Fractional-Order Predator–Prey Dynamics: Stability and Bifurcation Analysis

Abstract

In this paper, we investigate the dynamical behavior of a fractional-order prey–predator model incorporating the effect of hunting cooperation and logistic growth of prey. We have analyzed the existence and stability of the equilibrium points and determined the conditions for local bifurcations, including saddle-node, transcritical, and Hopf bifurcations. It is shown that the fractional order plays a crucial role in governing system stability: smaller fractional orders introduce stronger memory effects that enhance damping and stabilize the coexistence equilibrium, while larger orders weaken these effects and lead to oscillatory or unstable dynamics. Numerical simulations are performed to validate the theoretical results and to illustrate the influence of fractional order and hunting cooperation on the system’s long-term behavior. The study highlights that fractional-order modeling provides a more realistic framework for understanding complex ecological interactions, revealing that cooperative hunting and memory effects can jointly regulate stability, coexistence, and oscillatory population patterns.