Dynamic behaviors of a Holling type commensal symbiosis model with the first species subject to Allee effect

Abstract

A two species commensal symbiosis model with Holling type functional response and the first species subject to Allee effect takes the form

$$

\di\frac{dx}{dt}&=&x\Big(a_1-b_1x\Big)\di\frac{x}{\beta +x}+\di\frac{c_1xy^p}{1+y^p},

\di\frac{dy}{dt}&=&y(a_2-b_2y)

$$

is investigated, where $a_i, b_i, i=1,2$ $p$, $\beta$ and $c_1$ are all positive constants, $p\geq 1$ is a positive integer. Local and global stability property of the equilibria are investigated. Our study indicates that the unique positive equilibrium is globally stable. Also, the final density of the first species is increasing with the Allee effect, this result differs from that obtained for the Holling type commensal symbiosis model with the second species subject to Allee effect.