Dynamic behaviors of an obligate GilpinAyala system
 Danhong Wang^{1}Email author
Received: 21 June 2016
Accepted: 4 September 2016
Published: 25 October 2016
Abstract
In this paper, a nonautonomous obligate GilpinAyala system is proposed and studied. The persistence and extinction of the system are discussed by using the comparison theorem of differential equations. The results show that, depending on the cooperation intensity between the species, the first species will be driven extinct or be permanent. After that, by using the Lyapunov function method, series of sufficient conditions are obtained which ensure the global attractivity of the system. Finally, two examples are given to illustrate the feasibility of the main results.
Keywords
obligate GilpinAyala system extinction permanence Lyapunov function global attractivityMSC
34C25 92D25 34D20 34D401 Introduction
During the last decade, the dynamic behaviors of the mutualism model have been extensively investigated [1–8] and many excellent results were obtained, which concern the persistence, existence of a positive periodic solution, and stability of the system. However, there are only a few scholars to study the commensal symbiosis model.
The GilpinAyala competition system has been extensively studied, and many excellent results as regards the system have been obtained. However, to this day, still no scholar considered the GilpinAyala system with commensalism symbiosis.
The aim of the paper is, by using Lemma 2.3 and developing the analysis technique of Chen [20] to obtain a set of sufficient conditions to ensure the extinction, persistence, and global attractivity of system (1.5).
The paper is organized as follows: In Section 2, the necessary preliminaries are presented. In Section 3, the dynamic behaviors such as the permanence, extinction, and the globally attractivity of the system are investigated. In Section 4, some examples are given to illustrate the feasibility of main results. In the end, we finish this paper by a brief conclusion.
2 Preliminaries
Now we will state three lemmas which will be useful in proving the main theorems.
Lemma 2.1
Lemma 2.1 is a direct corollary of Lemma 2.2 of Chen [20].
Lemma 2.2
Let h be a real number and f be a nonnegative function defined on \([h;+\infty)\) such that f is integrable on \([h;+\infty)\) and is uniformly continuous on \([h;+\infty)\), then \(\lim_{t\rightarrow+\infty}f(t)=0\).
Lemma 2.2 is Lemma 2.4 of Chen [20].
Lemma 2.3
If \(m\leq x,y\leq M \), where \(m \leq M\) are positive constants, when \(\theta\geq1\), we have \(\frac{1}{\theta M^{\theta1}}x^{\theta}y^{\theta}\leqxy \leq \frac{1}{\theta m^{\theta1}}x^{\theta}y^{\theta}\), and when \(0\leq\theta< 1\), we have \(\frac{1}{\theta m^{\theta 1}}x^{\theta}y^{\theta}\leqxy \leq \frac{1}{\theta M^{\theta1}}x^{\theta}y^{\theta}\).
Proof

when \(\theta\geq1\), we have \(\frac{1}{\theta M^{\theta1}}x^{\theta}y^{\theta}\leqxy \leq \frac{1}{\theta m^{\theta1}}x^{\theta}y^{\theta}\),

when \(0\leq\theta< 1\), we have \(\frac{1}{\theta m^{\theta1}}x^{\theta}y^{\theta}\leqxy \leq \frac{1}{\theta M^{\theta1}}x^{\theta}y^{\theta}\).
3 Main results
Theorem 3.1
Proof
Theorem 3.2
Proof
Theorem 3.3
Proof
Let \((x_{1}(t),x_{2}(t))\) with \(x_{1}(0)>0\), \(x_{2}(0)>0\) and \((x_{1}^{*}(t),x_{2}^{*}(t))\) with \(x_{1}^{*}(0)>0\), \(x_{2}^{*}(0)>0\) be any two positive solutions of (1.5).
4 Examples
The following two examples show the feasibility of our main results.
Example 4.1
Example 4.2
5 Conclusion
In this paper, a nonautonomous obligate GilpinAyala system is considered. Some results as regards extinction, persistence, and global attractivity of the system are obtained. Theorem 3.1 and Theorem 3.2 show that, when (H_{1}) holds, that is, \(\frac{\alpha_{12}^{u}}{k_{1}^{l}}< \frac{1}{k_{2}^{u}}\), the cooperation between the species is small, then the first species could be driven extinct, while the second species has stability; when (H_{2}) holds, that is, \(\frac{\alpha_{12}^{l}}{k_{1}^{u}}> \frac{1}{k_{2}^{l}}\), the cooperation between species is big, then the two species could be permanent. These results show that the extinction or permanence of the first species is depending on the cooperation intensity between the two species.
Declarations
Acknowledgements
The author is grateful to anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper. This work was completed with the support of the Foundation of Fujian Provincial Department of Education (2015JA15431).
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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