Asymptotic Behavior For Three Types Of Ecological Models And The Hopf Bifurcation For A Model Of Hematopiesis

Asymptotic behavior for ecological mathematics models is an important research subject. In the present research results,some specified models are studied by many scholars.However,in real environment,the factors affect the variation of population density are many-sided,the relation between which is diversified and complicated.Therefore,the research of the generic ecological models is more valuable in theory and in practice.Some parameters' variation of ecological models will cause the variation of population's stability,which can produce periodic solution(or limit circle). Bifurcation phenomenon is a kind of it.This phenomenon is more common in the model of hematopoiesis, because the physiological mechanism of population are affected not only by their development but also by the environment.In this paper,the asymptotic behavior of three types of ecological models are investigated firstly,which includes the existence and stability of positive trivial solution, the attractivity of periodic solution,the existence of the permanence region,etc.Second,the bifurcation of a type of models of hematopoiesis are studied.The variation of population density is always complicated.lt is related with not only the density at the present moment but also the density at some previous moment .Moreover,the relations between them are not always linear but nonlinear.In order to approach real cases,in chapter 2,the more generic density restricted function is incorporated,and Logistic single species ecological models with generalized density restriction are discussed.The necessary and sufficient conditions for the unique positive trivial solution's local unconditional stability are obtained,by using the characteristic equation.The sufficient conditions for uniform persistence and global attractivity are got,by constructing Lyapunov-type functional and using differential inequalities.The feasibility of the theories' conditions is shown by examples.In real world,ecological systems and their parameters are affected by season variation,food's increase and decrease and the habit of animals' pregnancy,etc.Moreover,they don't all vary periodically. In order to reflect the variation more truely,in chapter 3,asymptotic predator-prey periodic system with diffusion is investigated,all the coefficients of which asymptotically approach some specified periodic function,respectively.Uniform persistence for the original system,the corresponding system and the assist system is proved,by constructing assist system and Lyapunov-type function.Using Brouwer fixed point theorem,it is proved that there exist unique global attractive positive periodic solutions for the corresponding system and the assist system,respectively.Using comparative theorem,the sufficient conditions which determine that all the positive solutions ofike original system asymptotically approach the periodic solution of the corresponding system are constructed.It is shown that two unstable pacthes can approach the population's uniform persistence by diffusion.In another side,delay phenomenon always happens together with diffusion in natural world.Therefore,in chapter 4,Machaelis-Menten type functional responsive functions are incorporated into predator-prey system with diffusion and delays.And two species nonautonomous ecological system with delays and generalized diffusion is investigated.By constructing Lyapunov-type functional,the existence of the positive invariant set and the boundness of the solution for the system are studied.In addition,by constructing persistant function.the uniform persistence region of the system is derived as well as the sufficient conditions for the uniform persistence.In chapter 5,a type of models of hematopoiesis with distributed delay and disturbance are investigaed. By using characteristic equation and bifurcation theory,the bifurcation value of the model is obtained.By using the sovability conditions and implicit function theorem,the form of the nontrivial periodic solution and the approximate periodic solution of the model is established.