Dynamics of a Predator‐Prey System With Fear Effect and Prey‐Dependent Predator Migrations in a Two‐Patch System

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ABSTRACT This work presents the formulation and analysis of a predator‐prey model based on a modified Leslie‐Gower predation approach in a heterogeneous habitat. The proposed model is studied in the presence of a fear effect induced by the predator's presence, which can lead to significant reductions in the prey's reproduction rate. The system is assumed to exist in a patchy environment consisting of two distinct patches (patch ). The prey migrates between these patches in search of food, while the predator moves between patches in pursuit of prey. The movement of both prey and predator is assumed to be fast compared to their growth and interactions. To capture the system's dynamics, slow and fast time scales are introduced. An aggregated model for the prey and predator populations is derived. The positive invariance and boundedness of all feasible solutions in the aggregated system are established. The necessary conditions for the feasibility and stability of various fixed points in the aggregated model are determined. Furthermore, the existence of periodic solutions is numerically verified through Hopf bifurcation and Hopf‐Hopf (double‐Hopf) bifurcation around coexistence points. The existence of a branch point (transcritical bifurcation) is also analyzed by satisfying all the conditions of Sotomayor's theorem. Additionally, a two‐parameter bifurcation analysis is conducted, including generalized Hopf bifurcation and Bogdanov‐Takens bifurcation. Several numerical simulations, performed using MATLAB and MATCONT, illustrate and validate the analytical results.

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