Dynamic Behavior Of The Predator-Prey Systems In Patchy Environment

The delayed Lotka-Volterra type predator-prey models between two species with stage structure for prey in patchy environment are considered and their dynamic behavior is investigated in this thesis. In the models, the species can live in two-patch environments. It is assumed that the prey species is divided into two stage, immature and mature, and confined to each of the patches but can not disperse between patches;in the mean time, it is always assumed that the immature preys can not be captured by predators and do not have the ability to reproduce. For predators, it is supposed that they can disperse between two patches: one patch with a low level of prey and one patch with a higher level of prey, both with predation.Under the hypotheses above, first, a delayed predator-prey model with constant coefficients and with stage structure for prey and dispersal for predator is studied. Based on the positivity and the boundedness of the solution for the model, the uniform persistence of the solution for this system with initial conditions is proved using the comparison principle, then the criterion for the local stability of the nonnegative equilibria is derived, and by constructing Lyapunov functions, sufficient condition is obtained for the global asymptotic stability of a positive equilibrium of this model.Next, a delayed ratio-dependent predator-prey model with constant coefficients and with stage structure for prey and dispersal for predator is discussed. Similar to the preceding process, the uniform persistence of the solution for this system with initial conditions is established using the comparison principle, then by constructing Lyapunov functions, sufficient condition is given for the global asymptotic stability of a positive equilibrium of this model.At last, a delayed ratio-dependent predator-prey model with periodic coefficients and with stage structure for prey and dispersal for predator is considered. On the basis of the uniform persistence of the solution for this system with initial conditions, using Gaines and Mawhin's coincidence degree theory, sufficient condition for the existence of positive periodic solution of the model is obtained.