The Asymptotic Behavior Of Three Kinds Of Ecological Models And The Oscillation For One Kind Of Ode With Impulse

The system's asymptotic behavior includs the stability of the solutions,the attrac-tivity of the solutions,the oscillation of the solutions and the Hopf-bifurcation , these behaviors reveal the long-term behavior of the species.since a lot of biological regular and phenomenon in the biological were discribed by using differential equation,which drew the attention of a lot of experts and scholars, and formed many new topics that have strong practicive background.We know that the study of the co-exist, stability,oscillation of the spices has very important practicive meaning to keep ecological equilibrition and protect ecological environment,even to save valuable and rare biologies which are on the verge of becoming extinct.In this paper,we investigate the asymptotical quality of three classes systems and the Hopf-bifurcation in a class model.The first system is the Nicholson-flowflies model, we studied the Hopf-bifurcation of this model, the sufficient condition for the existence of bifurcation periodic solution is obtioned,and by using the solvability conditions,the form of the approximate periodic solution is obtained.In natural world, many individual members have a life history that takes them through two stages: imature and mature,in particular,mammalia and some amphicious animals exhibit these two stages.and species have evident difference about their physiological character,in order to describe these physiological phenomenon,the species are always divided into two stages according to their physiological character when we set up biological models,meaning stage-structured sys-tems.The second model in this paper is a stage-structured two-species competitive system with cannibalism and refuges. Compared with the conclusion of the paper [10],it is shown that refuges have more effect on the equilibriums and the stability of this system,the necessary and sufficient condition for the only positive equilibrium was obtained,and the sufficient conditions for the globally asymptotical stability of the positive equilibrium and the boundary equilibrium were obtained.Since many natural phenomenon or the factors produced by mankind in population dynamics are impulsive, the research on the differential equation with impulse is an important branch to differentia] equation.The third model in this paper is a non-linear systems of differential equation with impulse.By taking transform,the uncontinuous impulse differential equation with constant coefficients was transformed to continuous un-impulse differential equation with changable coefficients.The sufficient condition for the oscillatory of every solution and the asymptotic behavior ofthe nonoscillatory solution are obtained.By using Laplace-transform and residue theory,the sufficient conditions for the generic oscillation and nonoscillation of the solutions are obtained.The persistence and stability of species is an important question which draw a lot of scholor.A series of processes:species birth,growth or death,predator-prey,competition or cooperatin among species,etc,can be presented in the reaction term in biological mathematics models.The forth model is a predator-prey system with diffusion.By using inequality and constructing a sutiable Lyapunov function,the sufficient conditions for uniform persistence and local asymptotic stability are obtained.