Refuge, Competition, and Diffusion: Pathways to Spatiotemporal Complexity in a Predator–Prey Model

The interaction between predators and prey is a fundamental aspect of ecology. As a result, studying predator–prey interactions has become more vital to understanding their habitat. Prey populations employ various mechanisms to avoid predation in predator–prey relationships. In this paper, we proposed and analyzed a nonlinear predator–prey interaction model in which the prey population seeks refuge, and predators exhibit interspecific competition. During the study, we explored the interplay between temporal bifurcation behavior and diffusion-driven pattern formation. We first analyzed the model’s temporal behavior within an ODE framework. We discussed positivity and boundedness, then analyzed the local stability of the existing equilibrium points. The local bifurcation analysis has been carried out for the bifurcation parameters prey refuge [Formula: see text] and conversion efficiency [Formula: see text]. The analysis demonstrated the existence of critical thresholds and illustrated how qualitative changes occur within the system. To explore the impact of random movement in an isolated area, the proposed model was extended to a reaction–diffusion model. A linear stability analysis was performed to determine the analytical conditions for Turing instability. By utilizing the prey diffusion coefficient as the spatial bifurcation parameter, we conducted a multiple-scale analysis and developed the amplitude equation. We validated the analytical findings with the MATLAB software MATCONT package to demonstrate several situations that arise when we alter the prey refuge and conversion efficiency. Numerical simulations demonstrated the formation of stationary spatial patterns on the surface of a sphere, such as spots, stripes, and mixed. These patterns resulted from nonlinear interactions between reaction and diffusion. The findings indicated that temporal stability in the nonspatial system does not guarantee spatial uniformity, as diffusion can cause instability and lead to the emergence of complex spatial structures. The research outcomes emphasize the importance of prey refuge, conversion efficiency, and random movement in system dynamics.