Allee effects in a predator--prey system with a saturated recovery function and harvesting

Abstract

In this paper, we will consider the Allee effects on predator--prey system with a saturated recovery function and harvesting. Local stability analysis of biologically feasible equilibrium points is worked out with help of ecological as well as disease basic reproduction numbers. We proved that the equilibrium $P_0=(0,0)$ of the predator--prey system  is (i) a saddle point in weak Allee effects (WAE) and (ii) asymptotically stable in strong Allee effects (SAE). We proved that the equilibrium $P_1=(\beta,0)$ of the system is a saddle point if $R_0(1)<1$ and unstable if $R_0(1)>1$ in SAE case. Also we proved that the equilibrium $P_2=(1,0)$ of the system  is a saddle point if $R_0(1)>1$ and asymptotically stability if $R_0(1)<1$ in SAE case. It is shown that the coexistence equilibria is  not asymptotically stable. Numerical simulations are carried out for a hypothetical set of parameter values to substantiate our
analytical findings.