Dynamics in a diffusive modified Leslie-Gower predator-prey model with time delay and prey harvesting

The dynamics of a diffusive predator-prey model with time delay and Michaelis-Menten-type harvesting subject to Neumann boundary condition is considered. Turing instability and Hopf bifurcation at positive equilibrium for the system without delay are investigated. Time delay-induced instability and Hopf bifurcation are also discussed. By the theory of normal form and center manifold, conditions for determining the bifurcation direction and the stability of bifurcating periodic solution are derived. Some numerical simulations are carried out for illustrating the theoretical results.