Limit Cycles in a Reduced Adaptive Predator–Prey Model (AV-PP) with Nonlinear Functional Responses

Abstract

This study investigates the emergence of limit cycles in adaptive predator–prey models by combining rigorous model reduction techniques with nonlinear functional responses. The analysis begins with a three-dimensional adaptive vulnerability predator–prey (AV-PP) model, in which prey susceptibility is treated as a dynamic variable to enhance biological realism. Due to the analytical complexity of the system, it is reduced to a two-dimensional planar model using a quasi-steady-state approximation (QSSA) supported by singular perturbation theory. This reduction is theoretically justified and preserves the essential qualitative dynamics of the original system. The reduced model with a linear interaction term is first examined, and the results show convergence toward a stable equilibrium without oscillatory behavior, indicating the absence of limit cycles. This highlights the limitation of linear formulations in representing realistic ecological dynamics. To address this, nonlinear functional responses are introduced. The incorporation of a Holling Type II functional response leads to equilibrium destabilization under specific parameter conditions, resulting in the emergence of a stable limit cycle. Further extension using a Holling Type III response enhances the robustness and persistence of oscillations by capturing ecological mechanisms such as predator learning and prey refuge effects. The qualitative analysis is carried out using the Poincaré–Bendixson theorem and the Elemental Limit Cycle (ELC) method. The results demonstrate that while the reduction simplifies the system without altering its fundamental behavior, the introduction of nonlinear functional responses is the key mechanism driving the transition from stable equilibrium to sustained oscillatory dynamics.