2+\epsilon )-c_1(\frac{b_1}{a_1}{M}_2^1+\epsilon )],n\ge n_3.$$

Since

$$0<r_1-a_{12}(M_2^2+\epsilon )-c_1(\frac{b_1}{a_1}{M}_2^1+\epsilon )<1,$$

similarly to the above discussion, we obtain

$$V_1=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{x}_1(n)\ge \underset{n\to \mathrm{\infty }}{lim}p(n)=\frac{1}{a_{11}}[r_1-a_{12}(M_2^2+\epsilon )-c_1(\frac{b_1}{a_1}{M}_2^1+\epsilon )].$$

From the arbitrariness of $\epsilon >0$, we conclude $V_1\ge m_2^1$, where

$$m_2^1=\frac{1}{a_{11}}[r_1-a_{12}{M}_2^2-c_1\frac{b_1}{a_1}{M}_2^1].$$

From Lemma 4 and the second equation of system (1.3), we further have

$$x_2(n+1)\ge x_2(n)exp[r_2-a_{22}{x}_2(n)-a_{21}(M_2^1+\epsilon )-c_2(\frac{b_2}{a_2}{M}_2^2+\epsilon )],n\ge n_3.$$

By a similar argument as that above, we can obtain

$$V_2=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{x}_2(n)\ge \frac{1}{a_{22}}[r_2-a_{21}{M}_2^1-c_2\frac{b_2}{a_2}{M}_2^2]:=m_2^2.$$

Continuing the above process, we can obtain four sequences $\{M_n^i\}$, $\{m_n^i\}$, $i=1,2$ such that

$$M_{n+1}^1=\frac{1}{a_{11}}[r_1-a_{12}{m}_n^2-c_1\frac{b_1}{a_1}{m}_n^1],M_{n+1}^2=\frac{1}{a_{22}}[r_2-a_{21}{m}_n^1-c_2\frac{b_2}{a_2}{m}_n^2]$$

(3.3)

and

$$m_n^1=\frac{1}{a_{11}}[r_1-a_{12}{M}_n^2-c_1\frac{b_1}{a_1}{M}_n^1],m_n^2=\frac{1}{a_{22}}[r_2-a_{21}{M}_n^1-c_2\frac{b_2}{a_2}{M}_n^2].$$

(3.4)

Clearly, we have

$$m_n^i\le V_i\le U_i\le M_n^i,i=1,2.$$

Now, by means of the inductive method, we prove $\{M_n^i\}$ is monotonically decreasing, $\{m_n^i\}$ is monotonically increasing, $i=1,2$.

Firstly, it is clear that $M_2^i\le M_1^i$, $m_2^i\ge m_1^i$, $i=1,2$. For $n=k$ ($k\ge 2$), we assume $M_k^i\le M_{k-1}^i$ and $m_k^i\ge m_{k-1}^i$, $i=1,2$, then we have

$$\begin{array}{c}{M}_{k+1}^1=\frac{1}{a_{11}}[r_1-a_{12}{m}_k^2-c_1\frac{b_1}{a_1}{m}_k^1]\le \frac{1}{a_{11}}[r_1-a_{12}{m}_{k-1}^2-c_1\frac{b_1}{a_1}{m}_{k-1}^1]=M_k^1,\hfill \\ M_{k+1}^2=\frac{1}{a_{22}}[r_2-a_{21}{m}_k^1-c_2\frac{b_2}{a_2}{m}_k^2]\le \frac{1}{a_{22}}[r_2-a_{21}{m}_{k-1}^1-c_2\frac{b_2}{a_2}{m}_{k-1}^2]=M_k^2\hfill \end{array}$$

and

$$\begin{array}{c}{m}_{k+1}^1=\frac{1}{a_{11}}[r_1-a_{12}{M}_{k+1}^2-c_1\frac{b_1}{a_1}{M}_{k+1}^1]\ge \frac{1}{a_{11}}[r_1-a_{12}{M}_k^2-c_1\frac{b_1}{a_1}{M}_k^1]=m_k^1,\hfill \\ m_{k+1}^2=\frac{1}{a_{22}}[r_2-a_{21}{M}_{k+1}^1-c_2\frac{b_2}{a_2}{M}_{k+1}^2]\ge \frac{1}{a_{22}}[r_2-a_{21}{M}_k^1-c_2\frac{b_2}{a_2}{M}_k^2]=m_k^2.\hfill \end{array}$$

Therefore, $\{M_n^i\}$ is monotonically decreasing, $\{m_n^i\}$ is monotonically increasing, $i=1,2$. Consequently, ${lim}_{n\to \mathrm{\infty }}{M}_n^i$ and ${lim}_{n\to \mathrm{\infty }}{m}_n^i$ both exist, $i=1,2$. Let

$$\underset{n\to \mathrm{\infty }}{lim}{M}_n^i={\overline{x}}_i,\underset{n\to \mathrm{\infty }}{lim}{m}_n^i={\overline{y}}_i,i=1,2.$$

From (3.3) and (3.4), we obtain

$$\{\begin{array}{c}{a}_{11}{\overline{x}}_1+a_{12}{\overline{y}}_2+\frac{b_1{c}_1}{a_1}{\overline{y}}_1=r_1,\hfill \\ a_{22}{\overline{x}}_2+a_{21}{\overline{y}}_1+\frac{b_2{c}_2}{a_2}{\overline{y}}_2=r_2,\hfill \\ a_{11}{\overline{y}}_1+a_{12}{\overline{x}}_2+\frac{b_1{c}_1}{a_1}{\overline{x}}_1=r_1,\hfill \\ a_{22}{\overline{y}}_2+a_{21}{\overline{x}}_1+\frac{b_2{c}_2}{a_2}{\overline{x}}_2=r_2.\hfill \end{array}$$

(3.5)

It is clear that $(x_1^{\ast },x_2^{\ast },{\mu }_1^{\ast },{\mu }_2^{\ast })$ is a unique solution of equations (3.5). Therefore,

$$U_i=V_i=\underset{n\to \mathrm{\infty }}{lim}{x}_i(n)=x_i^{\ast },i=1,2.$$

Further, by Lemma 6, we can obtain ${lim}_{n\to \mathrm{\infty }}{\mu }_i(n)={\mu }_i^{\ast }$, $i=1,2$. Thus, we complete the proof of Theorem 1. □

4 Example

The following example shows the feasibility of the main results.

Example 1 Choose $r_1=0.6$, $r_2=0.4$, $a_{12}=0.3$, $a_{21}=0.2$, $a_{11}=0.5$, $a_{22}=0.4$, $a_1=0.8$, $a_2=1.2$, $b_1=0.1$, $b_2=0.2$, $c_1=0.2$, $c_2=0.1$, $H_1(n)=H_4(n)=\frac{e-1}{e}{e}^{-n}$, $H_2(n)=H_3(n)=\frac{e^2-1}{e^2}{e}^{-2n}$, $H_5(n)=H_6(n)=\frac{e^4-1}{e^4}{e}^{-4n}$ in system (1.3). By calculating, we have that positive equilibrium $(0.8189,0.5669,0.1024,0.0945)$, $(r_1{a}_{21}-r_2{d}_1)(r_2{a}_{12}-r_1{d}_2)=0.0117$, $r_1-(\frac{r_2{a}_{12}}{a_{22}}+\frac{b_1{c}_1{r}_1}{a_1{a}_{11}})=0.27$, $r_2-(\frac{r_1{a}_{21}}{a_{11}}+\frac{b_2{c}_2{r}_2}{a_2{a}_{22}})=0.1433$. Since

$$\begin{array}{c}(r_1{a}_{21}-r_2{d}_1)(r_2{a}_{12}-r_1{d}_2)>0,(\frac{r_2{a}_{12}}{a_{22}}+\frac{b_1{c}_1{r}_1}{a_1{a}_{11}})<r_1\le 1,\hfill \\ (\frac{r_1{a}_{21}}{a_{11}}+\frac{b_2{c}_2{r}_2}{a_2{a}_{22}})<r_2\le 1,\hfill \end{array}$$

the conditions of Theorem 1 hold. So, equilibrium $(0.8189,0.5669,0.1024,0.0945)$ is globally attractive.

Choose initial values $(x_1(s),x_2(s),{\mu }_1(s),{\mu }_2(s))=(0.9,0.7,0.2,0.1)$, $s=\dots ,-n,-n+1,\dots ,-1,0$.

By the numerical simulation (see Figure 1), we find that the solution $(x_1(n),x_2(n),{\mu }_1(n),{\mu }_2(n))$ turns to equilibrium $(0.8189,0.5669,0.1024,0.0945)$ as $n\to \mathrm{\infty }$.

Figure 1

Dynamical behaviors of system ( 1.3 ) with ${\mathit{r}}_{\mathbf{1}}\mathbf{=}\mathbf{0.6}$ , ${\mathit{r}}_{\mathbf{2}}\mathbf{=}\mathbf{0.4}$ , ${\mathit{a}}_{\mathbf{12}}\mathbf{=}\mathbf{0.3}$ , ${\mathit{a}}_{\mathbf{21}}\mathbf{=}\mathbf{0.2}$ , ${\mathit{a}}_{\mathbf{11}}\mathbf{=}\mathbf{0.5}$ , ${\mathit{a}}_{\mathbf{22}}\mathbf{=}\mathbf{0.4}$ , ${\mathit{a}}_{\mathbf{1}}\mathbf{=}\mathbf{0.8}$ , ${\mathit{a}}_{\mathbf{2}}\mathbf{=}\mathbf{1.2}$ , ${\mathit{b}}_{\mathbf{1}}\mathbf{=}\mathbf{0.1}$ , ${\mathit{b}}_{\mathbf{2}}\mathbf{=}\mathbf{0.2}$ , ${\mathit{c}}_{\mathbf{1}}\mathbf{=}\mathbf{0.2}$ , ${\mathit{c}}_{\mathbf{2}}\mathbf{=}\mathbf{0.1}$ , ${\mathit{H}}_{\mathbf{1}}\mathbf{(}\mathit{n}\mathbf{)}\mathbf{=}{\mathit{H}}_{\mathbf{4}}\mathbf{(}\mathit{n}\mathbf{)}\mathbf{=}\frac{\mathit{e}\mathbf{-}\mathbf{1}}{\mathit{e}}{\mathit{e}}^{\mathbf{-}\mathit{n}}$ , ${\mathit{H}}_{\mathbf{2}}\mathbf{(}\mathit{n}\mathbf{)}\mathbf{=}{\mathit{H}}_{\mathbf{3}}\mathbf{(}\mathit{n}\mathbf{)}\mathbf{=}\frac{{\mathit{e}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}{{\mathit{e}}^{\mathbf{2}}}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathit{n}}$ , ${\mathit{H}}_{\mathbf{5}}\mathbf{(}\mathit{n}\mathbf{)}\mathbf{=}{\mathit{H}}_{\mathbf{6}}\mathbf{(}\mathit{n}\mathbf{)}\mathbf{=}\frac{{\mathit{e}}^{\mathbf{4}}\mathbf{-}\mathbf{1}}{{\mathit{e}}^{\mathbf{4}}}{\mathit{e}}^{\mathbf{-}\mathbf{4}\mathit{n}}$ .

Author’s contributions

The author carried out the proof of the theorem and approved the final manuscript.