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Optimal harvesting and stability analysis in a Leslie-Gower delayed predator-prey model

A delayed Leslie-Gower predator-prey model with continuous threshold prey harvesting is studied. Existence and local stability of the positive equilibrium of the system with or without delay are completely determined in the parameter plane. Considering delay as parameter, we investigate the effect of delay on stability of the coexisting equilibrium. It is observed that there are stability switches and a Hopf bifurcation occurs when the delay crosses some critical values. Employing the normal form theory, the direction and stability of the Hopf bifurcations are explicitly determined by the parameters of the system. Optimal harvesting is also investigated and some numerical simulations are given to support and extend our theoretical results.