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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 attractivity
MSC
 34C25
 92D25
 34D20
 34D40
1 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
References
 Fan, M, Wang, K: Periodic solutions of a discrete time nonautonomous ratiodependent predatorprey system. Math. Comput. Model. 35, 951961 (2002) MathSciNetView ArticleMATHGoogle Scholar
 Chen, FD, You, MS: Permanence for an integrodifferential model of mutualism. Appl. Math. Comput. 186, 3034 (2007) MathSciNetMATHGoogle Scholar
 Chen, FD: Permanence for the discrete mutualism model with time delays. Math. Comput. Model. 47, 431435 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Chen, FD, Yang, JH, Chen, LJ, Xie, XD: On a mutualism model with feedback controls. Appl. Math. Comput. 214, 581587 (2009) MathSciNetMATHGoogle Scholar
 Chen, LJ, Li, Z: Permanence of a delayed discrete mutualism model with feedback controls. Math. Comput. Model. 50, 10831089 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Sun, GC, Wei, WL: The qualitative analysis of commensal symbiosis model of two populations. Math. Theory Appl. 23(3), 6468 (2003) MathSciNetGoogle Scholar
 Zhu, ZF, Li, YA, Xu, F: Mathematical analysis on commensalism LotkaVolterra model of populations. J. Chongqing Inst. Tech. Nat. Sci. Ed. 21, 5962 (2007) Google Scholar
 Yang, YL, Han, RY, Xue, YL, Chen, FD: On a nonautonomous obligate LotkaVolterra model. J. Sanming Univ. 31(6), 1518 (2007) Google Scholar
 Xue, YL, Xie, XD, Chen, FD, Han, RY: Almost periodic solution of a discrete commensalism system. Discrete Dyn. Nat. Soc. 2015, Article ID 295483 (2015) MathSciNetGoogle Scholar
 Xie, XD, Miao, ZS, Xue, Yl: Positive periodic solution of a discrete LotkaVolterra commensal symbiosis model. Commun. Math. Biol. Neurosci. 2015, Article ID 2 (2015) Google Scholar
 Chen, FD, Pu, LQ, Yang, LY: Positive periodic solution of a discrete obligate LotkaVolterra model. Commun. Math. Biol. Neurosci. 2015, Article ID 9 (2015) Google Scholar
 Miao, ZS, Xie, XD, Pu, LQ: Dynamic behaviors of a periodic LotkaVolterra commensal symbiosis model with impulsive. Commun. Math. Biol. Neurosci. 2015, Article ID 3 (2015) Google Scholar
 Ayala, FJ, Gilpin, ME, Eherenfeld, JG: Competition between species: theoretical models and experimental tests. Theor. Popul. Biol. 4, 331356 (1973) MathSciNetView ArticleGoogle Scholar
 Chen, FD: Some new results on the permanence and extinction of nonautonomous GilpinAyala type competition model with delays. Nonlinear Anal., Real World Appl. 7(5), 12051222 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Chen, FD: Average conditions for permanence and extinction in nonautonomous GilpinAyala competition model. Nonlinear Anal., Real World Appl. 7(4), 895915 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Liao, XX, Li, J: Stability in GilpinAyala competition models with diffusion. Nonlinear Anal., Theory Methods Appl. 28(10), 17511758 (1997) MathSciNetView ArticleMATHGoogle Scholar
 Chen, LS: Mathematical Models and Methods in Ecology. Science Press, Beijing (1988) (in Chinese) Google Scholar
 Chen, FD, Chen, YM, Guo, SJ, Zhong, L: Global atrativitity of a generalized LotkaVolterra competition model. Differ. Equ. Dyn. Syst. 18(3), 303315 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Wang, DH, Xu, JY, Wang, HN: Extinction in a generalized GilpinAyala competition system. J. Fuzhou Univ. 41(6), 967971 (2013) (in Chinese) Google Scholar
 Chen, FD: On a nonlinear nonautonomous predatorprey model with diffusion and distributed delay. J. Comput. Appl. Math. 80(1), 3349 (2005) View ArticleMATHGoogle Scholar