Global Dynamics and Bifurcation of an Evolutionary Beverton-Holt Model with the Allee Effect

We study the global dynamics and bifurcation structure of an evolutionary Beverton–Holt model with Allee effects, a framework that couples ecological constraints with adaptive trait evolution. The model accounts for density dependence, mate limitation, and predator saturation, while traits evolve according to selection gradients that influence reproduction and competition. From an ecological perspective, we show that weak Allee effects create bistability between extinction and survival, while strong Allee effects generate a critical threshold below which populations collapse and above which they persist at carrying capacity. Evolutionary feedback further reshapes these outcomes by shifting thresholds, modifying stability regions, and producing multiple long-term attractors. Biologically, this reveals how demographic pressures such as scarce mates or high predation interact with trait evolution to determine persistence or extinction, and how adaptive responses may rescue populations facing critical density barriers. Our rigorous analysis and simulations demonstrate that eco–evolutionary processes not only alter classical Beverton–Holt outcomes but also provide insight into mechanisms underlying species persistence, extinction risk, and invasion success under Allee constraints.