Persistence and extinction in an Elk-Wolf prey-predator system with refuge and inter-regional movement

Data-driven predator-prey modeling remains a challenging problem in ecological dynamics. In this study, we develop a prey-predator model to examine the interactions between elk and wolves in Banff National Park, explicitly incorporating prey refuge, inter-regional movement, and predation pressure. The system consists of two prey populations-Banff townsite elk and Bow Valley elk and wolves as the predator population. Through linear stability analysis, we derive the critical threshold of the Bow Valley elk production rate (β) and determine both the existence and direction of Hopf bifurcations. Furthermore, global stability of the equilibrium point is established using a Lyapunov function approach. The model is parameterized with empirical population data to ensure biological realism. Through sensitivity analysis, we found β as the most influential parameter affecting the populations. Our analysis reveals that increasing β leads to the extinction of the Banff elk population after transient oscillations, while the Bow Valley elk and wolf populations undergo a transition from stable equilibria to sustained limit cycle oscillations. These findings highlight the role of prey refuge in shaping elk-wolf coexistence and provide new insights into population persistence under ecological constraints.