Asymptotic Behavior For Three Kinds Of Ecological Models And The Hopf Bifurcation For A Logistic Model

Abstract Asymptotic behavior of mathematical ecological models is an important conception with rich connotation,which mainly includes attractivity of the solutions,stability of the so-lutions(local and global),periodicity and oscillation and so on .In the population dynamics,people may make full use of nature and remold nature by means of study of these properties.These will have very important theoretical and practical meaning to protect and save valuable and rare species which is on the verge of becoming extinction,to keep the population diversity and sustainable development of ecosystem.In the paper,we investigate the asymptotic behavior of three ecological models and the Hopf bifurcation for a class of Logistic model with time delays.In the chapter 2,we study a Logistic model with the discrete and distributed delays which is based on the known model.Sufficient conditions of unconditional stability are obtained by using the theorey of characteristic value,and it is shown that the delay r is locally harmless.Furthermore conditions of existence of the Hopf bifurcation and bifucation value are gotten.The theoy of predation is an important subject in mathematical biology.The standard Lotka-Volterra type predator- prey model assumes that the per capita rate of predation depends on the prey numbers only.There is growing experiment and evidence that in many situations, especially when predators have to reserch for food,the per capita predator growth rate should be a function of the ratio of prey to predator abundance,and so should be the so-called predator functional response.In the next two chapters,we consider predator-prey models.Because of the common existence of delay and dispersion in the ecosystem,dispersion can save becoming extinct species that will transfer to change its existing enviroment.Therefore,in the chapter 3,we investigate a delayed Lotka-Volterra type nonautonomous diffusive model with two-predators and one-prey that is based on the above factors.Sufficient conditions for the existence of periodic solution are established by using the continuation theorem of coincidence degree theory, it indicates that the existence of positive periodic solution has nothing to do with delays.In ecosystem, global stability of the positive equilibrium can reflect eventually the system being coexistent state. In the chapter 4, we study the stabilities of three-species ratio-dependent predator-prey system with time delays, we dicuss the local and global asymptotical stability of the equilibrium by means of constructing suitable Liapunov functionals and the Cauchy-Schwitz inequility,and generalize some previous conclusions.