TWO-FOLD IMPACTS OF FEAR IN A SEASONALLY FORCED PREDATOR–PREY SYSTEM WITH COSNER FUNCTIONAL RESPONSE

Abstract

In this paper, we investigate the dynamics of a predator–prey system of an ecological community in which the fear instigated by the predators has an adverse effect on the reproduction rate of prey species, and also on the competition among themselves due to the limited environmental resources. To capture and handle the realistic scenario in a more meaningful way, we have mathematically built up the model system with the assumption that the predators predate on the prey items following Cosner functional response, which increases with increments in the prey and predator populations. The model system has been studied through noteworthy mathematical analysis and an extensive numerical simulation. Our simulation results demonstrate that the predator–prey system stabilizes due to predator’s induced fear suppressing/enhancing the birth/death of prey species. The competition among the predators for the available prey items also has a stabilizing role on the system’s dynamics. In contrast, the increasing growth rate of prey species or predation rate creates instability in the system by changing the stable phase to the limit cycle oscillations. Moreover, the effects of seasonality are also studied by extending the model system to its nonautonomous counterpart. Sufficient conditions are derived so that the seasonally driven system exhibits at least one positive periodic solution. Our numerical results show that the seasonally forced system exhibits periodic solution (globally attractive periodic solution), higher periodic solutions, bursting patterns and the extinction of predator species due to the seasonal variations of some parameters.