A commensal symbiosis model with Holling type functional response


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. Xiaoyan Zhou - Public Foundation Department, Fuzhou Polytechnic, Fuzhou, Fujian, 352300, P. R. China.


Abstract

A two species commensal symbiosis model with Holling type functional response takes the form \[\frac{dx}{dt}=x\left(a_1-b_1x+\frac{c_1y^p}{1+y^p}\right),\] \[\frac{dy}{dt}=y\left(a_2-b_2y\right)\] is investigated, where \(a_i, b_i, i = 1, 2, p\) and \(c_1\) are all positive constants, \(p \geq 1\). Local and global stability property of the equilibria is investigated. We also show that depending on the ratio of \(\frac{a_2}{b_2}\), the first component of the positive equilibrium \(x^*(p)\) may be the increasing or decreasing function of \(p\) or independent of \(p\). Our study indicates that the unique positive equilibrium is globally stable and the system always permanent.


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