Local Stability Analysis On Lotka-Volterra Predator-Prey Models With Prey Refuge And Harvesting

Abstract

We propose a predator-prey model by incorporating a constant harvesting rate into a Lotka-Volterra predator-prey model with prey refuge which was studied recently. All the positive equilibria and the local stability of the proposed model are studied and analyzed by sorting out the intervals of the parameters involved in the model. These intervals of the parameters exhibit the effects on the dynamical behaviors of prey and predators. The emphasis is put on the ranges of the prey refuge constant and harvesting rate. We show that the model has two positive boundary equilibria and one equilibrium. By using the qualitative theory for planar systems, we show that the two positive boundary equilibria can be saddles, saddle-nodes, topological saddles or stable or unstable nodes, and the interior positive equilibrium is locally asymptotically stable. Under suitable restrictions on the parameters, we prove that the positive interior equilibrium is a stable node.