Impact of physical and behavioral prey refuge on the stability and bifurcation of Gause type Filippov prey-predator system

Abstract

After the pioneering theoretical studies of Lotka and Volterra, Gause and his co-workers replace the previously used linear functional response by using a saturating functional response with a discontinuity at a threshold prey density. Here we assume that prey density at below this threshold value is effectively and successfully in a refuge patch. In this situation there is no food option for predator and go to extinction. But above this threshold value surplus density of prey is available to predator for its diet. But the system does not show any future activities when prey density is in the vicinity of the threshold density, system is ill posed because the trajectories are not well defined here. In the present study, we redefine and analyze the model by using Filippov regularization method. By this continuation method, the system becomes well posed and gives more results as predicted by Gause. Also predator fully depends upon alternative diet to survive from extinction risk when prey is in refuge patch and system largely varies with the availability of alternative diet resource but in the later case predator again switches to its primary (essential) food. When prey density is in the vicinity of the threshold density, then predator may choose its deit preferentially from essential or alternative resources according to its profit. Numerical examples support these hypothesis and analytical results.