Previously, we have discussed the local stability of the equilibria of system (1.1). In this section, we give a set of sufficient conditions that ensure the global attractivity of the unique positive equilibrium on the rectangle \((0, \frac{r_{1}}{a_{1}})\times(0, \frac {r_{2}}{a_{2}})\).
Theorem 3.1
In addition to
\((H_{1})\), further assume that
$$(H_{2})\quad r_{i}\leq1,\quad i=1,2. $$
Then system (1.1) admits a unique positive equilibrium
\((x^{*}_{1},x^{*}_{2})\), which is globally stable.
Now let us state several lemmas, which will be useful in the proof of Theorem 3.1.
Lemma 3.1
[1]
Let
\(f(u)=u\exp(\alpha \beta u)\), where
α
and
β
are positive constants. Then
\(f(u)\)
is nondecreasing for
\(u\in(0, \frac{1}{\beta}]\).
Lemma 3.2
[1]
Assume that the sequence
\({u(n)}\)
satisfies
$$u(n+1)=u(n)\exp\bigl(\alpha\beta u(n)\bigr),\quad n=1,2,\ldots, $$
where
α
and
β
are positive constants, and
\(u(0)>0\). Then:

(i)
If
\(\alpha<2\), then
\(\lim_{n\rightarrow+\infty} u(n)= \frac{\alpha}{\beta}\).

(ii)
If
\(\alpha\leq1\), then
\(u(n)\leq\frac{1}{\beta}, n=2,3,\ldots\) .
Lemma 3.3
[15]
Suppose that the functions
\(f,g: Z_{+}\times[0,\infty)\)
satisfy
\(f(n,x)\leq g(n,x) (f(n,x)\geq g(n,x))\)
for
\(n\in Z_{+}\)
and
\(g(n,x)\)
is nondecreasing with respect to
x. If
\({u(n)}\)
are the nonnegative solutions of the difference equations
$$\begin{aligned} x(n+1)=f\bigl(n,x(n)\bigr), \qquad u(n+1)=g\bigl(n,u(n)\bigr), \end{aligned}$$
respectively, and
\(x(0)\leq u(0)\)
\((x(0)\geq u(0))\), then
$$\begin{aligned} x(n)\leq u(n), \qquad\bigl(x(n)\geq u(n)\bigr)\quad \textit{for all } n\geq0. \end{aligned}$$
Proof of Theorem 3.1
Let \({(x_{1}(n),x_{2}(n))}\) be any positive solution of system (1.1). Denoting
$$\begin{aligned} &\liminf_{n\rightarrow+\infty}x_{1}(n)=m_{1},\qquad \limsup _{n\rightarrow+\infty} x_{1}(n)=M_{1}, \\ &\liminf_{t\rightarrow+\infty}x_{2}(n)=m_{2},\qquad \limsup _{n\rightarrow+\infty} x_{2}(n)=M_{2}, \end{aligned}$$
we claim that, under the assumptions of Theorem 3.1, \(M_{1}=m_{1}=x^{*}_{1}\) and \(M_{2}=m_{2}=x^{*}_{2}\).
From the first equation of system (1.1) we obtain
$$\begin{aligned} x_{1}(n+1)&=x_{1}(n)\exp{ \biggl[r_{1}a_{1}x_{1}(n) \frac {b_{1}x_{2}(n)}{1+c_{2}x_{2}(n)} \biggr]} \\ &\leq x_{1}(n)\exp{\bigl[r_{1}a_{1}x_{1}(n) \bigr]}. \end{aligned}$$
(3.1)
Consider the following auxiliary equation:
$$\begin{aligned} u(n+1)=u(n)\exp{\bigl[r_{1}a_{1}u(n) \bigr]}. \end{aligned}$$
(3.2)
By Lemma 3.2(ii), because of \(r_{1}\leq1\), we obtain \(u(n)\leq \frac{1}{a_{1}}\) for all \(n\geq2\), where \(u(n)\) is an arbitrary positive solution of (3.2) with initial value \(u(0)>0\). By Lemma 3.1, \(f(u)=u\exp(r_{1}a_{1}u)\) is nondecreasing for \(u\in (0, \frac{1}{a_{1}}]\). Based on Lemma 3.3, we obtain \(x_{1}(n)\leq u(n)\) for all \(n\geq2\), where \(u(n)\) is the solution of (3.2) with initial value \(u(2)=x_{1}(2)\). By Lemma 3.2(i) we obtain that
$$\begin{aligned} M_{1}=\limsup_{n\rightarrow+\infty} x_{1}(n)\leq\lim_{n\rightarrow+\infty} u(n)= \frac{r_{1}}{a_{1}}. \end{aligned}$$
(3.3)
From the second equation of system (1.1) we obtain
$$\begin{aligned} x_{2}(n+1)&=x_{2}(n)\exp{\biggl[r_{2}a_{2}x_{2}(n) \frac {b_{2}x_{1}(n)}{1+c_{1}x_{1}(n)}\biggr]} \\ &\leq x_{2}(n)\exp{\bigl[r_{2}a_{2}x_{2}(n) \bigr].} \end{aligned}$$
Similarly to the analysis of (3.1)(3.3), we have
$$\begin{aligned} M_{2}=\limsup_{n\rightarrow+\infty} x_{2}(n) \leq\frac {r_{2}}{a_{2}}. \end{aligned}$$
(3.4)
Then, for a sufficiently small constant \(\varepsilon>0\), without loss of generality, it follows from (3.3) and (3.4) that there exists an integer \(n_{1}>2\) such that, for all \(n>n_{1}\),
$$\begin{aligned} x_{1}(n)< \frac{r_{1}}{a_{1}}+\varepsilon\stackrel{ \mathrm {def}}{=} {M^{(1)}_{1}},\qquad x_{2}(n)< \frac{r_{2}}{a_{2}}+\varepsilon \stackrel{\mathrm{def}}{=} {M^{(1)}_{2}}. \end{aligned}$$
(3.5)
For \(n>n_{1}\), the second inequality of (3.5), combined with the first equation of system (1.1), leads to
$$\begin{aligned} x_{1}(n+1)&=x_{1}(n)\exp{ \biggl[r_{1}a_{1}x_{1}(n) \frac {b_{1}x_{2}(n)}{1+c_{2}x_{2}(n)} \biggr]} \\ &\geq x_{1}(n)\exp{\biggl[r_{1}a_{1}x_{1}(n) \frac {b_{1}M^{(1)}_{2}}{1+c_{2}M^{(1)}_{2}}\biggr]}. \end{aligned}$$
(3.6)
Consider the auxiliary equation
$$\begin{aligned} u(n+1)=u(n)\exp{\biggl[r_{1}a_{1}u(n) \frac {b_{1}M^{(1)}_{2}}{1+c_{2}M^{(1)}_{2}}\biggr]}. \end{aligned}$$
(3.7)
Since \(r_{1}\leq1\), according to Lemma 3.2(ii), we obtain \(u(n)\leq \frac{1}{a_{1}}\) for all \(n\geq n_{1}\), where \(u(n)\) is an arbitrary positive solution of (3.7) with initial value \(u(n_{1})>0\). By Lemma 3.1, \(f(u)=u\exp(r_{1}a_{1}u \frac {b_{1}M^{(1)}_{2}}{1+c_{2}M^{(1)}_{2}})\) is nondecreasing for \(u\in (0, \frac{1}{a_{1}}]\). According to Lemma 3.3, we obtain \(x_{1}(n)\geq u(n)\) for all \(n\geq n_{1}\), where \(u(n)\) is the solution of (3.7) with the initial value \(u(n_{1})=x_{1}(n_{1})\). According to Lemma 3.2(i), we have
$$\begin{aligned} m_{1}=\liminf_{n\rightarrow+\infty} x_{1}(n)\geq\lim_{n\rightarrow+\infty} u(n)= \frac{r_{1} \frac {b_{1}M^{(1)}_{2}}{1+c_{2}M^{(1)}_{2}}}{a_{1}}. \end{aligned}$$
(3.8)
The first inequality of (3.5), combined with the second equation of system (1.1), leads to
$$\begin{aligned} x_{2}(n+1)&=x_{2}(n)\exp{\biggl[r_{2}a_{2}x_{2}(n) \frac {b_{2}x_{1}(n)}{1+c_{1}x_{1}(n)}\biggr]} \\ &\geq x_{2}(n)\exp{\biggl[r_{2}a_{2}x_{2}(n) \frac {b_{2}M^{(1)}_{1}}{1+c_{1}M^{(1)}_{1}}\biggr].} \end{aligned}$$
Similarly to the analysis of (3.6)(3.8), we have
$$\begin{aligned} m_{2}=\liminf_{n\rightarrow+\infty} x_{2}(n) \geq\frac {r_{2} \frac{b_{2}M^{(1)}_{1}}{1+c_{1}M^{(1)}_{1}}}{a_{2}}. \end{aligned}$$
(3.9)
Then, for the above \(\varepsilon>0\), there exists an integer \(n_{2}>n_{1}\) such that, for all \(n>n_{2}\),
$$\begin{aligned} \begin{aligned} & x_{1}(n)> \frac{r_{1} \frac {b_{1}M^{(1)}_{2}}{1+c_{2}M^{(1)}_{2}}}{a_{1}}\varepsilon\stackrel{ \mathrm {def}}{=} {m^{(1)}_{1}}, \\ & x_{2}(n)> \frac{r_{2} \frac {b_{2}M^{(1)}_{1}}{1+c_{1}M^{(1)}_{1}}}{a_{2}}\varepsilon\stackrel{\mathrm {def}}{=} {m^{(1)}_{2}}. \end{aligned} \end{aligned}$$
(3.10)
For \(n>n_{2}\), the second inequality of (3.10), combined with the first equation of system (1.1), leads to
$$\begin{aligned} x_{1}(n+1)&=x_{1}(n)\exp{ \biggl[r_{1}a_{1}x_{1}(n) \frac {b_{1}x_{2}(n)}{1+c_{2}x_{2}(n)} \biggr]} \\ &\leq x_{1}(n)\exp{\biggl[r_{1}a_{1}x_{1}(n) \frac {b_{1}m^{(1)}_{2}}{1+c_{2}m^{(1)}_{2}}\biggr]}. \end{aligned}$$
(3.11)
Similarly to the analysis of (3.1)(3.3), we have
$$\begin{aligned} M_{1}=\limsup_{n\rightarrow+\infty} x_{1}(n)\leq\frac {r_{1} \frac{b_{1}m^{(1)}_{2}}{1+c_{2}m^{(1)}_{2}}}{a_{1}}. \end{aligned}$$
(3.12)
The first inequality of (3.10), combined with the second equation of system (1.1), leads to
$$\begin{aligned} x_{2}(n+1)&=x_{2}(n)\exp{\biggl[r_{2}a_{2}x_{2}(n) \frac {b_{2}x_{1}(n)}{1+c_{1}x_{1}(n)}\biggr]} \\ &\leq x_{2}(n)\exp{\biggl[r_{2}a_{2}x_{2}(n) \frac {b_{2}m^{(1)}_{1}}{1+c_{1}m^{(1)}_{1}}\biggr]}. \end{aligned}$$
(3.13)
Similarly to the analysis of (3.1)(3.3), we have
$$\begin{aligned} M_{2}=\limsup_{n\rightarrow+\infty} x_{2}(n) \leq\frac {r_{2} \frac{b_{2}m^{(1)}_{1}}{1+c_{1}m^{(1)}_{1}}}{a_{2}}. \end{aligned}$$
(3.14)
Then, for the above \(\varepsilon>0\), it follows from (3.12) and (3.14) that there exists an integer \(n_{3}>n_{2}\) such that, for all \(n>n_{3}\),
$$\begin{aligned} \begin{aligned}&x_{1}(n)< \frac{r_{1} \frac {b_{1}m^{(1)}_{2}}{1+c_{2}m^{(1)}_{2}}}{a_{1}}+ \frac{\varepsilon }{2} \stackrel{\mathrm{def}}{=} {M^{(2)}_{1}}, \\ &x_{2}(n)< \frac{r_{2} \frac {b_{2}m^{(1)}_{1}}{1+c_{1}m^{(1)}_{1}}}{a_{2}}+ \frac{\varepsilon }{2}\stackrel{ \mathrm{def}}{=} {M^{(2)}_{2}}. \end{aligned} \end{aligned}$$
(3.15)
Obviously,
$$\begin{aligned} M^{(2)}_{1}< M^{(1)}_{1},\qquad M^{(2)}_{2}< M^{(1)}_{2}. \end{aligned}$$
(3.16)
For \(n>n_{3}\), the second inequality of (3.15), combined with the first equation of system (1.1), leads to
$$\begin{aligned} x_{1}(n+1)&=x_{1}(n)\exp{ \biggl[r_{1}a_{1}x_{1}(n) \frac {b_{1}x_{2}(n)}{1+c_{2}x_{2}(n)} \biggr]} \\ &\geq x_{1}(n)\exp{\biggl[r_{1}a_{1}x_{1}(n) \frac {b_{1}M^{(2)}_{2}}{1+c_{2}M^{(2)}_{2}}\biggr]}. \end{aligned}$$
(3.17)
Similarly to the analysis of (3.6)(3.8), we have
$$\begin{aligned} m_{1}=\liminf_{n\rightarrow+\infty} x_{1}(n)\geq\frac {r_{1} \frac{b_{1}M^{(2)}_{2}}{1+c_{2}M^{(2)}_{2}}}{a_{1}}. \end{aligned}$$
(3.18)
The first inequality of (3.15), combined with the second equation of system (1.1), leads to
$$\begin{aligned} x_{2}(n+1)&=x_{2}(n)\exp{\biggl[r_{2}a_{2}x_{2}(n) \frac {b_{2}x_{1}(n)}{1+c_{1}x_{1}(n)}\biggr]} \\ &\geq x_{2}(n)\exp{\biggl[r_{2}a_{2}x_{2}(n) \frac {b_{2}M^{(2)}_{1}}{1+c_{1}M^{(2)}_{1}}\biggr].} \end{aligned}$$
Similarly to the analysis of (3.6)(3.8), we have
$$\begin{aligned} m_{2}=\liminf_{n\rightarrow+\infty} x_{2}(n) \geq\frac {r_{2} \frac{b_{2}M^{(2)}_{1}}{1+c_{1}M^{(2)}_{1}}}{a_{2}}. \end{aligned}$$
(3.19)
Then, for the above \(\varepsilon>0\), it follows from (3.18) and (3.19) that there exists an integer \(n_{4}>n_{3}\) such that, for all \(n>n_{4}\),
$$\begin{aligned} \begin{aligned}& x_{1}(n)> \frac{r_{1} \frac {b_{1}M^{(2)}_{2}}{1+c_{2}M^{(2)}_{2}}}{a_{1}} \frac{\varepsilon }{2} \stackrel{\mathrm{def}}{=} {m^{(2)}_{1}}, \\ &x_{2}(n)> \frac{r_{2} \frac {b_{2}M^{(2)}_{1}}{1+c_{1}M^{(2)}_{1}}}{a_{2}} \frac{\varepsilon }{2}\stackrel{ \mathrm{def}}{=} {m^{(2)}_{2}}. \end{aligned} \end{aligned}$$
(3.20)
Obviously,
$$\begin{aligned} m^{(1)}_{1}< m^{(2)}_{1},\qquad m^{(1)}_{2}< m^{(2)}_{2}. \end{aligned}$$
(3.21)
Continuing the above steps, we can get four sequences \(\{M^{(n)}_{i}\}, \{m^{(n)}_{i}\}, i=1,2, n=1,2,\ldots\) , such that, for \(n\geq2\),
$$\begin{aligned} \begin{aligned}& M^{(n)}_{1}= \frac{r_{1} \frac {b_{1}m^{(n1)}_{2}}{1+c_{2}m^{(n1)}_{2}}}{a_{1}}+ \frac {\varepsilon}{n}; \qquad M^{(n)}_{2}= \frac{r_{2} \frac{b_{2}m^{(n1)}_{1}}{1+c_{1}m^{(n1)}_{1}}}{a_{2}}+ \frac {\varepsilon}{n}; \\ &m^{(n)}_{1}= \frac{r_{1} \frac {b_{1}M^{(n)}_{2}}{1+c_{2}M^{(n)}_{2}}}{a_{1}} \frac{\varepsilon }{n};\qquad m^{(n)}_{2}= \frac{r_{2} \frac {b_{2}M^{(n)}_{1}}{1+c_{1}M^{(n)}_{1}}}{a_{2}} \frac{\varepsilon }{n}. \end{aligned} \end{aligned}$$
(3.22)
Clearly, we have
$$\begin{aligned} m^{(n)}_{i}\leq m_{i}\leq M_{i}\leq M^{(n)}_{i},\quad i=1,2, n=1,2,\ldots. \end{aligned}$$
(3.23)
Now, by means of the inductive method we will prove that \(\{ M^{(n)}_{1}\}, \{M^{(n)}_{2}\}\) are decreasing and \(\{m^{(n)}_{1}\} , \{m^{(n)}_{2}\}\) are increasing.
First of all, from (3.16) and (3.21) it is clear that
$$M^{(2)}_{i}< M^{(1)}_{i},\qquad m^{(2)}_{i}>m^{(1)}_{i},\quad i=1,2. $$
Let us assume that our claim is true for n, that is,
$$M^{(n)}_{i}< M^{(n1)}_{i},\qquad m^{(n)}_{i}>m^{(n1)}_{i},\quad i=1,2. $$
Again, since the function \(g(x)= \frac{bx}{1+cx}\ (b,c>0)\) is strictly increasing, we immediately obtain
$$\begin{aligned} &M^{(n+1)}_{1}= \frac{r_{1} \frac {b_{1}m^{(n)}_{2}}{1+c_{2}m^{(n)}_{2}}}{a_{1}}+ \frac{\varepsilon }{n+1}< \frac{r_{1} \frac {b_{1}m^{(n1)}_{2}}{1+c_{2}m^{(n1)}_{2}}}{a_{1}}+ \frac {\varepsilon}{n}\stackrel{\mathrm{def}}{=} {M^{(n)}_{1}}; \\ &M^{(n+1)}_{2}= \frac{r_{2} \frac {b_{2}m^{(n)}_{1}}{1+c_{1}m^{(n)}_{1}}}{a_{2}}+ \frac{\varepsilon }{n+1}< \frac{r_{2} \frac {b_{2}m^{(n1)}_{1}}{1+c_{1}m^{(n1)}_{1}}}{a_{2}}+ \frac {\varepsilon}{n}\stackrel{\mathrm{def}}{=} {M^{(n)}_{2}}; \\ &m^{(n+1)}_{1}= \frac{r_{1} \frac {b_{1}M^{(n+1)}_{2}}{1+c_{2}M^{(n+1)}_{2}}}{a_{1}} \frac {\varepsilon}{n+1}> \frac{r_{1} \frac {b_{1}M^{(n)}_{2}}{1+c_{2}M^{(n)}_{2}}}{a_{1}} \frac{\varepsilon }{n}\stackrel{\mathrm{def}}{=} {m^{(n)}_{1}}; \\ &m^{(n+1)}_{2}= \frac{r_{2} \frac {b_{2}M^{(n+1)}_{1}}{1+c_{1}M^{(n+1)}_{1}}}{a_{2}} \frac {\varepsilon}{n+1}> \frac{r_{2} \frac {b_{2}M^{(n)}_{1}}{1+c_{1}M^{(n)}_{1}}}{a_{2}} \frac{\varepsilon }{n}\stackrel{\mathrm{def}}{=} {m^{(n)}_{2}}. \end{aligned}$$
These inequalities show that \({\{M^{(n)}_{1}\}}\) and \({\{M^{(n)}_{2}\} }\) are decreasing, \({\{m^{(n)}_{1}\}}\) and \({\{m^{(n)}_{2}\}}\) are increasing. Let
$$\begin{aligned} & \lim_{n\rightarrow+\infty} {M^{(n)}_{1}}= \bar {x}_{1},\qquad \lim_{n\rightarrow+\infty} {M^{(n)}_{2}}= \bar {x}_{2}, \end{aligned}$$
(3.24)
$$\begin{aligned} & \lim_{n\rightarrow+\infty} {m^{(n)}_{1}}= \underline {x}_{1}, \qquad \lim_{n\rightarrow+\infty} {m^{(n)}_{2}}= \underline{x}_{2}. \end{aligned}$$
(3.25)
Letting \(n\rightarrow+\infty\) in (3.22), we obtain
$$\begin{aligned} & \bar{x}_{1}= \frac{r_{1} \frac{b_{1}\underline {x}_{2}}{1+c_{2}\underline{x}_{2}}}{a_{1}},\qquad \bar{x}_{2}= \frac {r_{2} \frac{b_{2}\underline{x}_{1}}{1+c_{1}\underline {x}_{1}}}{a_{2}}, \end{aligned}$$
(3.26)
$$\begin{aligned} &\underline{x}_{1}= \frac{r_{1} \frac{b_{1}\bar {x}_{2}}{1+c_{2}\bar{x}_{2}}}{a_{1}},\qquad \underline{x}_{2}= \frac{r_{2} \frac{b_{2}\bar{x}_{1}}{1+c_{1}\bar{x}_{1}}}{a_{2}}. \end{aligned}$$
(3.27)
Equations (3.26) and (3.27) are equivalent to
$$\begin{aligned} & a_{1}\bar{x}_{1}+ \frac{b_{1}\underline{x}_{2}}{1+c_{2}\underline {x}_{2}}=r_{1},\qquad a_{2}\bar{x}_{2}+ \frac{b_{2}\underline {x}_{1}}{1+c_{1}\underline{x}_{1}}=r_{2}, \end{aligned}$$
(3.28)
$$\begin{aligned} & a_{1}\underline{x}_{1}+ \frac{b_{1}\bar{x}_{2}}{1+c_{2}\bar {x}_{2}}=r_{1},\qquad a_{2}\underline{x}_{2}+ \frac{b_{2}\bar {x}_{1}}{1+c_{1}\bar{x}_{1}}=r_{2}. \end{aligned}$$
(3.29)
Equations (3.28) and (3.29) show that \((\bar {x}_{1},\bar {x}_{2})\) and \((\underline{x}_{1},\underline{x}_{2})\) are solutions of system (1.1). However, under the assumptions of Theorem 3.1, system (1.1) admits a unique positive solution \((x^{*}_{1},x^{*}_{2}) \). Therefore
$$\begin{aligned} M_{1}=m_{1}= \lim_{n\rightarrow+\infty }x_{1}(n)=x^{*}_{1},\qquad M_{2}=m_{2}= \lim_{n\rightarrow+\infty} x_{2}(n)=x^{*}_{2}. \end{aligned}$$
(3.30)
Thus, the unique interior equilibrium \(E(x^{*}_{1},x^{*}_{2})\) is globally attractive. This completes the proof of Theorem 3.1. □