1 Introduction

Recently, much attention has been paid to the competition systems. For example, Wang et al. [1] considered the following two-species competition system with nonlinear interinhibition terms:

$$ \textstyle\begin{cases} \dot{x_{1}}(t)=x_{1}(t) \{r_{1}(t)-a_{1}(t)x_{1}(t)-\frac {c_{2}(t)x_{2}(t)}{1+x_{2}(t)} \},\\ \dot{x_{2}}(t)=x_{2}(t) \{r_{2}(t)-a_{2}(t)x_{2}(t)-\frac {c_{1}(t)x_{1}(t)}{1+x_{1}(t)} \}, \end{cases} $$
(1.1)

where \(x_{1}(t)\), \(x_{2}(t)\) are the population densities of two competing species, \(a_{1}(t)\), \(a_{2}(t)\) are the intraspecific competition rate of the first and second species, \(c_{1}(t)\), \(c_{2}(t)\) represent the interspecific competing rates and \(r_{1}(t)\), \(r_{2}(t)\) are the intrinsic growth rates of species. Wang et al. [1] showed the existence and global asymptotic stability of positive almost periodic solutions of model (1.1). For the ecological sense of model (1.1), we refer to [2] and the references therein.

Considering that the discrete-time models governed by difference equations are more appropriate than the continuous ones when the populations have a short life expectancy and nonoverlapping generations, Qin et al. [3] introduced the following discrete analogue of system (1.1):

$$ \textstyle\begin{cases} x_{1}(n+1) = x_{1}(n)\operatorname{exp} \{r_{1}(n)-a_{1}(n)x_{1}(n)-\frac {c_{2}(n)x_{2}(n)}{1+x_{2}(n)} \},\\ x_{2}(n+1) = x_{2}(n)\operatorname{exp} \{r_{2}(n)-a_{2}(n)x_{2}(n)-\frac {c_{1}(n)x_{1}(n)}{1+x_{1}(n)} \}. \end{cases} $$
(1.2)

A good understanding of the permanence, existence, and global stability of positive periodic solutions was obtained in [3]. As for the almost periodic case, Wang and Liu [4] further studied the existence, uniqueness, and uniformly asymptotic stability of a positive almost periodic solution of system (1.2). Extinction of species and the stability property of another species were considered in [5]. Yue [6] investigated system (1.2) with one toxin producing species. Sufficient conditions that guarantee the extinction of one of the components and the global attractivity of the other one were obtained in [6].

Noting that ecosystems in the real world are often distributed by unpredictable forces that can result in changes in biological parameters, Wang et al. [7] proposed the following model, system (1.2) with feedback controls:

$$ \textstyle\begin{cases} x_{1}(n+1) = x_{1}(n)\operatorname{exp} \{r_{1}(n)-a_{1}(n)x_{1}(n)-\frac {c_{2}(n)x_{2}(n)}{1+x_{2}(n)}-e_{1}(n)u_{1}(n) \},\\ x_{2}(n+1)= x_{2}(n)\operatorname{exp} \{r_{2}(n)-a_{2}(n)x_{2}(n)-\frac {c_{1}(n)x_{1}(n)}{1+x_{1}(n)}-e_{2}(n)u_{2}(n) \},\\ \Delta u_{1}(n)=-b_{1}(n)u_{1}(n)+d_{1}(n)x_{1}(n), \Delta u_{2}(n)=-b_{2}(n)u_{2}(n)+d_{2}(n)x_{2}(n). \end{cases} $$
(1.3)

Wang et al. [7] established a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of system (1.3) with almost periodic parameters. Yu [8] further considered the influence of feedback control variables on the persistent property of the system. On the other hand, as we all know, the extinction property is also an important topic in the study of mathematical biology; however, until now there are still no scholars investigating this property of system (1.3). Indeed, in this paper, we apply the analysis technique of Chen et al. [9], Xu et al. [10], and Zhang et al. [11] to obtain a set of sufficient conditions that guarantee one of the two species and the corresponding feedback controls varieties will be driven to extinction. For more works in this direction, we refer to [12–21] and the references therein.

For any bounded sequence \(\{g(n)\}\), we denote \(g^{L}=\inf_{n\in Z} \{g(n)\}\), \(g^{M}=\sup_{n\in Z} \{g(n)\}\). For convenience, we introduce the following assumptions:

\((H_{1})\) :

\(\{r_{i}(n)\}\), \(i=1,2\), are bounded sequences defined on Z, and \(\{a_{i}(n)\}\), \(\{c_{i}(n)\}\), \(\{d_{i}(n)\}\), and \(\{e_{i}(n)\}\), \(i=1,2\), are bounded nonnegative sequences defined on Z.

\((H_{2})\) :

Sequences \(\{b_{i}(n)\}\) satisfy \(0< b_{i}^{L}\le b_{i}^{M}<1\) for all \(n\in Z\).

\((H_{3})\) :

There exists a positive integer ω such that, for each \(i=1,2\),

$$\liminf_{n\rightarrow\infty}\sum_{s=n}^{n+\omega-1}r_{i}(s)>0. $$
\((H_{4})\) :

There exists a positive integer ρ such that, for each \(i=1,2\),

$$\limsup_{n\rightarrow\infty}\prod_{s=n}^{n+\rho-1} \bigl(1-b_{i}(s) \bigr)< 1. $$

As regards the biological background, we focus our discussion on the positive solutions of system (1.3). So, we consider (1.3) together with the following initial conditions:

$$ x_{i}(0)>0,\qquad u_{i}(0)>0,\quad i=1,2. $$
(1.4)

It is obvious that the solutions of (1.3)-(1.4) are well defined and satisfy

$$ x_{i}(n)>0,\qquad u_{i}(n)>0,\quad i=1,2, \mbox{for } n\in Z. $$
(1.5)

The rest of this paper is organized as follows. In the next section, we study the extinction of one species and the corresponding feedback control varieties of system (1.3). Some examples together with their numerical simulations are carried out to show the feasibility of our results in Section 3. We end this paper with a brief discussion.

2 Extinction

In this section, we’ll establish sufficient conditions on the extinction of one of the two species and the corresponding feedback controls varieties of system (1.3). Wang et al. [7] showed that the positive solutions of system (1.3) were bounded eventually:

Lemma 2.1

see [7]

Any positive solution \((x_{1}(n),x_{2}(n),u_{1}(n),u_{2}(n))^{T}\) of system (1.3) satisfies

$$ \limsup_{n\rightarrow\infty}x_{i}(n)\le B_{i},\qquad \limsup_{n\rightarrow\infty}u_{i}(n)\le D_{i}, $$
(2.1)

where \(B_{i}=\frac{\exp(r_{i}^{M}-1)}{a_{i}^{L}}\) and \(D_{i}=\frac{B_{i}d_{i}^{M}}{b_{i}^{L}}\) for \(i=1, 2\).

We now come to study the extinction of species \(x_{2}\) and the feedback controls varieties \(u_{2}\) of system (1.3).

Theorem 2.1

In addition to \((H_{1})\)-\((H_{4})\), further suppose that:

$$(H_{5})\quad \limsup_{n\rightarrow\infty}\frac{\sum_{s=n}^{n+\omega-1}r_{2}(s)}{\sum_{s=n}^{n+\omega-1}r_{1}(s)}< \liminf _{n\rightarrow\infty}\frac{a_{2}(n)}{c_{2}(n)} $$

and

$$(H_{6})\quad \limsup_{n\rightarrow\infty}\frac {e_{1}(n)}{b_{1}(n)}< \liminf_{n\rightarrow\infty} \biggl(\frac {c_{1}(n)}{(1+B_{1})d_{1}(n)}\liminf _{n\rightarrow\infty}\frac{\sum_{s=n}^{n+\omega-1}r_{1}(s)}{\sum_{s=n}^{n+\omega-1}r_{2}(s)}-\frac {a_{1}(n)}{d_{1}(n)} \biggr), $$

where \(B_{1}\) is defined in Lemma 2.1. Then \(x_{2}\) and \(u_{2}\) will be driven to extinction, that is, for any positive solution \((x_{1}(n),x_{2}(n),u_{1}(n),u_{2}(n))^{T}\) of system (1.3), \(\lim_{n\rightarrow\infty}x_{2}(n)=0\) and \(\lim_{n\rightarrow\infty }u_{2}(n)=0\).

Proof

According to Lemma 2.1, for any \(\varepsilon>0\) small enough, there exists \(n_{1}>0\) large enough such that, for \(n\ge n_{1}\),

$$ x_{1}(n)\le B_{1}+\varepsilon,\qquad u_{1}(n) \le D,\qquad u_{2}(n)\le D, $$
(2.2)

where \(D=\operatorname{max}\{D_{1}+\varepsilon,D_{2}+\varepsilon\}\). Thus, it follows from \((H_{3})\) that there exist positive constants \(\eta_{0}\) and \(n_{2}\ge n_{1}\) such that

$$\sum_{s=n}^{n+\omega-1}r_{i}(s)\ge \eta_{0} \quad\mbox{for all } n\ge n_{2}. $$

By \((H_{1})\), \((H_{2})\), and \((H_{5})\) we can obtain that

$$ \liminf_{n\rightarrow\infty}\frac {e_{2}(n)}{b_{2}(n)}>\limsup_{n\rightarrow\infty} \biggl(\frac {c_{2}(n)}{d_{2}(n)}\limsup_{n\rightarrow\infty}\frac{\sum_{s=n}^{n+\omega-1}r_{2}(s)}{\sum_{s=n}^{n+\omega-1}r_{1}(s)}- \frac {a_{2}(n)}{d_{2}(n)} \biggr). $$
(2.3)

For the same ε, according to \((H_{5})\)-\((H_{6})\) and (2.3), we can choose positive constants \(\alpha, \beta, \gamma, \delta\), and \(n_{3}\ge n_{2}\) such that

$$\begin{aligned} &\frac{\sum_{s=n}^{n+\omega-1}r_{2}(s)}{\sum_{s=n}^{n+\omega -1}r_{1}(s)}< \frac{\alpha}{\beta}-\varepsilon< \frac{\alpha}{\beta }< \frac{a_{2}(n)}{c_{2}(n)}, \\ &\frac{e_{1}(n)}{b_{1}(n)}< \frac{\gamma}{\alpha}< \frac{\beta c_{1}(n)-\alpha(1+B_{1}+\varepsilon)a_{1}(n)}{\alpha(1+B_{1}+\varepsilon)d_{1}(n)} \end{aligned}$$

and

$$\frac{ e_{2}(n)}{b_{2}(n)}>\frac{\delta}{\beta}>\frac{\alpha c_{2}(n)-\beta a_{2}(n)}{\beta d_{2}(n)} $$

for all \(n\ge n_{3}\). Hence, we have:

$$\begin{aligned} &\sum_{s=n}^{n+\omega-1} \bigl(\beta r_{2}(s)-\alpha r_{1}(s) \bigr)< -\varepsilon \beta \eta_{0}, \end{aligned}$$
(2.4)
$$\begin{aligned} &\alpha e_{1}(n)-\gamma b_{1}(n)< 0, \end{aligned}$$
(2.5)
$$\begin{aligned} &\alpha a_{1}(n)-\frac{\beta c_{1}(n)}{1+B_{1}+\varepsilon}+\gamma d_{1}(n)< 0, \end{aligned}$$
(2.6)
$$\begin{aligned} &\delta b_{2}(n)-\beta e_{2}(n)< 0, \end{aligned}$$
(2.7)
$$\begin{aligned} &\alpha c_{2}(n)-\beta a_{2}(n)-\delta d_{2}(n)< 0. \end{aligned}$$
(2.8)

Consider the Lyapunov function

$$ V(n)=x_{1}^{-\alpha}(n)x_{2}^{\beta}(n) \operatorname{exp} \bigl\{ \gamma u_{1}(n)-\delta u_{2}(n) \bigr\} . $$
(2.9)

By calculating we get

$$\begin{aligned} \frac{V(n+1)}{V(n)}={}&\operatorname{exp} \biggl\{ \beta r_{2}(n)- \alpha r_{1}(n)+ \bigl(\alpha e_{1}(n)-\gamma b_{1}(n) \bigr)u_{1}(n) \\ &{}+ \bigl(\delta b_{2}(n)-\beta e_{2}(n) \bigr)u_{2}(n)+ \biggl(\alpha a_{1}(n)-\frac{\beta c_{1}(n)}{1+x_{1}(n)}+ \gamma d_{1}(n) \biggr)x_{1}(n) \\ &{}+ \biggl(\frac{\alpha c_{2}(n)}{1+x_{2}(n)}-\beta a_{2}(n)-\delta d_{2}(n) \biggr)x_{2}(n) \biggr\} \\ \le{}&\operatorname{exp} \biggl\{ \beta r_{2}(n)-\alpha r_{1}(n)+ \bigl(\alpha e_{1}(n)-\gamma b_{1}(n) \bigr)u_{1}(n) \\ &{}+ \bigl(\delta b_{2}(n)-\beta e_{2}(n) \bigr)u_{2}(n)+ \biggl(\alpha a_{1}(n)-\frac{\beta c_{1}(n)}{1+B_{1}+\varepsilon}+ \gamma d_{1}(n) \biggr)x_{1}(n) \\ &{}+ \bigl(\alpha c_{2}(n)-\beta a_{2}(n)-\delta d_{2}(n) \bigr)x_{2}(n) \biggr\} . \end{aligned}$$

It follows that from (2.5)-(2.8) that

$$ V(n+1)\le V(n)\operatorname{exp} \bigl\{ \beta r_{2}(n)-\alpha r_{1}(n) \bigr\} \quad \mbox{for all } n\ge n_{3}. $$
(2.10)

For any \(n\ge n_{3}\), we choose an integer \(m\ge0\) such that \(n\in [n_{3}+m\omega,n_{3}+(m+1)\omega)\). Integrating (2.10) from \(n_{3}\) to \(n-1\) leads to

$$\begin{aligned} V(n)&\le V(n_{3})\operatorname{exp} \Biggl\{ \sum_{s=n_{3}}^{n-1}\bigl(\beta r_{2}(s)-\alpha r_{1}(s)\bigr) \Biggr\} \\ &\le V(n_{3})\operatorname{exp} \Biggl\{ \sum _{s=n_{3}}^{n_{3}+m\omega-1}+\sum_{s=n_{3}+m\omega-1}^{n-1} \Biggr\} \bigl(\beta r_{2}(s)-\alpha r_{1}(s)\bigr) \\ &\le V(n_{3})\operatorname{exp}\{-\varepsilon\beta \eta_{0}m+M_{1}\} \\ &\le V(n_{3})\operatorname{exp}\biggl\{ -\varepsilon\beta \eta_{0} \biggl(\frac {n-n_{3}}{\omega}-1 \biggr)+M_{1}\biggr\} \\ &\le V(n_{3})\operatorname{exp} \biggl\{ -\frac{\varepsilon\beta\eta _{0}n}{\omega}+M_{1}^{*} \biggr\} , \end{aligned}$$
(2.11)

where \(M_{1}^{*}=\frac{\varepsilon\beta\eta_{0}n_{3}}{\omega}+\varepsilon \beta\eta_{0}+M_{1}\) and \(M_{1}=\sup_{n\in Z}\vert \beta r_{2}(n)-\alpha r_{1}(n)\vert \omega\). Relations (2.2), (2.9), and (2.11) imply that that, for \(n\ge n_{3}\),

$$ x_{2}(n)< \bigl[x_{1}^{-\alpha}(n_{3})x_{2}^{\beta}(n_{3}) \operatorname{exp} \bigl\{ (\gamma +\delta) D \bigr\} (B_{1}+ \varepsilon)^{\alpha} \operatorname{exp} \bigl\{ M_{1}^{*} \bigr\} \bigr]^{\frac{1}{\beta}}\operatorname{exp} \biggl\{ -\frac{\varepsilon\eta _{0}n}{\omega} \biggr\} . $$
(2.12)

Hence, \(x_{2}(n)\rightarrow0\) exponentially as \(n\rightarrow\infty\). Similarly to corresponding proof of Theorem 3.1 in Chen et al. [9], we can easily see that \(u_{2}(n)\rightarrow 0\) as \(n\rightarrow\infty\). This ends the proof of Theorem 2.1. □

Now, let us investigate the extinction property of species \(x_{1}\) and the feedback controls varieties \(u_{1}\) in system (1.3), which is also an interesting problem, and we obtain the following result.

Theorem 2.2

Let \((x_{1}(n),x_{2}(n),u_{1}(n),u_{2}(n))^{T}\) be any positive solution of system (1.3). Suppose that \((H_{1})\)-\((H_{4})\) and the following inequalities hold:

$$\begin{aligned} &(H_{7})\quad \liminf_{n\rightarrow\infty}\frac{\sum_{s=n}^{n+\omega-1}r_{2}(s)}{\sum_{s=n}^{n+\omega-1}r_{1}(s)}> \limsup_{n\rightarrow\infty}\frac{c_{1}(n)}{a_{1}(n)}, \\ &(H_{8})\quad \limsup_{n\rightarrow\infty}\frac {e_{2}(n)}{b_{2}(n)}< \liminf_{n\rightarrow\infty} \biggl(\frac {c_{2}(n)}{(1+B_{2})d_{2}(n)}\liminf _{n\rightarrow\infty}\frac{\sum_{s=n}^{n+\omega-1}r_{2}(s)}{\sum_{s=n}^{n+\omega-1}r_{1}(s)}-\frac {a_{2}(n)}{d_{2}(n)} \biggr), \end{aligned}$$

where \(B_{2} \) is defined in Lemma 2.1. Then \(\lim_{n\rightarrow\infty}x_{1}(n)=0\) and \(\lim_{n\rightarrow\infty }u_{1}(n)=0\).

Proof

According to Lemma 2.1, for any \(\varepsilon>0\) small enough, there exists a positive constant \(n_{4}>n_{3}\) such that, for \(n\ge n_{4}\),

$$ x_{2}(n)\le B_{2}+\varepsilon. $$
(2.13)

By \((H_{1})\), \((H_{2})\), and \((H_{7})\) we obtain that

$$ \liminf_{n\rightarrow\infty}\frac {e_{1}(n)}{b_{1}(n)}>\limsup_{n\rightarrow\infty} \biggl(\frac {c_{1}(n)}{d_{1}(n)}\limsup_{n\rightarrow\infty}\frac{\sum_{s=n}^{n+\omega-1}r_{1}(s)}{\sum_{s=n}^{n+\omega-1}r_{2}(s)}- \frac {a_{1}(n)}{d_{1}(n)} \biggr). $$
(2.14)

For the same ε, according to \((H_{7})\)-\((H_{8})\) and (2.14), we can choose positive constants \(\alpha, \beta, \gamma, \delta\), and \(n_{5}\ge n_{4}\) such that:

$$\begin{aligned} &\frac{\sum_{s=n}^{n+\omega-1}r_{2}(s)}{\sum_{s=n}^{n+\omega -1}r_{1}(s)}>\frac{\alpha}{\beta}+\varepsilon>\frac{\alpha}{\beta }> \frac{a_{2}(n)}{c_{2}(n)}, \\ &\frac{ e_{2}(n)}{b_{2}(n)}< \frac{\delta}{\beta}< \frac{\alpha c_{2}(n)-\beta(1+B_{2}+\varepsilon)a_{2}(n)}{\beta(1+B_{2}+\varepsilon)d_{2}(n)} \end{aligned}$$

and

$$\frac{e_{1}(n)}{b_{1}(n)}>\frac{\gamma}{\alpha}>\frac{\beta c_{1}(n)-\alpha a_{1}(n)}{\alpha d_{1}(n)} $$

for all \(n\ge n_{5}\). Hence, we have:

$$\begin{aligned} &\sum_{s=n}^{n+\omega-1} \bigl(\alpha r_{1}(s)-\beta r_{2}(s) \bigr)< -\varepsilon \beta \eta_{0}, \end{aligned}$$
(2.15)
$$\begin{aligned} &\beta e_{2}(n)-\delta b_{2}(n)< 0, \end{aligned}$$
(2.16)
$$\begin{aligned} &{-}\frac{\alpha c_{2}(n)}{1+B_{2}+\varepsilon}+\beta a_{2}(n)+\delta d_{2}(n)< 0, \end{aligned}$$
(2.17)
$$\begin{aligned} &\gamma b_{1}(n)-\alpha e_{1}(n)< 0, \end{aligned}$$
(2.18)
$$\begin{aligned} &{-}\alpha a_{1}(n)+\beta c_{1}(n)-\gamma d_{1}(n)< 0. \end{aligned}$$
(2.19)

Consider the Lyapunov function

$$ V(n)=x_{1}^{\alpha}(n)x_{2}^{-\beta}(n) \operatorname{exp} \bigl\{ \delta u_{2}(n)-\gamma u_{1}(n) \bigr\} . $$
(2.20)

By calculating and inequalities (2.16)-(2.19) we obtain that

$$\begin{aligned} \frac{V(n+1)}{V(n)}={}&\operatorname{exp} \biggl\{ \alpha r_{1}(n)-\beta r_{2}(n)+ \bigl(\gamma b_{1}(n)- \alpha e_{1}(n) \bigr)u_{1}(n) \\ &{}+ \bigl(\beta e_{2}(n)-\delta b_{2}(n) \bigr)u_{2}(n)+ \biggl(-\alpha a_{1}(n)+\frac{\beta c_{1}(n)}{1+x_{1}(n)}- \gamma d_{1}(n) \biggr)x_{1}(n) \\ &{}+ \biggl(-\frac{\alpha c_{2}(n)}{1+x_{2}(n)}+\beta a_{2}(n)+\delta d_{2}(n) \biggr)x_{2}(n) \biggr\} \\ \le{}&\operatorname{exp} \biggl\{ \alpha r_{1}(n)-\beta r_{2}(n)+ \bigl(\gamma b_{1}(n)-\alpha e_{1}(n) \bigr)u_{1}(n) \\ &{}+ \bigl(\beta e_{2}(n)-\delta b_{2}(n) \bigr)u_{2}(n)+ \bigl(-\alpha a_{1}(n)+\beta c_{1}(n)-\gamma d_{1}(n) \bigr)x_{1}(n) \\ &{}+ \biggl(-\frac{\alpha c_{2}(n)}{1+B_{2}+\varepsilon}+\beta a_{2}(n)+\delta d_{2}(n) \biggr)x_{2}(n) \biggr\} \\ \le{}&\operatorname{exp} \bigl\{ \alpha r_{1}(n)-\beta r_{2}(n) \bigr\} . \end{aligned}$$
(2.21)

From (2.21), similarly to the analysis of of Theorem 2.1, we can get the conclusion that \(x_{1}(n)\rightarrow0\) and \(u_{1}(n)\rightarrow0\) as \(n\rightarrow\infty\). This ends the proof of Theorem 2.1. □

When \(e_{i}(n)=b_{i}(n)=d_{i}(n)=0\ (i=1,2)\), (1.3) becomes (1.2), as discussed in [5]. Similarly to the proofs of Theorems 2.1 and 2.2, we can obtain the following:

Corrolary 2.1

In addition to \((H_{1})\)-\((H_{3})\), further suppose that

$$(H_{9})\quad \limsup_{n\rightarrow\infty}\frac{\sum_{s=n}^{n+\omega-1}r_{2}(s)}{\sum_{s=n}^{n+\omega-1}r_{1}(s)}< \min \biggl\{ \liminf_{n\rightarrow\infty}\frac{a_{2}(n)}{c_{2}(n)},\liminf _{n\rightarrow\infty}\frac{c_{1}(n)}{(1+B_{1})a_{1}(n)} \biggr\} , $$

where \(B_{i} \) \((i = 1, 2)\) are defined in Lemma 2.1. Then the species \(x_{2}\) will be driven to extinction, that is, for any positive solution \((x_{1}(n), x_{2}(n))^{T}\) of system (1.2), \(\lim_{n\rightarrow\infty}x_{2}(n)=0\).

Corrolary 2.2

Let \((x_{1}(n), x_{2}(n))^{T}\) be any positive solution of system (1.2). Suppose that

$$(H_{10})\quad \liminf_{n\rightarrow\infty}\frac{\sum_{s=n}^{n+\omega-1}r_{2}(s)}{\sum_{s=n}^{n+\omega-1}r_{1}(s)}>\max \biggl\{ \limsup_{n\rightarrow\infty}\frac{c_{1}(n)}{a_{1}(n)},\limsup _{n\rightarrow\infty}\frac{(1+B_{2})a_{2}(n)}{c_{2}(n)} \biggr\} , $$

where \(B_{2} \) is defined in Lemma 2.1. Then \(\lim_{n\rightarrow\infty}x_{1}(n)=0\).

Remark 2.1

Comparing with Assumptions \((H_{1})\) and \((H_{2})\) given in [5], we can see that our assumptions in Corollaries 2.1 and 2.2 are weaker.

3 Example and numeric simulation

In this section, we give the following two examples to illustrate our main results.

Example 3.1

Consider the following system:

$$ \textstyle\begin{cases} x_{1}(n+1) = x_{1}(n)\operatorname{exp} \{1.4-1.75x_{1}(n)-\frac{(1+0.3\sin (n))x_{2}(n)}{1+x_{2}(n)}-0.9u_{1}(n) \},\\ x_{2}(n+1) = x_{2}(n)\operatorname{exp} \{0.7-2.6x_{2}(n)-\frac{(6+2\cos (n))x_{1}(n)}{1+x_{1}(n)}-1.5u_{2}(n) \},\\ \Delta u_{1}(n)=-(0.8+0.1\sin(n))u_{1}(n)+0.5x_{1}(n),\\ \Delta u_{2}(n)=-(0.7+0.2\cos(n))u_{2}(n)+0.4x_{2}(n). \end{cases} $$
(3.1)

In this case, we have that \((H_{1})\)-\((H_{4})\) hold and \(B_{1}=\frac{\operatorname {exp}(r_{1}^{M}-1)}{a_{1}^{L}}\approx0.8525\), and hence

$$\begin{aligned} &\limsup_{n\rightarrow\infty}\frac{\sum_{s=n}^{n+\omega -1}r_{2}(s)}{\sum_{s=n}^{n+\omega-1}r_{1}(s)}=0.5< \liminf _{n\rightarrow \infty}\frac{a_{2}(n)}{c_{2}(n)}\approx2, \\ &\limsup_{n\rightarrow\infty}\frac{e_{1}(n)}{b_{1}(n)}\approx1.2857, \\ &\liminf_{n\rightarrow\infty} \biggl(\frac {c_{1}(n)}{(1+B_{1})d_{1}(n)}\liminf _{n\rightarrow\infty}\frac{\sum_{s=n}^{n+\omega-1}r_{1}(s)}{\sum_{s=n}^{n+\omega-1}r_{2}(s)}-\frac {a_{1}(n)}{d_{1}(n)} \biggr) \approx5.1370. \end{aligned}$$

So all conditions in Theorem 2.1 ares satisfied, and \(x_{2}\) and \(u_{2}\) in system (3.1) are extinct. Our numerical simulation supports this result (see Figure 1).

Figure 1
figure 1

Dynamic behavior of system ( 3.1 ) with the initial conditions \(\pmb{(x_{1}(0), x_{2}(0),u_{1}(0),u_{2}(0))=(0.1, 0.3, 0.2, 0.04)^{T}}\) and \(\pmb{(0.2, 0.1, 0.6, 0.5)^{T}}\) .

Full size image

Example 3.2

Consider the following system:

$$ \textstyle\begin{cases} x_{1}(n+1) = x_{1}(n)\operatorname{exp} \{0.4-1.75x_{1}(n)-\frac{(3.4+0.4\sin (n))x_{2}(n)}{1+x_{2}(n)}-0.9u_{1}(n) \},\\ x_{2}(n+1) = x_{2}(n)\operatorname{exp} \{1.6-1.3x_{2}(n)-\frac{(3+\cos (n))x_{1}(n)}{1+x_{1}(n)}-1.5u_{2}(n) \},\\ \Delta u_{1}(n)=-(0.8+0.1\sin(n))u_{1}(n)+0.5x_{1}(n),\\ \Delta u_{2}(n)=-(0.7+0.2\cos(n))u_{2}(n)+0.4x_{2}(n). \end{cases} $$
(3.2)

In this case, we have that \((H_{1})\)-\((H_{4})\) hold and \(B_{2}=\frac{\operatorname {exp}(r_{2}^{M}-1)}{a_{2}^{L}}\approx1.4016\), and hence

$$\begin{aligned} &\liminf_{n\rightarrow\infty}\frac{\sum_{s=n}^{n+\omega -1}r_{2}(s)}{\sum_{s=n}^{n+\omega-1}r_{1}(s)}=4>\limsup _{n\rightarrow \infty} \frac{c_{1}(n)}{a_{1}(n)}\approx2.2857, \\ &\limsup_{n\rightarrow\infty}\frac{e_{2}(n)}{b_{2}(n)}\approx3, \\ &\liminf_{n\rightarrow\infty} \biggl(\frac {c_{2}(n)}{(1+B_{2})d_{2}(n)}\liminf _{n\rightarrow\infty}\frac{\sum_{s=n}^{n+\omega-1}r_{2}(s)}{\sum_{s=n}^{n+\omega-1}r_{1}(s)}-\frac {a_{2}(n)}{d_{2}(n)} \biggr)\approx9.2417. \end{aligned}$$

So all conditions in Theorem 2.2 are satisfied, and \(x_{1}\) and \(u_{1}\) in system (3.2) are extinct. Numerical simulation also confirms our result (see Figure 2).

Figure 2
figure 2

Dynamic behavior of system ( 3.1 ) with the initial conditions \(\pmb{(x_{1}(0), x_{2}(0),u_{1}(0),u_{2}(0))=(0.1, 0.3, 0.2, 0.04)^{T}}\) and \(\pmb{(0.2, 0.1, 0.6, 0.5)^{T}}\) .

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4 Discussion

In this paper, we consider a two-species nonautonomous discrete competition system with nonlinear interinhibition terms and feedback controls, that is, (1.3) which was discussed in [7, 8]. However, until now, there are still no scholars investigating the extinction property of system (1.3), which is also an important topic in mathematical biology. By developing the analysis technique of Chen et al. [9], Xu et al. [10], and Zhang et al. [11] we obtain sufficient conditions that guarantee the extinction of one of the components and the corresponding feedback controls varieties. When \(e_{i}(n)=b_{i}(n)=d_{i}(n)=0\) \((i=1,2)\), (1.3) becomes (1.2), as discussed in [3–5]. As direct results of Theorems 2.1 and 2.2, Corollaries 2.1 and 2.2 improve and supplement those of [5, 7, 8]. Moreover, by comparing Theorem 2.1 with Corollary 2.1, and Theorem 2.2 with Corollary 2.2 we also found that, for such a kind of systems, feedback control variables play an important role in the extinction property of the system.