A model for the interactions of wild boars and park rangers

Boars, being one of the most widely spread ungulates worldwide, have a widely recognized important role in the balance of natural environment and forests. Since large boar populations severely damage crops and cause serious traffic accidents, they are widely hunted, thereby also representing a relevant economic resource. In the model presented here, the species is at times considered ravaging, enabling it to be kept in check, while on the other hand, it must be preserved from extinction as a protected species. We considered an idealized, relatively simple situation in which rangers of the park where the boars are hosted manage this animal population size when they extrude into the surrounding areas through the woods perimeter. Modeling this situation involves considering not the whole boar population, but only those that are involved in the spillover, i.e., those living in proximity of the woods edge. The theoretical investigation and the simulations revealed the existence of a transcritical bifurcation relating the two viable equilibria, coexistence, and the ranger-free point. Also, the possible onset of persistent oscillations via a Hopf bifurcation is shown, leading to periodic recalling of rangers to contain the spillovers. On the other hand, a better regime was obtained by reducing the environment's resources for the wild boars, which stabilized the the boar population at constant level, with a reduced presence of the rangers, reducing the costs of their periodic recalling.