Hopf bifurcation and Turing instability in a modified Leslie-Gower prey-predator model

In this paper, we study a modified Leslie-Gower prey-predator model in the presence of nonlinear harvesting in prey subject to the Neumann boundary condition. Our results reveal the conditions on the parameters so that the periodic solution exist surrounding the interior equilibrium. Furthermore, the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are investigated. For the model with the Neumann boundary condition, Turing instability of the interior equilibrium solution is studied. In particular, Turing instability region regarding the parameters is established. Numerical simulations are carried out to demonstrate the results obtained.