Dynamic behaviors of a commensal symbiosis model with ratio-dependent functional response and one party can not survive independently

Volume 16, Issue 4, pp 495-506 http://dx.doi.org/10.22436/jmcs.016.04.03

Publication Date: December 15, 2016 Submission Date: July 24, 2016

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Authors

Runxin Wu - College of Mathematics and Physics, Fujian University of Technology, Fuzhou, Fujian, 350014, P. R. China. Lin Li - College of Mathematics and Physics, Fujian University of Technology, Fuzhou, Fujian, 350014, P. R. China.

Abstract

We propose a two-species commensal symbiosis model with ratio-dependent functional response \[\frac{dx}{dt}=x\left(-a_1-b_1x+\frac{c_1y}{x+y}\right),\] \[\frac{dy}{dt}=y\left(a_2-b_2y\right),\] For autonomous case, we show that the unique positive equilibrium is globally stable if \(a_1 < c_1\) holds, and the boundary equilibrium \((0, \frac{a_2}{b_2})\) is globally stable if \(a_1 > c_1\) holds. For nonautonomous case, some sufficient conditions which ensure the permanence and global attractivity of the system are obtained. Numeric simulations are carried out to show the feasibility of the main results.

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Keywords

• Commensal symbiosis model
• stability.

MSC

•  34C25
•  92D25
•  34D20
•  34D40

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