Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton
- Xiangdong Xie^{1}Email author,
- Yalong Xue^{1},
- Runxin Wu^{2} and
- Liang Zhao^{3}
https://doi.org/10.1186/s13662-016-0974-4
© Xie et al. 2016
Received: 22 June 2016
Accepted: 13 September 2016
Published: 12 October 2016
Abstract
A two species non-autonomous competitive phytoplankton system with nonlinear inter-inhibition terms and one toxin producing phytoplankton is studied in this paper. Sufficient conditions which guarantee the extinction of a species and the global attractivity of the other one are obtained. Some parallel results corresponding to Yue (Adv. Differ. Equ. 2016:1, 2016, doi:10.1007/s11590-013-0708-4) are established. Numeric simulations are carried out to show the feasibility of our results.
Keywords
MSC
1 Introduction
Given a function \(g(t)\), let \(g_{L}\) and \(g_{M}\) denote \(\inf_{-\infty< t <\infty}g(t)\) and \(\sup_{-\infty< t<\infty}g(t)\), respectively.
It is well known that if the amount of the species is large enough, the continuous model is more appropriate, and this motivated us to propose the system (1.1). The aim of this paper is, by developing the analysis technique of [1, 8, 9], to investigate the extinction property of the system (1.1). The remaining part of this paper is organized as follows. In Section 2, we study the extinction of some species and the stability property of the rest of the species. Some examples together with their numerical simulations are presented in Section 3 to show the feasibility of our results. We give a brief discussion in the last section.
2 Main results
Following Lemma 2.1 is a direct corollary of Lemma 2.2 of Chen [10].
Lemma 2.1
Lemma 2.2
Proof
Lemma 2.3
(Fluctuation lemma [34])
Lemma 2.4
Suppose that \(r_{1}(t)\) and \(a_{1}(t)\) are continuous functions bounded above and below by positive constants, then any positive solutions of equation (2.5) are defined on \([0, +\infty)\), bounded above and below by positive constants and globally attractive.
Our main results are Theorems 2.1-2.5.
Theorem 2.1
Proof
Theorem 2.2
Proof
Theorem 2.3
Proof
Lemma 2.5
Theorem 2.4
Assume that the conditions of Theorem 2.1 or 2.2 or 2.3 hold, let \(x(t)=(x_{1}(t),x_{2}(t))^{T}\) be any positive solution of system (1.1), then the species \(x_{2}\) will be driven to extinction, that is, \(x_{2}(t)\rightarrow0 \) as \(t\rightarrow +\infty\), and \(x_{1}(t)\rightarrow x_{1}^{*}(t)\) as \(t\rightarrow+\infty\), where \(x_{1}^{*}(t)\) is any positive solution of system (2.5).
Proof
By applying Lemmas 2.3 and 2.4, the proof of Theorem 2.4 is similar to that of the proof of Theorem in [4]. We omit the details here. □
Another interesting thing is to investigate the extinction property of species \(x_{1}\) in system (1.1). For this case, we have the following.
Theorem 2.5
Proof
3 Numeric example
Now let us consider the following example.
Example 1
4 Conclusion
Stimulated by the work of Yue [1], in this paper, a two species non-autonomous competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton is proposed and studied. Series conditions which ensure the extinction of one species and the global attractivity of the other species are established.
We mention here that in system (1.1), we did not consider the influence of delay, we leave this for future investigation.
Declarations
Acknowledgements
The authors are grateful to anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. The research was supported by the Natural Science Foundation of Fujian Province (2015J01012, 2015J01019, 2015J05006) and the Scientific Research Foundation of Fuzhou University (XRC-1438).
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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