1 Introduction

In recent years, the dynamical behaviors of the discrete-time predator–prey systems have been widely investigated. Many important and interesting results can be found in many articles, such as in [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] and the references cited therein. Particularly, the discrete two-species predator–prey systems with ratio-dependent functional responses were studied in [10,11,12,13,14,15,16,17, 23, 25]. What interested them are the dynamical behaviors, such as, the study for the local and global stability of the equilibria, the persistence, permanence and extinction of species, the existence of positive periodic solutions and positive almost periodic solutions, the bifurcation and chaos phenomenon, etc.. Recently, Chen and Zhou [17] discussed the global stability for a nonautonomous two species discrete competition system. However, the conditions of their results in [17] is strong and complicated. Therefore, as an extension and improvement, we discuss in the present paper the following discrete-time two-species competition system:

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{ll} x(k+1)=x(k)\exp \left[ r_{1}\left( 1-\frac{x^{m}(k)}{K_{1}}-\mu _{2}y^{n}(k)\right) \right] ,\\ &{}\\ y(k+1)=y(k)\exp \left[ r_{2}\left( 1-\mu _{1}x^{m}(k)-\frac{y^{n}(k)}{K_{2}}\right) \right] . \end{array} \end{array}\right. } \end{aligned}$$
(1.1)

where x(k) and y(k) represent the sizes or the densities of species x and y at kth generation, respectively. Parameters \(r_{i}\), \(K_{i}\) and \(\mu _{i}\) \((i = 1, 2)\) are positive constants and represent the intrinsic growth rates, the carrying capacities, and the competition coefficients of species x and y, respectively. m and n are arbitrary positive integer.

In this paper, we will introduce a new method to discuss the global asymptotic stability of system (1.1). The main results of this paper is to establish the criteria on the existence and local asymptotic stability of equilibria for system (1.1) by using the linear approximation method, and obtain some new sufficient conditions on the global stability of the positive equilibrium for system (1.1) by using the iterative scheme method and the comparison principle of difference equations.

2 Preliminary Lemmas

Let (x(k), y(k)) be any solution of system (1.1) satisfying the initial value \(x(0)>0\) and \(y(0)>0\) considered the biological background of system (1.1). It is clear that any solution (x(k), y(k)) of system (1.1) is defined on \(Z_{+}\) and always remains positive, where \(Z_{+}\) denotes the set of all nonnegative integers.

What interested us is the positive equilibrium of system (1.1). By a simple computation, we directly obtain the following results.

Lemma 2.1

If \(1-\mu _{1}K_{1}>0\) and \(1-\mu _{2}K_{2}>0\), then system (1.1) has a unique positive equilibrium \(E_{+}(x_{0}, y_{0})\), where

$$\begin{aligned} x_{0}^{m}=\frac{K_{1}(1-\mu _{2}K_{2})}{1-\mu _{1}\mu _{2}K_{1}K_{2}},\quad y_{0}^{n}=\frac{K_{2}(1-\mu _{1}K_{1})}{1-\mu _{1}\mu _{2}K_{1}K_{2}}. \end{aligned}$$

Further, we need the following lemma, which can be easily proved by the relations between roots and coefficients of a quadratic equation.

Lemma 2.2

Consider the function \(F(\lambda )=\lambda ^{2}+p\lambda +q\), here, both p and q are constants. Suppose \(F(1)>0\) and \(\lambda _{1},\lambda _{2}\) are two roots of the quadratic equation \(F(\lambda )=0\). Then we can easily prove that

  1. 1.

    \(|\lambda _{1}|<1\) and \(|\lambda _{2}|<1\) if and only if \(F(-1)>0\) and \(q < 1\);

  2. 2.

    \(|\lambda _{1}|<1\) and \(|\lambda _{2}|>1\) if and only if \(F(-1)<0\);

  3. 3.

    \(|\lambda _{1}|>1\) and \(|\lambda _{2}|>1\) if and only if \(F(-1)>0\) and \(q > 1\);

  4. 4.

    \(\lambda _{1}=-1\) and \(|\lambda _{2}|\ne 1\) if and only if \(F(-1)=0\) and \(p\ne 0,2\);

  5. 5.

    \(\lambda _{1}\) and \(\lambda _{2}\) is a pair of conjugate complex root and \(|\lambda _{1}|=|\lambda _{2}|=1\) if and only if \(p^{2}-4q<0\) and \(q = 1\).

Here, with \(\lambda _{1}\) and \(\lambda _{2}\) be the two roots of the characteristic equation \(F(\lambda )=\lambda ^{2}+p\lambda +q=0\) of J(x, y), we have the following definitions.

  1. 1.

    If \(|\lambda _{1}|<1\) and \(|\lambda _{2}|<1\), then J(x, y) is called a sink and is locally asymptotic stable;

  2. 2.

    If \(|\lambda _{1}|>1\) and \(|\lambda _{2}|>1\) , then J(x, y) is called a source and is unstable;

  3. 3.

    If \(|\lambda _{1}|>1\) and \(|\lambda _{2}|<1\)(or \(|\lambda _{1}|<1\) and \(|\lambda _{2}|>1\)) , then J(x, y) is called a saddle and is unstable;

  4. 4.

    If \(|\lambda _{1}|=1\) or \(|\lambda _{2}|=1\), then J(x, y) is called non-hyperbolic.

Lemma 2.3

Let \(f(u)=u\exp (\alpha -\beta u^{n})\), where, \(\alpha \) and \(\beta \) are both positive constants, n is any a positive integer, then f(u) is nondecreasing on \(u\in \big (0,\root n \of {\frac{1}{n\beta }}\big ]\).

Lemma 2.4

If the sequence \(\{u(k)\}\) satisfies

$$\begin{aligned} u(k+1)=u(k)\exp \left( \alpha -\beta u^{n}(k)\right) ,\quad k=1,2,\ldots , \end{aligned}$$

here, \(\alpha \) and \(\beta \) are both positive constants, n is any a positive integer and \(u(0) > 0\). Then

  1. 1.

    If \(\alpha <\frac{2}{n}\), then \(\lim _{k\rightarrow \infty }u(k)=\root n \of {\frac{\alpha }{\beta }}\).

  2. 2.

    If \(\alpha \le \frac{1}{n}\), then \(u(k)\le \root n \of {\frac{1}{\beta n}}\) for all \(k=2,3,\ldots \).

Proof

Conclusion (1) can be proved using Theorem 2.8 in [4], so we omit it.

Note that the function \(x\exp (\alpha -\beta x^{n})\) has a unique maximum in \(x=\root n \of {\frac{1}{\beta n}}\), then

$$\begin{aligned} u(k+1)= & {} u(k)\exp \left( \alpha -\beta u^{n}(k)\right) \\\le & {} \root n \of {\frac{1}{\beta n}}\exp \left( \alpha -\frac{1}{n}\right) \le \root n \of {\frac{1}{\beta n}},\quad n=1,2,\ldots , \end{aligned}$$

then conclusion (2) is proved. This ends the proof. \(\square \)

Lemma 2.5

(see [23]) Assume that functions \(f, g: Z_{+} \times [0,\infty ) \rightarrow [0,\infty )\) satisfy \(f(n, x) \le g(n, x) (f(n, x) \ge g(n, x))\) for \(n \in Z_{+}\) and \(x \in [0,\infty )\), g(n, x) is nondecreasing for \(x > 0\). Let sequences \(\{x(n)\}\) and \(\{u(n)\}\) be the nonnegative solutions of the following difference equations

$$\begin{aligned} x(n + 1) = f(n, x(n)), \quad u(n+1) = g(n, u(n)), \quad n= 0, 1, 2,\ldots , \end{aligned}$$

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

$$\begin{aligned} x(n) \le u(n) (x(n) \ge u(n)). \end{aligned}$$

3 Local Stability

In this section, we use the eigenvalues of the variational matrix of system (1.1) at the equilibria \(E_{+}(x_{0}, y_{0})\) to study its local stability.

Let \(J(E_{+})\) be the variational matrix of system (1.1) at equilibrium \(E_{+}(x_{0}, y_{0})\), then

$$\begin{aligned} J(E_{+})=\left( \begin{array}{cc}1-\frac{mr_{1}x_{0}^{m}}{K_{1}} &{}\quad -nr_{1}\mu _{2}x_{0}y_{0}^{n-1} \\ -mr_{2}\mu _{1}x_{0}^{m-1}y_{0} &{}\quad 1-\frac{nr_{2}y_{0}^{n}}{K_{2}} \end{array}\right) . \end{aligned}$$

The corresponding characteristic equation of \(J(E_{+})\) can be written as

$$\begin{aligned} F(\lambda )=\lambda ^{2}+p\lambda +q=0, \end{aligned}$$
(3.1)

where

$$\begin{aligned} p= & {} -\left( 2-\frac{mr_{1}x_{0}^{m}}{K_{1}}-\frac{nr_{2}y_{0}^{n}}{K_{2}}\right) ,\\ q= & {} \left( 1-\frac{mr_{1}x_{0}^{m}}{K_{1}}\right) \left( 1-\frac{nr_{2}y_{0}^{n}}{K_{2}}\right) -mnr_{1}r_{2}\mu _{1}\mu _{2}x_{0}^{m}y_{0}^{n}. \end{aligned}$$

Then we have the following result.

Theorem 3.1

Assume that \(1-\mu _{1}K_{1}>0\) and \(1-\mu _{2}K_{2}>0\), then we have

  1. 1.

    \(E_{+}(x_{0}, y_{0})\) is a sink if one of the following conditions holds:

    1. (a)

      \(r_{1}<t_{2},r_{2}<t_{1},r_{2}\le \frac{1}{n(1-\mu _{1}K_{1})}\), where

      $$\begin{aligned} t_{1}=\frac{2(1-\mu _{1}\mu _{2}K_{1}K_{2})}{n(1-\mu _{1}K_{1})},\quad t_{2}=\frac{2[2(1-\mu _{1}\mu _{2}K_{1}K_{2})-nr_{2}(1 -\mu _{1}K_{1})]}{m(1-\mu _{2}K_{2})[2-nr_{2}(1-\mu _{1}K_{1})]}. \end{aligned}$$
    2. (b)

      \(t_{1}>r_{2}>\frac{1}{n(1-\mu _{1}K_{1})}\) and \(r_{1}<\min \{t_{2},t_{3}\}\), where

      $$\begin{aligned} t_{3}=\frac{nr_{2}(1-\mu _{1}K_{1})}{m(1-\mu _{2}K_{2})(nr_{2}(1 -\mu _{1}K_{1})-1)}. \end{aligned}$$
    3. (c)

      \(r_{2}>t_{4}\) and \(t_{3}>r_{1}>t_{2}\),where \(t_{4}=\frac{2}{n(1-\mu _{1}K_{1})}.\)

  2. 2.

    \(E_{+}(x_{0}, y_{0})\) is a source if one of the following conditions holds:

    1. (a)

      \(\frac{1}{1-\mu _{1}K_{1}}\le r_{2}<t_{1}\) and \(t_{3}<r_{1}< t_{2}\);

    2. (b)

      \(r_{2}>t_{4}\) and \(r_{1}> \max \{t_{2},t_{3}\}.\)

  3. 3.

    \(E_{+}(x_{0}, y_{0})\) is non-hyperbolic if one of the following conditions holds:

    1. (a)

      \(r_{1}= t_{2}\) and \(r_{2}= t_{1}\);

    2. (b)

      \(r_{1}= t_{2}\) and \(r_{2}>t_{4}.\);

  4. 4.

    \(E_{+}(x_{0}, y_{0})\) is a saddle if one of the following conditions holds:

    1. (a)

      \(r_{2}<t_{1}\) and \(r_{1}> t_{2}\);

    2. (b)

      \(t_{1}\le r_{2}\le t_{4}\);

    3. (c)

      \(r_{2}>t_{4}\) and \(r_{1}<t_{2}.\)

Proof

Here, we only prove conclusion (1) of Theorem 3.1. The others can also be proved by the same way.

From (3.1), we have

$$\begin{aligned} F(1)= & {} 1+p+q=mnr_{1}r_{2}x_{0}^{m}y_{0}^{n}\frac{1 -\mu _{1}\mu _{2}K_{1}K_{2}}{K_{1}K_{2}}> 0,\\ F(-1)= & {} 1-p+q=4-2\left( \frac{mr_{1}x_{0}^{m}}{K_{1}} +\frac{nr_{2}y_{0}^{n}}{K_{2}}\right) + mnr_{1}r_{2}x_{0}^{m}y_{0}^{n}\frac{1-\mu _{1}\mu _{2}K_{1}K_{2}}{K_{1}K_{2}}\\= & {} \frac{4(1-\mu _{1}\mu _{2}K_{1}K_{2})-2nr_{2}(1-\mu _{1}K_{1})}{1-\mu _{1}\mu _{2}K_{1}K_{2}}\\&-\frac{mr_{1}(1-\mu _{2}K_{2})[2-nr_{2}(1-\mu _{1}K_{1})]}{1-\mu _{1}\mu _{2}K_{1}K_{2}}, \end{aligned}$$

and

$$\begin{aligned} q-1 =\frac{mr_{1}(1-\mu _{2}K_{2})[nr_{2}(1-\mu _{1}K_{1})-1] -nr_{2}(1-\mu _{1}K_{1})}{1-\mu _{1}\mu _{2}K_{1}K_{2}}. \end{aligned}$$

If \(2(1-\mu _{1}\mu _{2}K_{1}K_{2})-nr_{2}(1-\mu _{1}K_{1})>0\), then we have \(r_{2} < t_{1}\) and \(2-nr_{2}(1-\mu _{1}K_{1})>0\). Hence, \(F(-1) > 0\) if

$$\begin{aligned} r_{1}<\frac{2[2(1-\mu _{1}\mu _{2}K_{1}K_{2})-nr_{2}(1 -\mu _{1}K_{1})]}{m(1-\mu _{2}K_{2})[2-nr_{2}(1 -\mu _{1}K_{1})]}\triangleq t_{2}. \end{aligned}$$

If \(nr_{2}(1-\mu _{1}K_{1})-1\le 0\), then \(q<1.\) If \(nr_{2}(1-\mu _{1}K_{1})-1> 0\), then \(q<1\) is equivalent to the following inequality

$$\begin{aligned} r_{1}<\frac{nr_{2}(1-\mu _{1}K_{1})}{m(1-\mu _{2}K_{2})(nr_{2}(1 -\mu _{1}K_{1})-1)}\triangleq t_{3}. \end{aligned}$$

Hence, if condition (a) or (b) of conclusion (1) of Theorem 3.1 holds, then we have \(F(-1) > 0\) and \(q <1\). From Lemma 2.2, we can obtain \(E_{+}(x_{0}, y_{0})\) in system (1.1) is a sink.

On the other hand, if \(r_{2}>\frac{2}{n(1-\mu _{1}K_{1})}\triangleq t_{4},\) then we have \(2(1-\mu _{1}\mu _{2}K_{1}K_{2})-nr_{2}(1-\mu _{1}K_{1})< 0.\) Hence, \(F(-1) > 0\) if \(r_{1} < t_{3}\). Since \(r_{2} > t_{4}\), a similar argument as in above we have \(q <1\) if \(r_{1} < t_{3}\). Hence, if condition (c) of conclusion (1) of Theorem 3.1 holds, then we have \(F(-1) > 0\) and \(q <1\). From Lemma 2.2, we obtain \(E_{+}(x_{0}, y_{0})\) in system (1.1) is also a sink. This completes the proof. \(\square \)

4 Global Stability

In this section, we will use the method of iteration scheme and the comparison principle of difference equations to study the global stability of the positive equilibrium of system (1.1).

Theorem 4.1

Assume that \(1-\mu _{1}K_{1}>0\) and \(1-\mu _{2}K_{2}>0\). If \(r_{1}\le \frac{1}{m}\) and \(r_{2}\le \frac{1}{n}\), then equilibrium \(E_{+}(x_{0}, y_{0})\) of system (1.1) is globally asymptotically stable.

Proof

Assume that (x(k), y(k)) is any a solution of system (1.1) with initial value \(x(0) > 0\) and \(y(0) > 0\). Let

$$\begin{aligned} U_{1}= & {} \limsup _{k\rightarrow \infty }x(k) \qquad V_{1}=\liminf _{k\rightarrow \infty }x(k),\\ U_{2}= & {} \limsup _{k\rightarrow \infty }y(k) \qquad V_{2}=\liminf _{k\rightarrow \infty }y(k). \end{aligned}$$

In the following, we will prove that \(U_{1} = V_{1} = x_{0}\) and \(U_{2} = V_{2} = y_{0}\).

From the first equation of system (1.1) we obtain

$$\begin{aligned} x(k+1)\le x(k)\exp \left( r_{1}-\frac{r_{1}}{K_{1}}x^{m}(k)\right) ,\quad k=0,1,2,\ldots . \end{aligned}$$

Consider the auxiliary equation

$$\begin{aligned} u(k+1)= u(k)\exp \left( r_{1}-\frac{r_{1}}{K_{1}}u^{m}(k)\right) . \end{aligned}$$
(4.1)

Let u(k) be any a solution of Eq. (4.1) with initial value \(u(0) > 0 \). For \(0<r_{1}\le \frac{1}{m}\), by conclusion (2) of Lemma 2.4, we have that \(u(k) \le \root m \of {\frac{K_{1}}{mr_{1}}}\) for all \(n \ge 2 \). From Lemma 2.3, we have \(f(u) = u \exp (r_{1}-\frac{r_{1}}{K_{1}}u^{m}) \) is nondecreasing for \(u \in \big (0, \root m \of {\frac{K_{1}}{mr_{1}}}\big ]\).

Hence, from Lemma 2.5, we have \(x(k) \le u(k)\) for all \(k \ge 2\), where u(k) is the solution of Eq. (4.1) with \(u(2) = x(2)\). By conclusion (1) of Lemma 2.4, we further obtain

$$\begin{aligned} U_{1}=\limsup _{k\rightarrow \infty }x(k)\le \lim _{k \rightarrow \infty }u(k)=\root m \of {K_{1}}\triangleq M_{1}^{x}. \end{aligned}$$

Hence, for any \(\varepsilon > 0\) small enough, there exists a \(N_{1}> 2\) such that if \(n \ge N_{1}\), then \(x(k) \le M_{1}^{x} + \varepsilon .\)

From the second equation of system (1.1) we have

$$\begin{aligned} y(k+1)\le y(k)\exp \left( r_{2}-\frac{r_{2}}{K_{2}}y^{n}(k)\right) ,\quad k\ge N_{1}. \end{aligned}$$

By the same way, we can obtain

$$\begin{aligned} U_{2}=\limsup _{k\rightarrow \infty }y(k)\le \root n \of {K_{2}} \triangleq M_{1}^{y}. \end{aligned}$$

Hence, for any \(\varepsilon > 0\) small enough, there exists a \(N_{2}> N_{1}\) such that if \(k \ge N_{2}\), then \(y(k) \le M_{1}^{y} + \varepsilon .\)

From the first equations of system (1.1) again, we further have

$$\begin{aligned} x(k+1)\ge x(k)\exp \left[ r_{1}\left( 1-\frac{1}{K_{1}}x^{m}(k) -\mu _{2}\left( M_{1}^{y}+\varepsilon \right) ^{n}\right) \right] ,\quad k\ge N_{2}. \end{aligned}$$

Consider the auxiliary equation

$$\begin{aligned} u(k+1)=u(k)\exp \left[ r_{1}\left( 1-\frac{1}{K_{1}}u^{m}(k) -\mu _{2}\left( M_{1}^{y}+\varepsilon \right) ^{n}\right) \right] . \end{aligned}$$
(4.2)

From the arbitrariness of \(\varepsilon \), we can let \(\varepsilon <\frac{1-\root n \of {\mu _{2}}M_{1}^{y}}{\root n \of {\mu _{2}}}\). From \(1-\mu _{2}K_{2}> 0\), we have \(0< r_{1}(1-\mu _{2}(M_{1}^{y}+\varepsilon )^{n}) < \frac{1}{m}.\) By conclusion (2) of Lemma 2.4, we conclude that \(u(k) \le \root m \of { \frac{K_{1} }{mr_{1}}}\) for all \(k \ge N_{2}\), where u(k) is any solution of Eq. (4.2) with initial value \(u(0) > 0\). From Lemma 2.3, we have that \(f(u) = u \exp (r_{1}-r_{1}\mu _{2}(M_{1}^{y}+\varepsilon )^{n}-\frac{r_{1}}{K_{1}}u^{m})\) is nondecreasing for \(u \in \left( 0,\root m \of { \frac{K_{1} }{mr_{1}}}\right] \). Hence from Lemma 2.5 we have that \(x(k) \ge u(k)\) for all \(k \ge N_{2}\), where u(k) is the solution of Eq. (4.2) with \(u(N_{2}) = x(N_{2})\). From conclusion (1) of Lemma 2.4 again, we have

$$\begin{aligned} V_{1}=\liminf _{n\rightarrow \infty }x(k)\ge \lim _{k\rightarrow \infty }u(k)=\root m \of {K_{1}\left( 1-\mu _{2}\left( M_{1}^{y}+\varepsilon \right) ^{n}\right) }. \end{aligned}$$

From the arbitrariness of \(\varepsilon > 0\), we have \(V_{1} \ge N^{x} _{1}\), where

$$\begin{aligned} N^{x} _{1}=\root m \of {K_{1}\left( 1-\mu _{2}(M_{1}^{y})^{n}\right) } . \end{aligned}$$

Hence, for \(\varepsilon > 0\) small enough, there exists a \(N_{3} > N_{2}\) such that if \(k \ge N_{3}\), then \(x(k) \ge N^{x} _{1}-\varepsilon .\)

From the second equations of system (1.1) we further have

$$\begin{aligned} y(k+1)\ge y(k)\exp \left[ r_{2}\left( 1-\frac{1}{K_{2}}y^{n}(k) -\mu _{1}\left( M_{1}^{x}+\varepsilon \right) ^{m}\right) \right] ,\quad k\ge N_{3}. \end{aligned}$$

By the same way, we can obtain

$$\begin{aligned} V_{2}=\liminf _{k\rightarrow \infty }y(k)\ge \root n \of {K_{2}\left( 1-\mu _{1}\left( M_{1}^{x}+\varepsilon \right) ^{m}\right) }. \end{aligned}$$

From the arbitrariness of \(\varepsilon > 0\), we get \(V_{2} \ge N^{y} _{1} \), where

$$\begin{aligned} N^{y} _{1}=\root n \of {K_{2}\left( 1-\mu _{1}\left( M_{1}^{x}\right) ^{m}\right) }<\root n \of {K_{2}}. \end{aligned}$$

Hence, for \(\varepsilon > 0\) small enough, there exists a \(N_{4} \ge N_{3}\) such that if \(k \ge N_{4}\), then \(y(k) \ge N^{y} _{1}-\varepsilon >0.\)

Further, from the first equations of system (1.1) we have

$$\begin{aligned} x(k+1)\le x(k)\exp \left[ r_{1}\left( 1-\mu _{2}\left( N_{1}^{y}-\varepsilon \right) ^{n} -\frac{x^{m}(k)}{K_{1}}\right) \right] ,\quad k\ge N_{4}. \end{aligned}$$

Using the similar argument as in above, we can get

$$\begin{aligned} U_{1}=\limsup _{k\rightarrow \infty }x(k)\le \root m \of {K_{1}\left( 1-\mu _{2}\left( N_{1}^{y}-\varepsilon \right) ^{n}\right) }. \end{aligned}$$

From the arbitrariness of \(\varepsilon > 0\), we claim that \(U_{1} \le M_{2}^{x}\), where

$$\begin{aligned} M_{2}^{x}=\root m \of {K_{1}\left( 1-\mu _{2}\left( N_{1}^{y}\right) ^{n}\right) }<\root m \of {K_{1}}. \end{aligned}$$

Hence, for any \(\varepsilon > 0\) small enough, there exists a \(N_{5} \ge N_{4}\) such that if \(k \ge N_{5}\), then \(x(k) \le M_{2}^{x} + \varepsilon \).

From the second equations of system (1.1) we further obtain

$$\begin{aligned} y(k+1)\le y(k)\exp \left[ r_{2}\left( 1-\mu _{1}\left( N_{1}^{x} -\varepsilon \right) ^{m}-\frac{y^{n}(k)}{K_{2}}\right) \right] ,\quad k\ge N_{5}. \end{aligned}$$

Similarly to the above argument, we can obtain

$$\begin{aligned} U_{2}=\limsup _{k\rightarrow \infty }y(k)\le \root n \of {K_{2}\left( 1-\mu _{1}\left( N_{1}^{x}-\varepsilon \right) ^{m}\right) }. \end{aligned}$$

From the arbitrariness of \(\varepsilon > 0\), we obtain \(U_{2} \le M^{y} _{2}\), where

$$\begin{aligned} M^{y} _{2}=\root n \of {K_{2}\left( 1-\mu _{1}\left( N_{1}^{x}\right) ^{m}\right) }<\root n \of {K_{2}}. \end{aligned}$$

Hence, for \(\varepsilon > 0\) small enough, there exists a \(N_{6} > N_{5}\) such that if \(k \ge N_{6}\), \(y(k) \le M^{y} _{2} + \varepsilon .\)

Further, from the first equations of system (1.1) we obtain

$$\begin{aligned} x(k+1)\ge x(k)\exp \left[ r_{1}\left( 1-\mu _{2}\left( M_{2}^{y}+\varepsilon \right) ^{n} -\frac{x^{m}(k)}{K_{1}}\right) \right] ,\quad k\ge N_{6}. \end{aligned}$$

Using a similar argument, we again can obtain

$$\begin{aligned} V_{1}=\liminf _{k\rightarrow \infty }x(k)\ge \root m \of {K_{1}\left( 1-\mu _{2}\left( M_{2}^{y}+\varepsilon \right) ^{n}\right) }. \end{aligned}$$

From the arbitrariness of \(\varepsilon > 0\), we get that \(V_{1} \ge N_{2}^{x}\), where

$$\begin{aligned} N_{2}^{x}=\root m \of {K_{1}\left( 1-\mu _{2}\left( M_{2}^{y} \right) ^{n}\right) }>\root m \of {K_{1}\left( 1 -\mu _{2}\left( M_{1}^{y}\right) ^{n}\right) }=N_{1}^{x}. \end{aligned}$$

Hence, for any \(\varepsilon > 0\) small enough, there exists a \(N_{7} > N_{6}\) such that if \(k \ge N_{7}\), \(x(k) \ge N_{2}^{x}-\varepsilon >0.\)

From the second equations of system (1.1) we further have

$$\begin{aligned} y(k+1)\ge y(k)\exp \left[ r_{2}\left( 1-\mu _{1}\left( M_{2}^{x} +\varepsilon \right) ^{m}-\frac{y^{n}(k)}{K_{2}}\right) \right] ,\quad k\ge N_{7}. \end{aligned}$$

Using a similar discussion, we again can obtain

$$\begin{aligned} V_{2}=\liminf _{k\rightarrow \infty }y(k)\ge \root n \of {K_{2}\left( 1-\mu _{1}\left( M_{2}^{x}+\varepsilon \right) ^{m}\right) }. \end{aligned}$$

From the arbitrariness of \(\varepsilon > 0\), we claim that \(V_{2} \ge N_{2}^{y}\), where

$$\begin{aligned} N_{2}^{y}=\root n \of {K_{2}\left( 1-\mu _{1}\left( M_{2}^{x}\right) ^{m}\right) }>\root n \of {K_{2}\left( 1 -\mu _{1}\left( M_{1}^{x}\right) ^{m}\right) }=N_{1}^{y}. \end{aligned}$$

Repeating the above process, we can finally obtain four sequences \(\{M_{k}^{x}\}\), \(\{N_{k}^{x}\}\), \(\{M_{k}^{y}\}\) and \(\{N_{k}^{y}\}\) such that

$$\begin{aligned} M_{k}^{x}&= \root m \of {K_{1}\left( 1-\mu _{2}\left( N_{k-1}^{y}\right) ^{n}\right) }&M_{k}^{y}&= \root n \of {K_{2}\left( 1-\mu _{1}\left( N_{k-1}^{x}\right) ^{m}\right) }, \end{aligned}$$
(4.3)

and

$$\begin{aligned} N_{k}^{x}&= \root m \of {K_{1}\left( 1-\mu _{2}\left( M_{k}^{y}\right) ^{n}\right) }&N_{k}^{y}&= \root n \of {K_{2}\left( 1-\mu _{1}\left( M_{k}^{x}\right) ^{m}\right) }. \end{aligned}$$
(4.4)

Clearly, we have for any integer \(k>0\)

$$\begin{aligned} N_{k}^{x}\le V_{1}\le U_{1}&\le M_{k}^{x}&N_{k}^{y}\le V_{2}\le U_{2}&\le M_{k}^{y}. \end{aligned}$$
(4.5)

In the following, we will prove that \(\{M_{k}^{x}\}\) and \(\{M_{k}^{y}\}\) are monotonically decreasing, \(\{N_{k}^{x}\}\) and \(\{N_{k}^{y}\}\) are monotonically increasing, by means of inductive method.

Firstly, it is clear that

$$\begin{aligned} M_{2}^{x}\le M_{1}^{x},\quad M_{2}^{y}\le M_{1}^{y},\quad N_{2}^{x}\ge N_{1}^{x},\quad N_{2}^{y}\ge N_{1}^{y}. \end{aligned}$$

For \(k (k\ge 2)\),we assume that \(M_{k}^{x}\le M_{k-1}^{x}\) and \(N_{k}^{x}\ge N_{k-1}^{x}\), then we further have

$$\begin{aligned} M_{k}^{y} = \root n \of {K_{2}\left( 1-\mu _{1}\left( N_{k-1}^{x}\right) ^{m}\right) }\le \root n \of {K_{2}\left( 1-\mu _{1}\left( N_{k}^{x}\right) ^{m}\right) }=M_{k-1}^{y}, \end{aligned}$$
(4.6)

and

$$\begin{aligned} N_{k}^{y}= \root n \of {K_{2}\left( 1-\mu _{1}\left( M_{k}^{x}\right) ^{m}\right) } \ge \root n \of {K_{2}\left( 1-\mu _{1}\left( M_{k-1}^{x}\right) ^{m}\right) }=N_{k-1}^{y}. \end{aligned}$$
(4.7)

From (4.6) and (4.7) we have

$$\begin{aligned} \left[ M_{k+1}^{x}\right] ^{m}-\left[ M_{k}^{x}\right] ^{m}= & {} K_{1}\left( 1-\mu _{2}\left( N_{k}^{y}\right) ^{n}\right) -K_{1}\left( 1-\mu _{2}\left( N_{k-1}^{y}\right) ^{n}\right) \nonumber \\= & {} -K_{1}\mu _{2}\left[ N_{k}^{y})^{n}-N_{k-1}^{y})^{n}\right] \nonumber \\\le & {} 0. \end{aligned}$$
(4.8)
$$\begin{aligned} \left[ M_{k+1}^{y}\right] ^{n}-\left[ M_{k}^{y}\right] ^{n}= & {} K_{2}\left( 1-\mu _{1}\left( N_{k}^{x}\right) ^{m}\right) -K_{2}\left( 1-\mu _{1}\left( N_{k-1}^{x}\right) ^{m}\right) \nonumber \\= & {} -K_{2}\mu _{1}\left[ N_{k}^{x})^{m}-N_{k-1}^{x})^{m}\right] \nonumber \\\le & {} 0. \end{aligned}$$
(4.9)

Note that \(a^{n}-b^{n}\) and \(a-b\) have the same sign, when both a and b are positive constants. Therefore, from (4.8) and (4.9), we have \(M_{k+1}^{x}\le M_{k}^{x}\) and \(M_{k+1}^{y}\le M_{k}^{y}.\)

From (4.8) and (4.9) we further have

$$\begin{aligned} \left[ N_{k+1}^{x}\right] ^{m}-\left[ N_{k}^{x}\right] ^{m}= & {} K_{1}\left( 1-\mu _{2}\left( M_{k+1}^{y}\right) ^{n}\right) -K_{1}\left( 1-\mu _{2}\left( M_{k}^{y}\right) ^{n}\right) \\= & {} -K_{1}\mu _{2}\left[ \left( M_{k+1}^{y}\right) ^{n}-\left( M_{k}^{y}\right) ^{n}\right] \\\ge & {} 0. \end{aligned}$$

and

$$\begin{aligned} \left[ N_{k+1}^{y}\right] ^{n}-\left[ N_{k}^{y}\right] ^{n}= & {} K_{2}\left( 1-\mu _{1}\left( M_{k+1}^{x}\right) ^{m}\right) -K_{2}\left( 1-\mu _{1}\left( M_{k}^{x}\right) ^{m}\right) \\= & {} -K_{2}\mu _{1}\left[ \left( M_{k+1}^{x}\right) ^{m}-\left( M_{k}^{x}\right) ^{m}\right] \\\ge & {} 0. \end{aligned}$$

This means that \(\{M_{k}^{x}\}\) and \(\{M_{k}^{y}\}\) are monotonically decreasing, \(\{N_{k}^{x}\}\) and \(\{N_{k}^{y}\}\) are monotonically increasing. Therefore, by the criterion of monotone bounded, we have proved that every one of this four sequences has a limit.

From (4.3) and (4.4), we can obtain

$$\begin{aligned} \left( M_{k}^{x}\right) ^{m}=K_{1}\left[ 1-\mu _{2}\left( N_{k-1}^{y}\right) ^{n}\right] =K_{1}\left[ 1-\mu _{2}K_{2}\left( 1-\mu _{1}\left( M_{k-1}^{x}\right) ^{m}\right) \right] \end{aligned}$$

and

$$\begin{aligned} \left( M_{k}^{y}\right) ^{n}=K_{2}\left[ 1-\mu _{1}\left( N_{k-1}^{x}\right) ^{m}\right] =K_{2}\left[ 1-\mu _{1}K_{1}\left( 1-\mu _{2}\left( M_{k-1}^{y}\right) ^{n}\right) \right] . \end{aligned}$$

Taking \(k \rightarrow \infty \) in both sides of the above two equations, respectively, then we have

$$\begin{aligned} \lim _{k \rightarrow \infty }M_{k}^{x}=x_{0},\quad \quad \lim _{k \rightarrow \infty }M_{k}^{y}=y_{0}. \end{aligned}$$

By the same way, we also can obtain

$$\begin{aligned} \lim _{k \rightarrow \infty }N_{k}^{x}=x_{0},\quad \quad \lim _{k \rightarrow \infty }N_{k}^{y}=y_{0}. \end{aligned}$$

It follows from (4.5) that

$$\begin{aligned} U_{1}=V_{1}=x_{0},\quad \quad U_{2}=V_{2}=y_{0}. \end{aligned}$$

Therefore, we finally have

$$\begin{aligned} \lim _{k \rightarrow \infty }x(k)=x_{0},\quad \quad \lim _{k \rightarrow \infty }y(k)=y_{0}. \end{aligned}$$

This shows that equilibrium \(E_{+}(x_{0}, y_{0})\) of system (1.1) is globally attractive.

From Theorem 3.1, we can obtain that equilibrium \(E_{+}(x_{0}, y_{0})\) of system (1.1) is locally asymptotically stable. Therefore, we finally obtain that \(E_{+}(x_{0}, y_{0})\) is globally asymptotically stable. This completes the proof. \(\square \)

Remark 1

The main results obtained in the present paper is for any positive integer m and n, which generalizes what paper [7] has obtained. The method given in this paper is new and very resultful comparing with articles [6, 9, 10, 14, 16, 19, 22] on the study of global stability for discrete predator–prey systems. Note that our conditions is more better than the conditions of theorem 3 in paper [7]. For example, the conditions of theorem 3 in paper [7] has been obtained as follows:

\((H_{1})\) :

   \(1-\mu _{1}x^{*}>0\) and \(1-\mu _{2}y^{*}>0\), where

$$\begin{aligned} x^{*}=\frac{K_{1}}{r_{1}}\exp (r_{1}-1), \quad \quad y^{*}=\frac{K_{2}}{r_{2}}\exp (r_{2}-1). \end{aligned}$$
\((H_{2})\) :
$$\begin{aligned} \lambda _{1}=\max \left\{ \left| 1-\frac{r_{1}}{K_{1}}x^{*}\right| , \left| 1 -\frac{r_{1}}{K_{1}}x_{*}\right| \right\} +\mu _{2}r_{1}y^{*}<1 \end{aligned}$$

and

$$\begin{aligned} \lambda _{2}=\max \left\{ \left| 1-\frac{r_{2}}{K_{2}}y^{*}\right| , \left| 1 -\frac{r_{2}}{K_{2}}y_{*}\right| \right\} +\mu _{1}r_{2}x^{*}<1, \end{aligned}$$

where

$$\begin{aligned} x_{*}=K_{1}(1-\mu _{2}y^{*})\exp \left[ r_{1}\left( 1-\mu _{2}y^{*} -\frac{x^{*}}{K_{1}}\right) \right] \end{aligned}$$

and

$$\begin{aligned} y_{*}=K_{2}(1-\mu _{1}x^{*})\exp \left[ r_{2}\left( 1 -\mu _{1}x^{*}-\frac{y^{*}}{K_{2}}\right) \right] . \end{aligned}$$

Note that \(\frac{\exp (r-1)}{r}>1\) for \(r>0\), therefore, it is easy to see that condition \((H_{1})\) is stronger than \(1-\mu _{1}K_{1} > 0\) and \(1-\mu _{2}K_{2}> 0.\)

We can also see that condition \((H_{2})\) is complicated comparing with our conditions \(r_{1} \le \frac{1}{m}\) and \(r_{2} \le \frac{1}{n}\), and not easy to verify. Furthermore, if taking \(r_{1}=r_{2}=1\), then we have \(x^{*}= K_{1}\), \(y^{*}= K_{2}\), \(x_{*}= K_{1}(1-\mu _{2}K_{2})\exp (-\mu _{2}K_{2})\) and \(y_{*}= K_{2}(1-\mu _{1}K_{1})\exp (-\mu _{1}K_{1}).\) Then

$$\begin{aligned} \lambda _{1}=1+\mu _{2}K_{2}\exp (-\mu _{2}K_{2})-\exp (-\mu _{2}K_{2}) +\mu _{2}K_{2}. \end{aligned}$$

It is clear to see that \(\lambda _{1}>1\) for \(\mu _{2}K_{2}>\frac{1}{2}\). This shows that \((H_{2})\) is stronger than \(r_{1} = r_{2} = 1\), here \(m = n = 1\).

Remark 2

According to Theorem 4.1 of this paper, we have known that the equilibrium \(E_{+}(x_{0}, y_{0})\) of system (1.1) is globally asymptotically stable for \(r_{1}\le \frac{1}{m},r_{2}\le \frac{1}{n}\), and is locally asymptotically stable for \(r_{1}<t_{2},r_{2}<t_{1}\) and \(r_{2}\le \frac{1}{n(1-\mu _{1}K_{1})}\) (Theorem 3.1). However, whether the equilibrium \(E_{+}(x_{0}, y_{0})\) is also globally asymptotically stable for \(\frac{1}{m}< r_{1}< t_{2}, \frac{1}{n}< r_{2} < t_{1}\) and \(r_{2} \le \frac{1}{1-\mu _{1}K_{1}}\), it is still open.

Remark 3

Another important and interesting open question is whether we can also obtain the same inequality (4.5) but do not apply the comparison principle. If it is possible, then the conditions on the global stability of positive equilibrium of system (1.1) may be extended.

Remark 4

The condition in Theorem 3.1 is to guarantee the existence of positive equilibrium \(E_{+}(x_{0}, y_{0})\) of system (1.1), and the possibility of how the two species can coexist. If the conditions in conclusion (1) of Theorem 3.1 do not hold, then the positive equilibrium of system (1.1) will be unstable.

Remark 5

The approach can also be devoted to studying the global asymptotic stability of positive equilibrium for the other general multiple species discrete population systems. We would like to do some valuable research about the subject.