Extinction for a discrete competition system with the effect of toxic substances
- Qin Yue^{1}Email author
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
MSC
1 Introduction
For any bounded sequence \(\{f(n)\}\), \(f^{L}=\inf_{n\in N} \{f(n)\}\), \(f^{M}=\sup_{n\in N} \{f(n)\}\).
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
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
Proof
Theorem 2.2
Proof
Theorem 2.3
Proof
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
Proof
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])
Lemma 3.2
Proof
Lemma 3.3
Proof
The proof of Lemma 3.3 is similar to that of the proof of Lemma 3.2, we omit the details here. □
Lemma 3.4
(see [8])
Lemma 3.5
(see [8])
Now, we come to showing the main results of this section.
Theorem 3.1
Proof
Similarly, by using Lemmas 3.3 and 3.5, we have the following theorem.
Theorem 3.2
As a direct corollary of Theorem 3.1 and Theorem 3.2, we have the following corollary.
Corollary 3.1
Corollary 3.2
4 Examples and numeric simulation
In this section, we give the following two examples to verify the feasibilities of our results.
Example 4.1
Case 1. \(b_{1}(n)=0.2\).
Case 2. \(b_{1}(n)=0\).
Example 4.2
Case 1. \(b_{1}(n)=0.3\).
Case 2. \(b_{1}(n)=0\).
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
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