First, a discrete stage-structured and harvested predator-prey model is established, which is based on a predator-prey model with Type III functional response. Then the~ oretical methods are used to investigate existe...First, a discrete stage-structured and harvested predator-prey model is established, which is based on a predator-prey model with Type III functional response. Then the~ oretical methods are used to investigate existence of equilibria and their local proper- ties. Third, it is shown that the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of R~_, by using the normal form of discrete systems, the center manifold theorem and the bifurcation theory, as varying the model parameters in some range. In particular, the direction and the stability of the flip bifurcation and the Neimark -Sacker bifurcation are showed. Finally, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the com- plex dynamical behaviors, such as cascades of period-doubling bifurcation and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm fur- ther the complexity of the dynamical behaviors. In addition, we show also the stabilizing effect of the harvesting by using numerical simulations.展开更多
文摘First, a discrete stage-structured and harvested predator-prey model is established, which is based on a predator-prey model with Type III functional response. Then the~ oretical methods are used to investigate existence of equilibria and their local proper- ties. Third, it is shown that the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of R~_, by using the normal form of discrete systems, the center manifold theorem and the bifurcation theory, as varying the model parameters in some range. In particular, the direction and the stability of the flip bifurcation and the Neimark -Sacker bifurcation are showed. Finally, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the com- plex dynamical behaviors, such as cascades of period-doubling bifurcation and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm fur- ther the complexity of the dynamical behaviors. In addition, we show also the stabilizing effect of the harvesting by using numerical simulations.
