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Extinction for a discrete competition system with the effect of toxic substances

Advances in Difference Equations20162016:1

https://doi.org/10.1186/s13662-015-0739-5

Received: 10 October 2015

Accepted: 21 December 2015

Published: 4 January 2016

Abstract

A nonautonomous discrete competitive system with nonlinear inter-inhibition terms and one toxin producing species is studied in this paper. Sufficient conditions which guarantee the extinction of one of the components are obtained and the global attractivity of the other one is proved. Our results supplement some existing ones. Numerical simulations show the feasibility of our results.

Keywords

discreteextinctioncompetitivetoxicology

MSC

34C0534C25

1 Introduction

For any bounded sequence \(\{f(n)\}\), \(f^{L}=\inf_{n\in N} \{f(n)\}\), \(f^{M}=\sup_{n\in N} \{f(n)\}\).

Recently, many authors considered the following discrete two species competitive system with nonlinear inter-inhibition terms (see [14]):
$$ \begin{aligned} &x_{1}(n+1) = x_{1}(n)\exp \biggl\{ r_{1}(n)-a_{1}(n)x_{1}(n)-\frac {c_{2}(n)x_{2}(n)}{1+x_{2}(n)} \biggr\} , \\ &x_{2}(n+1) = x_{2}(n)\exp \biggl\{ r_{2}(n)-a_{2}(n)x_{2}(n)- \frac {c_{1}(n)x_{1}(n)}{1+x_{1}(n)} \biggr\} , \end{aligned} $$
(1.1)
where \(r_{i}(n)\), \(a_{i}(n)\), \(c_{i}(n)\) (\(i=1,2\)) are assumed to be bounded positive sequences and \(x_{1}(n)\), \(x_{2}(n)\) are population density of species \(x_{1}\) and \(x_{2}\) at the nth generation, respectively. For the ecological meaning of model (1.1), see [1]. Sufficient conditions which guarantee the permanence, existence, and global stability of positive periodic solutions of system (1.1) were established by Qin et al. [1]. By using the Lyapunov function, some analysis techniques, and preliminary lemmas, Wang and Liu [2] further established a criterion for the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solution of system (1.1) with almost periodic parameters. Noting that ecosystems in the real world are often distributed by unpredictable forces which can result in changes in biological parameters, Wang et al. [3] investigated the existence and uniformly asymptotic stability of the unique positive almost periodic solution of system (1.1) with almost periodic parameters and feedback controls. Yu [4] further showed that feedback control variables have no influence 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 scholar investigations of this property of system (1.1). One aim of this work is to obtain a set of sufficient conditions which guarantee the extinction of system (1.1).
In recent years, the competition system with toxic substance has been widely studied. Li and Chen [5] studied the extinction property and global attractivity of the following two species discrete competitive system:
$$ \begin{aligned} &x_{1}(n+1) = x_{1}(n)\exp \bigl\{ r_{1}(n)-a_{11}(n)x_{1}(n)-a_{12}(n)x_{2}(n)-b_{1}(n)x_{1}(n)x_{2}(n) \bigr\} , \\ &x_{2}(n+1) = x_{2}(n)\exp \bigl\{ r_{2}(n)-a_{21}(n)x_{1}(n)-a_{22}(n)x_{2}(n)-b_{2}(n)x_{1}(n)x_{2}(n) \bigr\} . \end{aligned} $$
(1.2)
Solé et al. [6] and Bandyopadhyay [7] considered the Lotka-Volterra system of two interacting phytoplankton species with one species that could be toxic, while the other one is non-toxic. The stability property of the equilibrium of the system is obtained. Motivated by the above ideas, Chen et al. [8] introduced the following system:
$$ \begin{aligned} &x_{1}(n+1) = x_{1}(n)\exp \bigl\{ r_{1}(n)-a_{11}(n)x_{1}(n)-a_{12}(n)x_{2}(n)-b_{1}(n)x_{1}(n)x_{2}(n) \bigr\} , \\ &x_{2}(n+1) = x_{2}(n)\exp \bigl\{ r_{2}(n)-a_{21}(n)x_{1}(n)-a_{22}x_{2}(n) \bigr\} , \end{aligned} $$
(1.3)
where \(x_{1}(n)\) represents the density of non-toxic phytoplankton and \(x_{2}(n)\) is the toxic liberating phytoplankton. The coefficients in system (1.3) have the same restriction as that in system (1.2) and system (1.3) is a special case of (1.2) with \(b_{2}(n)\equiv0\), i.e., the first species could not be toxic. They obtain several sets of sufficient conditions which guarantee the extinction of the one species and the global stability of the other species. To the best of the author’s knowledge, to this day, no work has been done previously on the discrete competitive system with nonlinear inter-inhibition terms and one toxin producing species. Hence, we consider the following system:
$$ \begin{aligned} &x_{1}(n+1) = x_{1}(n)\exp \biggl\{ r_{1}(n)-a_{11}(n)x_{1}(n)-\frac {a_{12}(n)x_{2}(n)}{1+x_{2}(n)}-b_{1}(n)x_{1}(n)x_{2}(n) \biggr\} , \\ &x_{2}(n+1) = x_{2}(n)\exp \biggl\{ r_{2}(n)-a_{22}(n)x_{2}(n)- \frac {a_{21}(n)x_{1}(n)}{1+x_{1}(n)} \biggr\} , \end{aligned} $$
(1.4)
where all the coefficients have the same meaning as that of systems (1.1)-(1.3), and for \(i, j = 1, 2\), \({r_{i}(n)}\), \({a_{ij}(n)}\), and \({b_{1}(n)}\) are bounded nonnegative sequences defined for \(n\in N = \{0, 1, 2, \ldots\}\) such that
$$ 0< r_{i}^{L}\le r_{i}(n)\le r_{i}^{M}, \qquad 0< a^{L}_{ij}\le a_{ij}(n)\le a^{M}_{ij}, \qquad 0< b_{1}^{L}\le b_{1}(n)\le b_{1}^{M}. $$
(1.5)
As regards the biological meaning, we assume (1.4) together with the initial conditions: \(x_{1}(0) > 0\) and \(x_{2}(0) > 0\). It is not difficult to see that the solutions of (1.4) are defined and remain positive for all \(n\in N\). For more relevant work, one could refer to [519] and the references cited therein.

The remaining part of this paper is organized as follows. In Section 2, we study the extinction of some one species. The global stability of the other species when the previous species is eventually in extinction both for systems (1.4) and (1.1) is then studied in Section 3. Some examples together with their numerical simulations are presented in Section 4 to show the feasibility of our results. We give a brief discussion in the last section.

2 Extinction

In this section, we will establish sufficient conditions on the extinction of species \(x_{2}\) or \(x_{1}\). By a similar proof to Lemma 2.1 in Li and Chen [5], we can obtain the following result.

Lemma 2.1

Any positive solution \((x_{1}(n), x_{2}(n))^{T}\) of system (1.4) satisfies
$$ \limsup_{n\rightarrow+\infty}x_{i}(n)\le B_{i}, $$
(2.1)
where \(B_{i}=\frac{\exp(r_{i}^{M}-1)}{a_{ii}^{L}}\), \(i=1,2\).

Lemma 2.1 shows that the positive solutions of system (1.4) are bounded eventually. We now come to the study of the extinction of species \(x_{2}\) of system (1.4).

Theorem 2.1

Assume
$$ (\mathrm{H}_{1})\quad \frac{r_{2}^{M}}{r_{1}^{L}}< \min \biggl\{ \frac {a_{21}^{L}}{a_{11}^{M}(1+B_{1})},\frac{a_{22}^{L}}{a_{12}^{M}} \biggr\} $$
and
$$ (\mathrm{H}_{2})\quad b_{1}^{M}< \frac{1}{B_{1}B_{2}}\min \biggl\{ r_{1}^{L}-r_{2}^{M} \frac {a_{11}^{M}(1+B_{1})}{a_{21}^{L}},r_{1}^{L}-r_{2}^{M} \frac{a_{12}^{M}}{a_{22}^{L}} \biggr\} $$
hold, where \(B_{i} \) (\(i = 1, 2\)) is 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 (1.4), \(\lim_{n\rightarrow+\infty}x_{2}(n)=0\).

Proof

Conditions (H1) and (H2) can be rewritten as
$$ \frac{r_{2}^{M}}{r_{1}^{L}-b_{1}^{M}B_{1}B_{2}}< \min \biggl\{ \frac {a_{21}^{L}}{a_{11}^{M}(1+B_{1})},\frac{a_{22}^{L}}{a_{12}^{M}} \biggr\} . $$
(2.2)
According to (2.2), one can choose a small enough positive constant \(\varepsilon_{1}\) such that
$$ \frac{r_{2}^{M}}{r_{1}^{L}-b_{1}^{M}(B_{1}+\varepsilon_{1})(B_{2}+\varepsilon_{1})}< \min \biggl\{ \frac{a_{21}^{L}}{a_{11}^{M}(1+B_{1}+\varepsilon_{1})},\frac {a_{22}^{L}}{a_{12}^{M}} \biggr\} . $$
(2.3)
By (2.3), there exist positive constants α and β such that
$$ \frac{r_{2}^{M}}{r_{1}^{L}-b_{1}^{M}(B_{1}+\varepsilon_{1})(B_{2}+\varepsilon _{1})}< \frac{\beta}{\alpha}< \min \biggl\{ \frac {a_{21}^{L}}{a_{11}^{M}(1+B_{1}+\varepsilon_{1})}, \frac {a_{22}^{L}}{a_{12}^{M}} \biggr\} . $$
(2.4)
Thus,
$$ \beta a_{11}^{M}-\frac{\alpha a_{21}^{L}}{1+B_{1}}< 0,\qquad \beta a_{12}^{M}-\alpha a_{22}^{L}< 0 $$
(2.5)
and we can choose a constant \(\delta_{1}> 0\), such that
$$ \alpha r_{2}^{M}-\beta r_{1}^{L}+\beta b_{1}^{M}(B_{1}+\varepsilon) (B_{2}+ \varepsilon )< -\delta_{1}< 0. $$
(2.6)
For the above \(\varepsilon_{1}\), it follows from Lemma 2.1 that there exists a large enough N such that
$$ x_{i}(n)< B_{i}+\varepsilon_{1},\quad n>N. $$
(2.7)
For any \(p>N\), according to the equations of system (1.4) and (2.7), we can get
$$ \begin{aligned} &\ln\frac{x_{1}(p+1)}{x_{1}(p)} = r_{1}(p)-a_{11}(p)x_{1}(p)- \frac {a_{12}(p)x_{2}(p)}{1+x_{2}(p)}-b_{1}(p)x_{1}(p)x_{2}(p) \\ &\hphantom{\ln\frac{x_{1}(p+1)}{x_{1}(p)}}\ge r_{1}^{L}-a_{11}^{M}x_{1}(p)-a_{12}^{M}x_{2}(p)-b_{1}^{M}x_{1}(p)x_{2}(p), \\ &\ln\frac{x_{2}(p+1)}{x_{2}(p)} = r_{2}(p)-a_{22}(p)x_{2}(p)- \frac{a_{21}(p)x_{1}(p)}{1+x_{1}(p)} \\ &\hphantom{\ln\frac{x_{2}(p+1)}{x_{2}(p)}}\le r_{2}^{M}-a_{22}^{L}x_{2}(p)- \frac{a_{21}^{L}x_{1}(p)}{1+B_{1}+\varepsilon _{1}}. \end{aligned} $$
(2.8)
Therefore, inequalities (2.5)-(2.8) show that
$$\begin{aligned}& \alpha\ln\frac{x_{2}(p+1)}{x_{2}(p)}-\beta\ln\frac {x_{1}(p+1)}{x_{1}(p)} \\& \quad \le \bigl(\alpha r_{2}^{M}-\beta r_{1}^{L} \bigr)+\biggl(\beta a_{11}^{M}-\frac{\alpha a_{21}^{L}}{1+B_{1}+\varepsilon_{1}} \biggr)x_{1}(p)+\bigl(\beta a_{12}^{M}-\alpha a_{22}^{L}\bigr)x_{2}(p) \\& \qquad {}+ \beta b_{1}^{M}x_{1}(p)x_{2}(p) \\& \quad < \alpha r_{2}^{M}-\beta r_{1}^{L}+ \beta b_{1}^{M}(B_{1}+\varepsilon ) (B_{2}+\varepsilon) \\& \quad < -\delta_{1}< 0,\quad p> N. \end{aligned}$$
(2.9)
Summing both sides of the above inequalities from \(N+1\) to \(n-1\) leads to
$$ \alpha\ln\frac{x_{2}(n)}{x_{2}(N+1)}-\beta\ln\frac {x_{1}(n)}{x_{1}(N+1)}< -\delta_{1}(n-N-1), $$
(2.10)
hence
$$ x_{2}(n)< \biggl[ \biggl(\frac{x_{1}(n)}{x_{1}(N+1)} \biggr)^{\beta}\bigl(x_{2}(N+1) \bigr)^{\alpha}\biggr]^{\frac{1}{\alpha}}\exp \biggl(-\frac {\delta_{1}}{\alpha}(n-N-1) \biggr). $$
(2.11)
The above inequality together with the ultimate boundedness of \(x_{1}(n)\) shows that \(\lim_{n\rightarrow+\infty}x_{2}(n)=0\). The proof is completed. □

Theorem 2.2

In addition to (H1), further suppose that
$$ (\mathrm{H}_{3})\quad b_{1}^{M}< \frac{r_{1}^{L}a_{21}^{L}-(1+B_{1})a_{11}^{M}r_{2}^{M}}{(1+B_{1})B_{2}r_{2}^{M}} $$
holds, where \(B_{i}\) (\(i = 1, 2\)) is defined in Lemma  2.1, then for any positive solution \((x_{1}(n), x_{2}(n))^{T}\) of system (1.4), \(\lim_{n\rightarrow+\infty}x_{2}(n)=0\).

Proof

It follows from conditions (H1) and (H3) that
$$ \frac{r_{2}^{M}}{r_{1}^{L}}< \min \biggl\{ \frac{a_{21}^{L}}{(a_{11}^{M}+ b_{1}^{M}B_{2})(1+B_{1})},\frac{a_{22}^{L}}{a_{12}^{M}} \biggr\} . $$
(2.12)
Thus, one can choose a small enough positive constant \(\varepsilon_{2}\) such that
$$ \frac{r_{2}^{M}}{r_{1}^{L}}< \min \biggl\{ \frac{a_{21}^{L}}{(a_{11}^{M}+ b_{1}^{M}(B_{2}+\varepsilon_{2}))(1+B_{1}+\varepsilon_{2})},\frac {a_{22}^{L}}{a_{12}^{M}} \biggr\} . $$
(2.13)
By (2.13), there exist positive constants α and β such that
$$ \frac{r_{2}^{M}}{r_{1}^{L}}< \frac{\beta}{\alpha}< \min \biggl\{ \frac {a_{21}^{L}}{(a_{11}^{M}+ b_{1}^{M}(B_{2}+\varepsilon_{2}))(1+B_{1}+\varepsilon _{2})}, \frac{a_{22}^{L}}{a_{12}^{M}} \biggr\} . $$
(2.14)
So
$$ \beta a_{11}^{M}+ \beta b_{1}^{M}(B_{2}+ \varepsilon_{2})-\frac{\alpha a_{21}^{L}}{1+B_{1}+\varepsilon_{2}}< 0,\qquad \beta a_{12}^{M}- \alpha a_{22}^{L}< 0 $$
(2.15)
and we can choose a constant \(\delta_{2}> 0\), such that
$$ \alpha r_{2}^{M}-\beta r_{1}^{L}< - \delta_{2}< 0. $$
(2.16)
Therefore, inequalities (2.15), (2.16), (2.7), and (2.8) show that
$$\begin{aligned}& \alpha\ln\frac{x_{2}(p+1)}{x_{2}(p)}-\beta\ln\frac {x_{1}(p+1)}{x_{1}(p)} \\& \quad \le \bigl(\alpha r_{2}^{M}-\beta r_{1}^{L} \bigr)+\biggl(\beta a_{11}^{M}-\frac{\alpha a_{21}^{L}}{1+B_{1}+\varepsilon_{2}} \biggr)x_{1}(p)+\bigl(\beta a_{12}^{M}-\alpha a_{22}^{L}\bigr)x_{2}(p) \\& \qquad {}+ \beta b_{1}^{M}x_{1}(p)x_{2}(p) \\& \quad \le \bigl(\alpha r_{2}^{M}-\beta r_{1}^{L} \bigr)+ \biggl(\beta a_{11}^{M}-\frac{\alpha a_{21}^{L}}{1+B_{1}+\varepsilon_{2}}+ \beta b_{1}^{M}(B_{2}+\varepsilon_{2}) \biggr)x_{1}(p) \\& \qquad {}+\bigl(\beta a_{12}^{M}-\alpha a_{22}^{L} \bigr)x_{2}(p) \\& \quad < \alpha r_{2}^{M}-\beta r_{1}^{L}< - \delta_{2}< 0,\quad p> N. \end{aligned}$$
(2.17)
The rest of the proof is similar to that of the corresponding proof of Theorem 2.1, we omit the details here. This ends the proof of Theorem 2.2. □

Theorem 2.3

Let \((x_{1}(n), x_{2}(n))^{T}\) be any positive solution of system (1.4), in addition to (H1), further suppose that
$$ (\mathrm{H}_{4})\quad b_{1}^{M}< \frac{r_{1}^{L}a_{22}^{L}-r_{2}^{M}a_{12}^{M}}{B_{1}r_{2}^{M}} $$
holds, where \(B_{1}\) is defined in Lemma  2.1, then \(\lim_{n\rightarrow+\infty}x_{2}(n)=0\).

Proof

It follows from conditions (H1) and (H4) that
$$ \frac{r_{2}^{M}}{r_{1}^{L}}< \min \biggl\{ \frac {a_{21}^{L}}{a_{11}^{M}(1+B_{1})},\frac{a_{22}^{L}}{a_{12}^{M}+b_{1}^{M}B_{1}} \biggr\} . $$
(2.18)
Thus, one can choose a small enough positive constant \(\varepsilon_{3}\) such that
$$ \frac{r_{2}^{M}}{r_{1}^{L}}< \min \biggl\{ \frac {a_{21}^{L}}{a_{11}^{M}(1+B_{1}+\varepsilon_{3})},\frac {a_{22}^{L}}{a_{12}^{M}+b_{1}^{M}(B_{1}+\varepsilon_{3})} \biggr\} . $$
(2.19)
By (2.19), there exist positive constants α and β such that
$$ \frac{r_{2}^{M}}{r_{1}^{L}}< \frac{\beta}{\alpha}< \min \biggl\{ \frac {a_{21}^{L}}{a_{11}^{M}(1+B_{1}+\varepsilon_{3})}, \frac {a_{22}^{L}}{a_{12}^{M}+b_{1}^{M}(B_{1}+\varepsilon_{3})} \biggr\} . $$
(2.20)
Thus,
$$ \beta a_{11}^{M}-\frac{\alpha a_{21}^{L}}{1+B_{1}+\varepsilon_{3}}< 0, \qquad \beta a_{12}^{M}-\alpha a_{22}^{L}+\beta b_{1}^{M}(B_{1}+\varepsilon_{3})< 0, $$
(2.21)
and we can choose a constant \(\delta_{3}> 0\), such that
$$ \alpha r_{2}^{M}-\beta r_{1}^{L}< - \delta_{3}< 0. $$
(2.22)
Therefore, inequalities (2.21), (2.22), (2.7), and (2.8) show that
$$\begin{aligned}& \alpha\ln\frac{x_{2}(p+1)}{x_{2}(p)}-\beta\ln\frac {x_{1}(p+1)}{x_{1}(p)} \\& \quad \le \bigl(\alpha r_{2}^{M}-\beta r_{1}^{L} \bigr)+ \biggl(\beta a_{11}^{M}-\frac{\alpha a_{21}^{L}}{1+B_{1}+\varepsilon_{3}} \biggr)x_{1}(p) \\& \qquad {}+ \bigl(\beta a_{12}^{M}-\alpha a_{22}^{L}+\beta b_{1}^{M}(B_{1}+ \varepsilon_{3}) \bigr)x_{2}(p) \\& \quad < \alpha r_{2}^{M}-\beta r_{1}^{L}< - \delta_{3}< 0, \quad p> N. \end{aligned}$$
(2.23)
The rest of the proof is similar to that of the corresponding proof of Theorem 2.1, we omit the details here. This ends the proof of Theorem 2.3. □

Now, let us investigate the extinction property of species \(x_{1}\) in system (1.4) which is also an interesting problem and we obtain the following result.

Theorem 2.4

Let \((x_{1}(n), x_{2}(n))^{T}\) be any positive solution of system (1.4). Suppose
$$ (\mathrm{H}_{5})\quad \frac{r_{2}^{L}}{r_{1}^{M}}>\max \biggl\{ \frac{a_{21}^{M}}{a_{11}^{L}}, \frac {a_{22}^{M}(1+B_{2})}{a_{12}^{L}} \biggr\} $$
holds, where \(B_{2}\) is defined in Lemma  2.1, then the species \(x_{1}\) will be driven to extinction, that is, \(\lim_{n\rightarrow+\infty}x_{1}(n)=0\).

Proof

According to (H5), one can choose a small enough positive constant \(\varepsilon_{4}\) such that
$$ \frac{r_{2}^{L}}{r_{1}^{M}}>\max \biggl\{ \frac{a_{21}^{M}}{a_{11}^{L}},\frac {a_{22}^{M}(1+B_{2}+\varepsilon_{4})}{a_{12}^{L}} \biggr\} . $$
(2.24)
By (2.24), there exist positive constants α and β such that
$$ \frac{r_{2}^{L}}{r_{1}^{M}}>\frac{\beta}{\alpha}>\max \biggl\{ \frac {a_{21}^{M}}{a_{11}^{L}}, \frac{a_{22}^{M}(1+B_{2}+\varepsilon _{4})}{a_{12}^{L}} \biggr\} . $$
(2.25)
Thus,
$$ \alpha a_{21}^{M}-\beta a_{11}^{L}< 0, \qquad \alpha a_{22}^{M}-\beta\frac {a_{12}^{L}}{1+B_{2}+\varepsilon_{4}}< 0 $$
(2.26)
and we can choose a constant \(\delta_{4}> 0\), such that
$$ \beta r_{1}^{M}-\alpha r_{2}^{L}< - \delta_{4}< 0. $$
(2.27)
For any \(p>N\), according to the equations of system (1.4) and (2.7), we can get
$$ \begin{aligned} &\ln\frac{x_{1}(p+1)}{x_{1}(p)} \le r_{1}^{M}-a_{11}^{L}x_{1}(p)- \frac{a_{12}^{L}}{1+B_{2}+\varepsilon _{4}}x_{2}(p)-b_{1}^{L}x_{1}(p)x_{2}(p), \\ &\ln\frac{x_{2}(p+1)}{x_{2}(p)} \ge r_{2}^{L}-a_{22}^{M}x_{2}(p)-a_{21}^{M}x_{1}(p). \end{aligned} $$
(2.28)
Therefore, inequalities (2.26)-(2.28) show that
$$\begin{aligned}& \beta\ln\frac{x_{1}(p+1)}{x_{1}(p)}-\alpha\ln\frac {x_{2}(p+1)}{x_{2}(p)} \\& \quad \le \bigl(\beta r_{1}^{M}-\alpha r_{2}^{L} \bigr)+\bigl(\alpha a_{21}^{M}-\beta a_{11}^{L} \bigr)x_{1}(p)+\biggl(\alpha a_{22}^{M}-\beta \frac {a_{12}^{L}}{1+B_{2}+\varepsilon_{4}}\biggr)x_{2}(p) \\& \qquad {}-\beta b_{1}^{L}x_{1}(p)x_{2}(p) \\& \quad < \beta r_{1}^{M}-\alpha r_{2}^{L}< - \delta_{4}< 0,\quad p>N. \end{aligned}$$
(2.29)
The rest of the proof is similar to that of the corresponding proof of Theorem 2.1, we omit the details here. This ends the proof of Theorem 2.4. □

3 Global stability

In Section 2, we get sufficient conditions which guarantee the extinction of the first or second species in system which motives us to investigate the stability property of the rest species. Let us first state several lemmas which will be useful in the proof of the main result of this section.

Lemma 3.1

(see [20])

Assume that \(\{x(n)\}\) satisfies
$$x(n+1)\ge x(n)\exp\bigl\{ a(n)-b(n)x(n)\bigr\} ,\quad n\ge N_{0}, $$
\(\limsup_{n\rightarrow+\infty}x(n)\leq x^{*}\), and \(x(N_{0})>0\), where \(a(n)\) and \(b(n)\) are nonnegative sequences bounded above and below by positive constants and \(N_{0}\in N\). Then
$$\liminf_{n\rightarrow+\infty}x(n)\geq \min \biggl\{ \frac{a^{L}}{b^{U}}\exp\bigl\{ a^{L}-b^{U}x^{*}\bigr\} ,\frac{a^{L}}{b^{U}} \biggr\} . $$

Lemma 3.2

Suppose conditions in Theorem  2.1 or 2.2, or 2.3 hold, let \((x_{1}(n), x_{2}(n))^{T}\) be any positive solution of system (1.4), then
$$A_{1}\le\liminf_{n\rightarrow+\infty}x_{1}(n)\le\limsup _{n\rightarrow +\infty}x_{1}(n)\le B_{1}, $$
where \(A_{1}=\frac{r_{1}^{L}}{a_{11}^{M}}\exp\{r_{1}^{L}-a_{11}^{M}B_{1}\}\) and \(B_{1}\) is defined in Lemma  2.1.

Proof

It follows from Lemma 2.1 and Theorem 2.1 or 2.2, or 2.3 that
$$ \lim_{n\rightarrow+\infty}x_{2}(n)=0,\qquad \limsup _{n\rightarrow+\infty }x_{1}(n)\le B_{1}. $$
(3.1)
To end the proof of Lemma 3.2, it is enough to show that
$$ \liminf_{n\rightarrow+\infty}x(n)\ge A_{1}. $$
(3.2)
Since \(r_{1}^{L}>0\), there exists a small enough \(\varepsilon>0\) such that
$$ A_{\varepsilon}\triangleq r_{1}^{L}-a_{12}^{M} \varepsilon-b_{1}^{M}(B_{1}+\varepsilon) \varepsilon>0. $$
(3.3)
According to (3.1), for the above \(\varepsilon>0\), there exists a large enough \(N_{1}>0\), such that, for \(n\ge N_{1}\),
$$ x_{1}(n)\le B_{1}+\varepsilon,\qquad x_{2}(n)\le \varepsilon. $$
(3.4)
Thus, it follows from (3.4) and the first equation of system (1.4) that
$$ x_{1}(n+1) \ge x_{1}(n)\exp \bigl\{ r_{1}^{L}-a_{11}^{M}x_{1}(n)-a_{12}^{M} \varepsilon -b_{1}^{M}(B_{1}+\varepsilon) \varepsilon \bigr\} . $$
(3.5)
Since \(A_{\varepsilon}>0\), by applying Lemma 3.1 to (3.5), it immediately follows that
$$\liminf_{n\rightarrow+\infty}x_{1}(n)\geq \min \biggl\{ \frac{A_{\varepsilon}}{a_{11}^{M}}\exp\bigl\{ A_{\varepsilon }-a_{11}^{M}B_{1} \bigr\} ,\frac{A_{\varepsilon}}{a_{11}^{M}} \biggr\} . $$
Setting \(\varepsilon\rightarrow0\) in the above inequality, one can obtain
$$ \liminf_{n\rightarrow+\infty}x_{1}(n)\geq \min \biggl\{ \frac{r_{1}^{L}}{a_{11}^{M}}\exp\bigl\{ r_{1}^{L}-a_{11}^{M}B_{1} \bigr\} ,\frac {r_{1}^{L}}{a_{11}^{M}} \biggr\} . $$
(3.6)
By calculation, one can easily get
$$ r_{1}^{L}-a_{11}^{M}B_{1}=r_{1}^{L}-a_{11}^{M} \frac{\exp (r_{1}^{M}-1)}{a_{11}^{L}}\le r_{1}^{L}-\exp\bigl(r_{1}^{M}-1 \bigr)\le r_{1}^{L}-r_{1}^{M} \le0. $$
(3.7)
Inequality (3.6) together with (3.7) leads to
$$ \liminf_{n\rightarrow+\infty}x_{1}(n)\geq \frac{r_{1}^{L}}{a_{11}^{M}}\exp \bigl\{ r_{1}^{L}-a_{11}^{M}B_{1} \bigr\} \triangleq A_{1}, $$
(3.8)
that is to say, (3.2) holds. This ends the proof of Lemma 3.2. □

Lemma 3.3

Suppose conditions in Theorem  2.4 hold, let \((x_{1}(n), x_{2}(n))^{T}\) be any positive solution of system (1.4), then
$$A_{2}\le\liminf_{n\rightarrow+\infty}x_{2}(n)\le\limsup _{n\rightarrow +\infty}x_{2}(n)\le B_{2}, $$
where \(A_{2}=\frac{r_{2}^{L}}{a_{22}^{M}}\exp\{r_{2}^{L}-a_{22}^{M}B_{2}\}\) and \(B_{2}\) is defined in Lemma  2.1.

Proof

The proof of Lemma 3.3 is similar to that of the proof of Lemma 3.2, we omit the details here. □

Consider the following discrete logistic equation:
$$ x(n + 1) = x(n)\exp\bigl(r_{1}(n)- a_{11}(n)x(n)\bigr), \quad n\in N, $$
(3.9)
where \(r_{1}(n)\) and \(a_{11}(n)\) are bounded nonnegative sequences.

Lemma 3.4

(see [8])

For any positive solution \(x(n)\) of (3.9), we have
$$A_{1}\le\liminf_{n\rightarrow+\infty}x(n)\le\limsup _{n\rightarrow +\infty}x(n)\le B_{1}, $$
where \(A_{1}\), \(B_{1}\) are defined by Lemma  3.2.
Consider the following discrete logistic equation:
$$ x(n + 1) = x(n)\exp\bigl(r_{2}(n)- a_{22}(n)x(n)\bigr), \quad n\in N, $$
(3.10)
where \(r_{2}(n)\) and \(a_{22}(n)\) are bounded nonnegative sequences.

Lemma 3.5

(see [8])

For any positive solution \(\tilde{x}(n)\) of (3.10), we have
$$A_{2}\le\liminf_{n\rightarrow+\infty}\tilde{x}(n)\le\limsup _{n\rightarrow+\infty}\tilde{x}(n)\le B_{2}, $$
where \(A_{2}\), \(B_{2}\) are defined by Lemma  3.3.

Now, we come to showing the main results of this section.

Theorem 3.1

Suppose in addition the conditions of Theorem  2.1 or 2.2, or 2.3 hold, further suppose that
$$ (\mathrm{H}_{6})\quad \frac{a_{11}^{M}}{a_{11}^{L}}\exp\bigl(r_{1}^{M}-1 \bigr)< 2. $$
Then for any positive solution \((x_{1}(n), x_{2}(n))^{T}\) of system (1.4), we have
$$\lim_{n\rightarrow+\infty}\bigl(x_{1}(n)-x(n)\bigr)=0,\qquad \lim _{n\rightarrow +\infty}x_{2}(n)=0, $$
where \(x(n)\) is any positive solution of system (3.9).

Proof

It follows from Theorem 2.1 or 2.2, or 2.3 that
$$ \lim_{n\rightarrow+\infty}x_{2}(n)=0. $$
(3.11)
Set \(y(n)=\ln x_{1}(n)-\ln x(n)\), then it follows from the first equation of system (1.4) and (3.9) that
$$ y(n+1)=y(n)-a_{11}(n)x(n) \bigl(\exp\bigl(y(n)\bigr)-1\bigr)- \frac {a_{12}(n)x_{2}(n)}{1+x_{2}(n)}-b_{1}(n)x_{1}(n)x_{2}(n). $$
(3.12)
Using the mean value theorem, we can obtain
$$ \exp\bigl(y(n)\bigr)-1=\exp\bigl(\theta(n)y(n)\bigr)y(n),\quad \theta(n)\in (0,1). $$
(3.13)
Substituting (3.13) into the right side of equation (3.12), we can get
$$ y(n+1)= \bigl(1-a_{11}(n)x(n)\exp\bigl(\theta(n)y(n)\bigr) \bigr)y(n)- \biggl(\frac{a_{12}(n)}{1+x_{2}(n)}+b_{1}(n)x_{1}(n) \biggr)x_{2}(n). $$
(3.14)
Considering (H6) implies that \(-1<1-a_{11}^{M}B_{1}\), there exists a small enough \(\varepsilon>0\) such that
$$ -1< 1-a_{11}^{M}(B_{1}+\varepsilon). $$
(3.15)
According to Lemma 3.2, Lemma 3.4, and (3.11), for the above \(\varepsilon>0\), there exists large enough \(N>0\), such that, for \(n\ge N\),
$$ A_{1}-\varepsilon\le x_{1}(n)\le B_{1}+ \varepsilon,\qquad x_{2}(n)\le \varepsilon, \qquad A_{1}- \varepsilon\le x(n)\le B_{1}+\varepsilon. $$
(3.16)
Note that \(\theta(n)\in(0,1)\) implies that \(x(n)\exp(\theta (n)y(n))\) lies between \(x(n)\) and \(x_{1}(n)\). From (3.14) and (3.16), for \(n\ge N\), one can get
$$\begin{aligned} \bigl\vert y(n+1)\bigr\vert \le&\max \bigl\{ \bigl\vert 1-a_{11}^{M}(B_{1}+\varepsilon )\bigr\vert , \bigl\vert 1-a_{11}^{L}(A_{1}-\varepsilon)\bigr\vert \bigr\} \bigl\vert y(n)\bigr\vert \\ &{}+ \bigl(a_{12}^{M}+b_{1}^{M}(B_{1}+ \varepsilon) \bigr)\varepsilon \\ \triangleq&\lambda_{\varepsilon}\bigl\vert y(n)\bigr\vert +M_{\varepsilon}\varepsilon, \end{aligned}$$
(3.17)
where \(\lambda_{\varepsilon}=\max \{|1-a_{11}^{M}(B_{1}+\varepsilon )|,|1-a_{11}^{L}(A_{1}-\varepsilon)| \}\), \(M_{\varepsilon}=a_{12}^{M}+b_{1}^{M}(B_{1}+\varepsilon)\). This implies that
$$ \bigl\vert y(n)\bigr\vert \le\lambda_{\varepsilon}^{n-N}\bigl\vert y(N)\bigr\vert +\frac{1-\lambda _{\varepsilon}^{n-N}}{1-\lambda_{\varepsilon}}M_{\varepsilon}\varepsilon , \quad \mbox{for } n\ge N. $$
(3.18)
Note that \(1-a_{11}^{M}(B_{1}+\varepsilon)\le1-a_{11}^{L}(A_{1}-\varepsilon )<1\), hence \(0<\lambda_{\varepsilon}<1\) according to (3.15). Thus, \(\lim_{n\rightarrow+\infty}y(n)=0\) can be immediately obtained by (3.18), and so \(\lim_{n\rightarrow+\infty }(x_{1}(n)-x(n))=0\). This ends the proof of Theorem 3.1. □

Similarly, by using Lemmas 3.3 and 3.5, we have the following theorem.

Theorem 3.2

In addition to the conditions of Theorem  2.4, further suppose that
$$ (\mathrm{H}_{7})\quad \frac{a_{22}^{M}}{a_{22}^{L}}\exp\bigl(r_{2}^{M}-1 \bigr)< 2.$$
Then for any positive solution \((x_{1}(n), x_{2}(n))^{T}\) of system (1.4) and any positive solution \(\tilde{x}(n)\) of system (3.10), we have
$$\lim_{n\rightarrow+\infty}x_{1}(n)=0,\qquad \lim _{n\rightarrow+\infty }\bigl(x_{2}(n)-\tilde{x}(n)\bigr)=0. $$

As a direct corollary of Theorem 3.1 and Theorem 3.2, we have the following corollary.

Corollary 3.1

In addition to (H1), further suppose that
$$ (\mathrm{H}_{6})\quad \frac{a_{11}^{M}}{a_{11}^{L}}\exp\bigl(r_{1}^{M}-1 \bigr)< 2.$$
Then for any positive solution \((x_{1}(n), x_{2}(n))^{T}\) of system (1.1) and any positive solution \(x(n)\) of system (3.9), we have
$$\lim_{n\rightarrow+\infty}\bigl(x_{1}(n)-x(n)\bigr)=0,\qquad \lim _{n\rightarrow +\infty}x_{2}(n)=0. $$
That is to say, the species \(x_{2}\) will be driven to extinction.

Corollary 3.2

Assume that the conditions of Theorem  2.4 hold, also
$$ (\mathrm{H}_{7})\quad \frac{a_{22}^{M}}{a_{22}^{L}}\exp\bigl(r_{2}^{M}-1 \bigr)< 2.$$
Then for any positive solution \((x_{1}(n), x_{2}(n))^{T}\) of system (1.1) and any positive solution \(\tilde{x}(n)\) of system (3.10), we have
$$\lim_{n\rightarrow+\infty}x_{1}(n)=0,\qquad \lim _{n\rightarrow+\infty }\bigl(x_{2}(n)-\tilde{x}(n)\bigr)=0. $$
That is to say, the species \(x_{1}\) will be driven to extinction.

4 Examples and numeric simulation

In this section, we give the following two examples to verify the feasibilities of our results.

Example 4.1

Consider the following system:
$$ \begin{aligned} &x_{1}(n+1) = x_{1}(n)\exp \biggl\{ 1.2-1.5x_{1}(n)-\frac{(1+0.3\sin(\sqrt {7}n))x_{2}(n)}{1+x_{2}(n)}-b_{1}(n)x_{1}(n)x_{2}(n) \biggr\} , \\ &x_{2}(n+1) = x_{2}(n)\exp \biggl\{ 0.6-1.3x_{2}(n)- \frac{(4+\cos(\sqrt {3}n))x_{1}(n)}{1+x_{1}(n)} \biggr\} . \end{aligned} $$
(4.1)

Case 1. \(b_{1}(n)=0.2\).

Take easy calculation, we have \(\frac{r_{2}^{M}}{r_{1}^{L}}=0.5\), \(B_{1}\approx0.8143\), \(B_{2}\approx0.5156\), \(\frac {a_{21}^{L}}{a_{11}^{M}(1+B_{1})}\approx1.1024\), \(\frac{a_{22}^{L}}{a_{12}^{M}}=1\), \(r_{1}^{L}-r_{2}^{M}\frac{a_{11}^{M}(1+B_{1})}{a_{21}^{L}}\approx0.6476\), \(r_{1}^{L}-r_{2}^{M}\frac{a_{12}^{M}}{a_{22}^{L}}=0.6\), thus
$$ \frac{r_{2}^{M}}{r_{1}^{L}}=0.5< \min \biggl\{ \frac {a_{21}^{L}}{a_{11}^{M}(1+B_{1})},\frac{a_{22}^{L}}{a_{12}^{M}} \biggr\} =1 $$
(4.2)
and
$$\begin{aligned}& b_{1}^{M}=0.2 < \frac{1}{B_{1}B_{2}}\min \biggl\{ r_{1}^{L}-r_{2}^{M} \frac {a_{11}^{M}(1+B_{1})}{a_{21}^{L}},r_{1}^{L}-r_{2}^{M} \frac{a_{12}^{M}}{a_{22}^{L}} \biggr\} \approx1.4291, \\& b_{1}^{M}=0.2 < \frac {r_{1}^{L}a_{21}^{L}-(1+B_{1})a_{11}^{M}r_{2}^{M}}{(1+B_{1})B_{2}r_{2}^{M}}\approx3.5048, \\& b_{1}^{M}=0.2 < \frac{r_{1}^{L}a_{22}^{L}-r_{2}^{M}a_{12}^{M}}{B_{1}r_{2}^{M}}\approx 1.5965, \end{aligned}$$
(4.3)
(4.2)-(4.3) show that the coefficients of the system (4.1) satisfy the conditions of Theorems 2.1, 2.2, and 2.3. Moreover,
$$\frac{a_{11}^{M}}{a_{11}^{L}}\exp\bigl(r_{1}^{M}-1\bigr)=\exp(1.2-1) \approx1.2214< 2. $$
Hence, condition (H6) is also satisfied. It follows from Theorem 3.1 that, for any positive solution \((x_{1}(n), x_{2}(n))^{T}\) of system (4.1), we have \(\lim_{n\rightarrow+\infty}(x_{1}(n)-x(n))=0\), \(\lim_{n\rightarrow +\infty}x_{2}(n)=0\), where \(\{x(n)\}\) is any positive solution of the system
$$x(n+1) = x(n)\exp \bigl\{ 1.2-1.5x(n) \bigr\} . $$
Our numerical simulation supports our result (see Figure 1).
Figure 1
Figure 1

Dynamical behaviors of the system ( 4.1 ) with \(\pmb{b_{1}(n)=0.2 }\) and the initial conditions \(\pmb{(x_{1}(0), x_{2}(0))=(1,0.4)^{T}}\) , \(\pmb{(0.33,0.14)^{T}}\) , \(\pmb{(0.68,0.9)^{T}}\) , and \(\pmb{(1.3,0.5)^{T}}\) , respectively.

Case 2. \(b_{1}(n)=0\).

\(b_{1}(n)=0\) shows that the two species are all non-toxic. One can easily find that the conditions in Corollary 3.1 are satisfied, so for any positive solution \((x_{1}(n), x_{2}(n))^{T}\) of system (4.1) with \(b_{1}(n)=0\), \(x_{2}(n)\) is in extinction while \(x_{1}(n)\) will be globally attractive (see Figure 2).
Figure 2
Figure 2

Dynamical behaviors of the system ( 4.1 ) with \(\pmb{b_{1}(n)=0 }\) and the initial conditions \(\pmb{(x_{1}(0), x_{2}(0))=(1,0.4)^{T}}\) , \(\pmb{(0.33,0.14)^{T}}\) , \(\pmb{(0.68,0.9)^{T}}\) , and \(\pmb{(1.3,0.5)^{T}}\) , respectively.

Example 4.2

Consider the following system:
$$\begin{aligned}& x_{1}(n+1) = x_{1}(n)\exp \biggl\{ 0.3-1.5x_{1}(n)-\frac{(2.2+0.2\sin(\sqrt {5}n))x_{2}(n)}{1+x_{2}(n)} \\& \hphantom{x_{1}(n+1) ={}}{}-b_{1}(n)x_{1}(n)x_{2}(n) \biggr\} , \\& x_{2}(n+1) = x_{2}(n)\exp \biggl\{ 1.2-1.3x_{2}(n)- \frac{(2+\cos(\sqrt {7}n))x_{1}(n)}{1+x_{1}(n)} \biggr\} . \end{aligned}$$
(4.4)

Case 1. \(b_{1}(n)=0.3\).

In this case, one could easily see that \(\frac{r_{2}^{L}}{r_{1}^{M}}=4\), \(B_{2}\approx0.9395\), \(\frac{a_{21}^{M}}{a_{11}^{L}}=2\), \(\frac {a_{22}^{M}(1+B_{2})}{a_{12}^{L}}\approx1.2607\), so
$$ \frac{r_{2}^{L}}{r_{1}^{M}}=4>\max \biggl\{ \frac {a_{21}^{M}}{a_{11}^{L}},\frac{a_{22}^{M}(1+B_{2})}{a_{12}^{L}} \biggr\} =2 $$
(4.5)
and
$$ \frac{a_{22}^{M}}{a_{22}^{L}}\exp\bigl(r_{2}^{M}-1\bigr)=\exp (1.2-1) \approx1.2214< 2, $$
(4.6)
(4.5) and (4.6) mean that all conditions of Theorem 3.2 are satisfied in system (4.4). Thus, for any positive solution \((x_{1}(n), x_{2}(n))^{T}\) of system (4.4) and any positive solution \(\{\tilde{x}(n)\}\) of system (3.10), we have \(\lim_{n\rightarrow+\infty}x_{1}(n)=0\), \(\lim_{n\rightarrow+\infty }(x_{2}(n)-\tilde{x}(n))=0\), where \(\{\tilde{x}(n)\}\) is any positive solution of the system
$$x(n+1) = x(n)\exp \bigl\{ 1.2-1.3x(n) \bigr\} . $$
Figure 3 shows the dynamical behavior of system (4.4) with \(b_{1}(n)=0.3\).
Figure 3
Figure 3

Dynamical behaviors of the system ( 4.2 ) with \(\pmb{b_{1}(n)=0.3}\) and the initial conditions \(\pmb{(x_{1}(0), x_{2}(0))=(0.9,0.3)^{T}}\) , \(\pmb{(0.2,0.1)^{T}}\) , \(\pmb{(0.62,0.8)^{T}}\) , and \(\pmb{(0.5,0.5)^{T}}\) , respectively.

Case 2. \(b_{1}(n)=0\).

\(b_{1}(n)=0\) shows that the two species are all non-toxic. One can easily find that the conditions in Corollary 3.2 are satisfied, so for any positive solution \((x_{1}(n), x_{2}(n))^{T}\) of system (4.4) with \(b_{1}(n)=0\), \(x_{1}(n)\) is in extinction while \(x_{2}(n)\) will be globally attractive (see Figure 4).
Figure 4
Figure 4

Dynamical behaviors of the system ( 4.2 ) with \(\pmb{b_{1}(n)=0}\) and the initial conditions \(\pmb{(x_{1}(0), x_{2}(0))=(0.9,0.3)^{T}}\) , \(\pmb{(0.2,0.1)^{T}}\) , \(\pmb{(0.62,0.8)^{T}}\) , and \(\pmb{(0.5,0.5)^{T}}\) , respectively.

5 Discussion

In this paper, we consider a two species nonautonomous discrete competitive system with nonlinear inter-inhibition terms and one toxin producing species, i.e., (1.4). By developing the analysis technique of Chen et al. [8], sufficient conditions which guarantee the extinction of one of the two species are obtained and the stability property of the other species are proved. As direct results of Theorem 3.1 and Theorem 3.2, Corollaries 3.1 and 3.2 show the same conclusions for a non-toxic system, which supplements the results of [1, 2]. Moreover, by comparing Theorem 3.1 with Corollary 3.1, and Theorem 3.2 with Corollary 3.2, we also found that, for such a kind of system, a lower rate of toxic production has no influence on the extinction property of the system.

Declarations

Acknowledgements

The author would like to thank the anonymous referees and the editor for their constructive suggestions on improving the presentation of the paper. Also, this research was supported by Anhui Province College Excellent Young Talents Support Plan Key Projects (No. gxyqZD2016240).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Finance and Mathematics, West Anhui University, Liuan, P.R. China

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