Computational Modeling and Stochastic Analysis of a Three-Species Prey-Predator System Incorporating Beddington-DeAngelis Functional Response and Refuge Effect

Background: We propose and analyze a three-species prey-predator system incorporating Lotka-Volterra competition between two prey species. The model captures a symbiotic interaction between species x and the predator z, while predation on species y follows a Beddington-DeAngelis functional response. To enhance ecological realism, a constant proportion of species y is assumed to occupy a refuge, reducing predation pressure. The study aims to investigate the qualitative and quantitative dynamics of the system under both deterministic and stochastic environmental conditions. Results: The well-posedness of the system is established by proving positivity and boundedness of solutions. Multiple biologically feasible equilibria are identified, and their local stability properties are rigorously examined. The system undergoes critical dynamical transitions, including transcritical bifurcation and Hopf bifurcation, driven by key competition parameters. Numerical investigations reveal rich dynamical behaviors, including the emergence of stable and unstable limit cycle oscillations. Furthermore, a stochastic extension of the model is formulated to incorporate environmental variability, and sufficient conditions ensuring existence, uniqueness, and exponential stability of solutions are derived. These analytical findings are substantiated through comprehensive numerical simulations. Conclusions: This study highlights the intricate interplay between competition, predation, refuge mechanisms, and environmental fluctuations in shaping ecosystem dynamics. The results demonstrate that refuge and stochastic perturbations can significantly influence system stability and persistence. The proposed framework provides deeper theoretical insights into ecological interactions and may inform practical strategies for ecosystem management, species conservation, and biodiversity preservation.