Nonlinear Dynamics of Allee Effect and Fear in a Delayed Diffusive Predator-Prey Model

Abstract

This paper investigates the dynamics of a predator-prey model incorporating both the Allee effect and predator-induced fear, alongside a delay representing the time required for prey to develop anti-predation defenses. The model also integrates diffusion terms to account for species movement and includes harvesting pressure on both populations. We first establish the existence and local stability of a coexistence equilibrium, then analyze the conditions under which a delay-induced Hopf bifurcation occurs. Using center manifold theory and normal form analysis, we characterize the direction, stability, and periodicity of the bifurcating solutions. Numerical simulations are performed to validate the theoretical predictions and reveal rich dynamics, including transitions from stability to periodic oscillations and the emergence of spatial patterns due to asymmetric diffusion. These results underscore the critical role of delayed behavioral responses and spatial heterogeneity in shaping ecological stability.