Fear and its carry over effects in a delay-induced predator-prey model with additional food to predator

Abstract

Interaction between prey and predator is a natural phenomenon in ecology that significantly contributes to the structure of ecological variety. Recent studies indicate that the presence of predator can influence the physiological behaviour of prey species to such an extent that it can be more efficient than direct predation in decreasing the prey biomass. Moreover, such non-lethal effects can be carried over through seasons or generations. In this present article, we analyze the impact of predator-induced fear and its carry-over effect in a predator-prey model in which the predator species can access some alternative or additional food sources. Well-posedness of the system and some basic dynamical properties such as extinction criteria, stability analysis with global stability, uniform persistence etc. are discussed thoroughly. From the bifurcation analyses, we can observe that fear and its carry-over effect can not switch the stability from one equilibrium state to other equilibrium state. However once the coexistence equilibrium state occurs in the system, a higher levels of fear can stabilize it. On the other hand, higher levels of carry-over effect promote the oscillatory dynamics around the coexistence state. Therefore, fear and its carry-over effects have two opposite roles in the stability of the coexistence equilibrium. Predator species may go extinct if the effective quantity of additional food is sufficiently low. Next we study the model system in presence of gestation delay and observe some interesting dynamics by taking the delay parameter as a bifurcation parameter. Our study demonstrates how non-lethal effects alter the dynamics of a prey-predator model and provides valuable biological insights, particularly into the dynamics of small food web.