A Discrete Predator-Prey Model with Cannibalism, Refuge, and Memory Effect: Implementation of Piecewise Constant Argument (PWCA) Method

Predator-prey models are essential for understanding ecological dynamics, and fractional-order models provide a more realistic approach by considering memory effects. This study aims to analyze the discrete dynamics of a predator-prey model, incorporating predator cannibalism, refuge, and memory effects with a Caputo-type fractional-order. The Piecewise Constant Argument (PWCA) method was employed for discretization, followed by an analysis of the equilibrium points and their stability. Four equilibrium points were identified: the origin, prey extinction, predator extinction, and coexistence. It was found that the origin point was unstable, while the prey extinction, predator extinction, and coexistence points were conditionally locally asymptotically stable, depending on the parameter values. The order of the fractional derivative and step size significantly influenced the stability of these equilibrium points. Numerical simulations confirmed the theoretical findings, showing how parameter variations affect system behavior.