Stability For The Predator-prey System In Patches Environment With Stage Structure

The delayed Lotka-Volterra type predator-prey models between two species with stage structure in patchy environment are considered and their dynamic behavior is investigated in this thesis,the study for them are great theoretical and practical significance.First,suppose that the species can live in two-patch environments,and 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,and capture prey with higher level of Prey in the patch.Under the hypotheses above,a delayed Holling-2 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,and by constructing Lyapunov functions,sufficient condition is obtained for the global stability of a positive equilibrium of this model.Numerical simulations are given to illustrate theoretical results.Next,suppose that the species can live in two-patch environments,and the predator 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 predators can not capture the preys and do not have the ability to reproduce.For preys,it is supposed that they can disperse between two patches:one patch with a low level of population density and one patch with a higher level of population density.Under the hypotheses above,first, a delayed Leslie-Gower predator-prey model with constant coefficients and with stage structure for predator and dispersal for prey is studied.Finally,we consider the same conditions,in which mature predators can capture prey in both patch. Similar to the preceding process,the uniform persistence of the solution for this system with initial conditions is established using the comparison principle,they by constructing Lyapunov functions,sufficient condition is obtained for the global stability of a positive equilibrium of this model.Numerical simulations are given to illustrate theoretical results.