Bifurcation Structure and Chaos Control in a Discrete-Time Fractional Predator–Prey Model with Double Allee Effect

This paper investigates a discrete-time fractional-order predator–prey model incorporating a double Allee effect in the prey population, derived from a fractional differential system via the piecewise constant argument method to capture both memory effects and density-dependent constraints. We establish the existence and local stability of all biologically meaningful equilibria and show that the interaction between fractional memory and the double Allee threshold significantly influences the stability of the coexistence state. Through the integration of linear stability analysis and center manifold reduction, we are able to obtain explicit conditions for Neimark–Sacker and period-doubling bifurcations. The system exhibits rich dynamics, including periodic oscillations, quasi-periodicity, and chaos. The double Allee effect plays a key role in shaping system stability. To suppress instability and chaotic behavior, feedback and hybrid control strategies are applied and shown to be effective. Numerical simulations are given to confirm the results obtained by the theoretical analysis and to show the transitions among different dynamical states, in which the fractional-order memory and multiple Allee effects play important roles.