A Fractal-Fractional Approach to Non-linear Predator-Prey Models with Logistic Growth, Holling Type II Functional Response and Immigration Effects

Abstract

This paper develops a nonlinear fractal-fractional predator-prey model that incorporates logistic prey growth and immigration effects. The predator-prey interaction is characterized by a Holling type II functional response, capturing the saturation phenomenon in the predator's feeding rate. Using the Caputo-Fabrizio (CF) fractional operator, the model integrates memory effects into the population dynamics. Theoretical investigations establish the existence and uniqueness of solutions by applying Krasnoselskii's fixed point theorem and Banach's contraction principle, followed by the stability analysis of equilibrium points. For the numerical approximation, a modified Adams-Bashforth method adapted to the CF operator is employed. Simulation results reveal that small yet positive immigration rates promote asymptotically stable coexistence between prey and predator populations, emphasizing the stabilizing influence of immigration on ecosystem dynamics. The study demonstrates how fractal-fractional calculus can provide deeper insight into ecological stability and long-term behavior of interacting species.