Dynamics and Bifurcation Analysis of a Discrete Predator–Prey Model with Dual Allee Effects

This study investigates a discrete-time predator–prey model incorporating Allee effects in both species. The prey follows logistic growth with a prey-specific Allee threshold, while the predator grows through prey consumption but suffers its own Allee effect at low densities. The model is formulated as a nonlinear system of difference equations, capturing key ecological interactions and intra-species constraints. A detailed mathematical analysis explores the local stability of equilibria and the emergence of flip and Neimark–Sacker bifurcations, leading to periodic and quasi-periodic dynamics. Special cases with vanishing Allee constants reveal critical impacts on species persistence and extinction. Numerical simulations support the analytical results and illustrate diverse regimes, from stability to complex fractal trajectories. Furthermore, a chaos control strategy based on bifurcation analysis is proposed to stabilize the system dynamics near desired states. The findings highlight the intricate interplay between predation and dual Allee effects in shaping long-term ecological outcomes. In addition, our analysis is supported by several foundational works on nonlinear dynamics, bifurcation theory, and discrete-time predator–prey modeling, which provide important theoretical tools for understanding complex ecological systems [ Gakkhar & Singh , 2012 ; Jing & Yang , 2006 ; Kuznetsov , 1997 ; Liu & Xiao , 2007 ; MacDonald , 2008 ; Wan , 1978 ; Wiggins , 2003 ; Rocha & Taha , 2023 ; Singh et al. , 2019 ; Hassan & Goldar , 2023 ; Goldar , 2025 ; Goldar et al. , 2025b ].