Dynamics of Allee and Refuge Effects in Prey–Predator System With Holling's Type III Functional Response

This study explores the dynamics of a prey–predator system incorporating both the Allee and refuge effects in the prey population. A deterministic mathematical model is formulated to capture these ecological phenomena, in which both analytical and numerical properties are investigated. The study establishes the existence of three biologically relevant equilibria, namely, extinction, prey only, and coexistence. Moreover, the conditions for their local and global stability were derived. A Monte Carlo simulations approach is applied to account for parameter uncertainty and random fluctuations, revealing key numerical behaviors. The results demonstrate that the system may exhibit stable coexistence, oscillatory dynamics, or limit cycles depending on the parameter values, highlighting the interplay between Allee and refuge effects. Low Allee thresholds enhance population resilience, while high thresholds increase vulnerability to collapse. These results provide insights into the stability and long-term dynamics of prey–predator interactions and highlight the importance of considering Allee and refuge effects in ecological management.