Mathematical Analysis of Discrete Fractional Prey-Predator Model with Fear Effect and Square Root Functional Response

Abstract

This paper investigates the dynamics of a discrete fractional prey-predator system. The prey-predator interaction is modelled using the square root functional response, which appropriately models systems in which the prey exhibits a strong herd structure, implying that the predator generally interacts with the prey along the herd's outer corridor. Some recent field experiments and studies show that predators affect prey by directly killing and inducing fear in prey, reducing prey species' reproduction rate. Considering these facts, we propose a mathematical model to study herd behaviour and fear effect in the prey-predator system. We show algebraically equilibrium points and their stability condition. Condition for Neimark-Sacker bifurcation, Flip bifurcation and Fold bifurcation are given. Phase portraits and bifurcation diagrams are portraits that depict the model's behaviour based on some hypothetical data. Numerical simulations reveal the model's rich dynamics as a result of fear and fractional order.