Asymptotic Behavior Of Positive Periodic Solutions Of Three Kinds Of Ecological Models And The Synchronous Bifurcation For A Weakly Coupled Model With Time Delay

Owing to the rapid increase of population quantities and the constant development of industy, the global ecology environment has been destoried heavyly. While the human recognition improves continuously,people start to realise the consequence to which their actions going after benefits lead. Therefore researches of the population dynamics can guide people to exploit and protect natural rescources reasonably. There are four parts in this paper.Firstly,we investigate the asymptotic behavior of positive periodic solutions of three kinds of ecological models,including the existence, oscillation, global attractivity of the solutions.Secondly,the synchronous bifurcating periodic solutions for a weakly coupled model with time delay is studied.All kinds of species,who live in the same environment,often have the cruel competition with each other. A species preys on other to live, however preys reproduce their offspring quickly or escape to exist and develop. So dispersion is a general phenomenon among population growth.Because of most of environment, having periodic changes, such as seasonal variation. food's resource and the habit of animal's pregnancy etc. In order to describe these change rules and physiological phenomenon, we study three species ratio-dependent predator-prey model with time delays and dispersion in the chapter 2.Sufficient conditions for the existence of positive periodic solution are established by using the continuation theorem of coincidence degree theory.In real world,species can have more chance to find food and reproduction by dispersing between different kinds of patches.Inferior species always try their best to avoid being preyed by looking for refuges.This action can maintain the balance of the ecological system.In the chapter 3,we study three species predator-prey model with dispersion and refuge.Firstly,we prove uniform persistence and global attractivity by using the comparison theorem and Lyapunov function method.Then the existence and uniqueness of the positive periodic solutions and positive almost periodic solutions for corresponding periodic system and almost periodic system are also discussed by using Brouwer fixed point theorem and constructing auxiliary system and Lyapunov function.The variation of the environment is very important in ecological models.so it is very reasonable that we think of paramaters in models change periodically with time.In the chapter 4,we study a delay single-species model.In the nondelay case,the existence and global attractivity of positive periodic solutions are proved by using Brouwer fixed point theorem.In the delay case.sufficientconditions of oscillation and global attractivity about positive periodic solutions are obtained.In the chapter 5.we discuss the synchronous bifurcating periodic solutions for a weakly coupled model with time delay.Sufficient conditions of the existence of bifurcating periodic solutions and approximate form are investigated by using Hopf bifurcating theorem in the noncoupled model.The unstability of synchronous bifurcating periodic solution for a weakly coupled model by applying the theory of Floquet exponent is proved,so a weak coupling can change the stability of bifurcating periodic solutions.