Asymptotic Behavior Of Three Differential Systems

ASYMPTOTIC BEHAVIOR OF THREE DIFFERENTIALSYSTEMSTang Hong-JiAbstract: The system's asymptotic behavior is the solutions' behavior at / -> -I -xxlt iiicluds the stability of the solutions ,the attractivity of the solutions, the oscillation of the solutions.The purpose of this paper is to investigate the asymptotic behavior of throe differential systems.In the study of species dynamics,the permanence of the species is one of the most interesting problems.In recent years,people have destroyed the enviorment,which leads to some biologies distinction.In some documents on the permanence of the species ,there are two ways which can be used to save the distincting species. One is human's interference,the other is species' adjust ion. There are documents that showed two instable patches can reach stability by diffusion.Delay is generally exists in the nature.And it has different impacts on different models.While both delay and diffusion exist at the same time.So single species nonautonomous delay diffusion model .i.e.are studied. By using differential inequality, comparable theory and the Lyapunov functional we obtain some sufficient conditions that guarantee the permanence of the species and some sufficient conditions that guarantee every positive solution global attractivity.And if the system is a. periodic systcm,then there is a periodic solution which is globally attractiv.Many species have two stages,immature and mature ,in the nature world.There is distinctive physiological difference between the two stages.In order to react the physiological pheiiominon. stage structure ecological models i.eis considered. It is a two-species system ,where the both species have two stages.immature and mature. And the mature species arc aquated. The growth of the both species is of Logistic naluf. The mature species-y preys on the mature-x. By using the lyapunov function we obtain the-conditions for globally asymptotic stablility of positive equilibria without harvesting .conditions for global asymptotic stablility of four positive equilibria with harvesting and the threhold of hai vi-sting mature population. The experience and the lessen of people's exploitation for the nature show chat the utilization of nature should have the unitary of economic benefits and the envioiiiinent benefits.There is close relation between the proper exploitation of biologies and the la-sl ing development. So it has been laid stress on by the sphere of learuing.Tho optimal harvesting of mature population in the model is also considered.Lots of models can be depicted by functional differential equation. Asymptotic behavior and oscillation are of momentous theorical significance and realistic significance. So researching them becomes one of interesting problem.There is distinctive difference between linear system and nonlinear system. In this paper,a higher nonlinear differential systemis studied. We get some new criteria for the oscillation of it and the solution's asymptotic behavior.