A Mathematical Study of Wild Boar–Ranger Interaction Dynamics

Wild boars are among the most widely distributed ungulate species worldwide and play a significant role in maintaining ecological balance within forest ecosystems. Nevertheless, uncontrolled population growth leads to severe agricultural damage and increased traffic accidents, making population regulation necessary. At the same time, wild boars represent an important economic resource and, in some regions, a protected species, requiring management strategies that balance control and conservation.

In this work, we present a mathematical model describing the dynamics of wild boar populations at forest boundaries, with particular emphasis on spillover phenomena into surrounding human-dominated areas. Rather than modeling the entire population, we focus on individuals living near the forest edge, as these are primarily responsible for spillover events. The model incorporates ranger intervention as a control mechanism activated when spillovers occur, representing hunting or containment actions aimed at regulating population size without driving the species toward extinction.

The qualitative analysis of the model reveals the existence of biologically feasible equilibrium points, including a coexistence equilibrium and a ranger-free equilibrium. Using bifurcation theory, we demonstrate the presence of a transcritical bifurcation that governs the stability exchange between these equilibria. Furthermore, under certain parameter conditions, the system undergoes a Hopf bifurcation, leading to the emergence of stable periodic solutions. These oscillations correspond to recurrent spillover events that necessitate periodic deployment of rangers, resulting in increased management costs.

Numerical simulations are provided to support the analytical results and to illustrate the long-term behavior of the system under different management strategies. In particular, the proposed model provides useful insights into sustainable wild boar management and highlights the effectiveness of resource-based control strategies in stabilizing population dynamics.