Predator-prey interactions with harvesting of predator with prey in refuge

Abstract

In this paper, we study predator prey interactions where the predator is exposed to the risk of disease and harvesting while the prey has the ability to use a refuge. We consider two models: the ability of prey to use a constant refuge and the ability to use random refuge. We found bounded, non-periodic solutions and the equilibrium points for both models. We then show the role of the refuges in the stability of the systems. The equilibrium was stable locally, but not globally, and we found some basin to these equilibrium points. Numerical simulations show several types of oscillations that occur due to the kinds of refuges and prey's ability to use these refuges. For both models, there exist an invariant region; the invariant region in the constant refuge is better than the invariant region in random refuge because it ensures the continuity of all populations and sustainability of the harvested species and controlling the disease without it becoming endemic. Finally the low density prey in refuges makes a limit cycle around the equilibrium in refuges whiles the small density is stable.