A Fractional Lotka–Volterra Predator–Prey Model with Strong Allee Effects, Nonlinear Harvesting, and Memory

This work develops a fractional Lotka–Volterra predator–prey model incorporating a strong Allee effect, nonlinear anthropogenic harvesting, and memory-dependent dynamics through the Atangana–Baleanu–Caputo (ABC) fractional operator. The model describes the interactions between two prey populations and a predator under competition, predation, and nonlinear harvesting. Theoretical analysis establishes the existence and uniqueness, non-negativity, ultimate boundedness, and Ulam–Hyers stability of the solutions. Numerical simulations examine the effects of the fractional order ϖ, the characteristic time-scale τ, the Allee-effect parameters, and harvesting effort. The results show that the fractional order modifies transient population dynamics, while τ regulates the fractional response through the factor τ1−ϖ. A systematic analysis of the Allee threshold C1 reveals a sharp numerical persistence and extinction transition for the stronger prey, bracketed within 12.448<C1,tr<12.449 for the prescribed parameter set and initial conditions, while the Allee severity parameter influences its temporal recovery. Increasing harvesting effort suppresses the prey populations and consequently affects the predator through trophic coupling, with fractional memory modifying the transient response. The proposed framework provides a flexible approach for investigating predator–prey dynamics under Allee effects, anthropogenic exploitation, and memory-dependent processes.