The Effects of Harvesting and Time Delay on Predator-prey Systems with Holling Type II Functional Response

In this paper, the effects of harvesting and time delay on two different types of predator-prey systems with delayed predator specific growth and Holling type II functional response are studied by applying the normal form theory of retarded functional differential equations developed by Faria and Magalhaes [J. Differential Equations, 122 (1995), pp. 181–200, J. Differential Equations, 122 (1995), pp. 201–224]. Hopf bifurcations are demonstrated in models with harvesting of the prey at a constant rate by taking the delay as a bifurcation parameter, and numerical examples supporting our theoretical prediction are also given. Furthermore, bifurcation analysis indicates that delayed predator-prey systems with predator harvesting exhibit Bogdanov–Takens bifurcation. The versal unfoldings of the models at the Bogdanov–Takens singularity are obtained, and numerical simulations and bifurcation diagrams are given to illustrate the obtained results.

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