A Holling type commensal symbiosis model involving Allee effect

Abstract

A two species commensal symbiosis model with Holling type functional response and Allee effect on the second species takes the form

$$

\di\frac{dx}{dt}&=&x\Big(a_1-b_1x+\di\frac{c_1y^p}{1+y^p}\Big),

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

$$

is investigated, where $a_i, b_i, i=1,2$ $p$, $u$ and $c_1$ are all positive constants, $p\geq 1$. Local and global stability property of the equilibria is investigated.Our study indicates that the unique positive equilibrium is globally stable and the system always permanent, consequently, Allee effect has no influence on the final density of the species. However, numeric simulations show that the stronger the Allee effect, the longer the for the system to reach its stable steady-state solution.