Analysis of a Fractional Predator–Prey System Incorporating Strong Allee Effects and Nonlinear Defensive Feedback
Keywords:
Fractional calculus; Predator–prey model; Allee effect; Anti predator feedback; Stability analysisAbstract
This paper presents a fractional-order predator–prey model incorporating a strong Allee effect in the prey population and a nonlinear anti-predator feedback mechanism in the predator population. The model is formulated using the Caputo fractional derivative of order α (0 < α ⩽ 1), allowing the inclusion of memory and hereditary effects that are frequently observed in ecological systems. The prey population is assumed to experience a strong Allee effect, while predator dynamics are regulated through a density-dependent feedback response that reflects behavioural adaptation and defensive interactions. Fundamental mathematical proper ties of the model, including existence, uniqueness, positivity, and boundedness of solutions, are established to ensure biological feasibility. Equilibrium points of the system are determined, and their local stability is investigated through Jacobian linearization and eigenvalue analysis. For the fractional-order model, stabil ity conditions are examined using the fractional eigenvalue criterion, highlighting the influence of memory effects on the asymptotic behaviour of the system. To illustrate the theoretical findings, extensive numer ical simulations are performed using the fractional Adams–Bashforth–Moulton predictor–corrector scheme. Phase-plane analysis, equilibrium investigations, and two-parameter stability maps are employed to examine the influence of the carrying capacity, Allee threshold, interaction strength, feedback intensity, and fractional order on the system dynamics. The results demonstrate that increasing the carrying capacity enlarges the biologically feasible region and promotes persistence, whereas stronger Allee effects reduce the coexistence domain and increase the risk of extinction. Furthermore, the nonlinear anti-predator feedback significantly modifies predator survival and stability characteristics. A comparative study between the classical integer order model (α = 1) and the fractional-order formulation (0 < α < 1) reveals that ecological memory plays a stabilizing role. Fractional dynamics generate smoother trajectories, reduced oscillatory behaviour, and slower convergence toward equilibrium states when compared with the corresponding classical system. These memory-induced effects moderate abrupt population fluctuations and promote more realistic ecolog ical responses. Overall, the proposed model provides a flexible framework for understanding the combined roles of memory, Allee effects, and behavioural feedback in predator–prey interactions, offering new insights into the stability and persistence of ecological systems..
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