Dynamical behavior of a harvested prey-predator model with stage structure and discrete time delay

A prey-predator model with stage structure for prey and selective harvest effort on predator is proposed, in which gestation delay is considered and taxation is used as a control instrument to protect the population from overexploitation. It is established that when the discrete time delay is zero, the model system is stable around the interior equilibrium and an optimal harvesting policy is discussed with the help of Pontryagin's maximum principle; On the other hand, stability switch of the model system due to the variation of discrete time delay is also studied, which reveals that the discrete time delay has a destabilizing effect. As the discrete time delay increases through a certain threshold, a phenomenon of Hopf bifurcation occurs and a limit cycle corresponding to the periodic solution of model system is also observed. Numerical simulations are carried out to show the consistency with theoretical analysis.