A predator–prey model with prey refuge: under a stochastic and deterministic environment

This study aims to thoroughly investigate the dynamics of a predator–prey model with a Beddington-De Angelis functional response. We assume that the prey refuge is proportional to both species. We establish the standard properties of boundedness, permanence, and local stability. We show that under certain parameter conditions, transcritical bifurcation and Hopf bifurcation occur. To understand the nature of the limit cycle, we determine the direction of the Hopf bifurcation. We focus on the significant ranges of the predators’ prey capturing rate and examine how the level of prey fear and the predator’s mutual interference affect the system’s stability. Through numerical analysis, we study the behavior of the Lyapunov exponent and observe multiple self-repeating shrimp-shaped patterns that indicate periodic attractors in discrete-time predator–prey system. These structures appear across a broad region associated with chaotic dynamics. Additionally, if the intensity of white noise is kept below a specific threshold, the deterministic control approach is equally effective in environmental fluctuation. Numerical simulations support these findings.