Two Types Of Ecological Models And Asymptotic Properties Of A Class Of Difference Equations

In this paper the asymptotic properties of two ecological models and a class of difference equation are studied, including the persistence, asympotic stability, the existence of Hopf bifurcation, the oscillation of the difference equation etc. People may make full use of nature by mesns of study of these properties. These will have very important practical meaning to better know the natural, use the natural resources, improve the ecological environment, protect eeological systems, and keep the populations sustainble development.The single speice is the basic unit of the entire ecosystem. The construction of the single-speice model and the study of it is very useful to research the whole ecosystem. The permanence and global asymptotic stability of a non-autonomous dispersal model with time delays and feedback controls are discussed in the second chapter. Sufficient conditions for the permanence of the system are obtained by applying the differential equation comparable theory and estimating of value by using inequalities:and if the model is a periodic one. it has only one periodic solution, and the global asymptotic stability of it is derived by constructing the suitable Lyapunov functions; Finally the feasibility of the theo-rys and corollarys are shown by some examples and the solution figures drown by Mat lab.In the nature world, change of biological group density is extremely complex. In actual ecosystems, delays and interference usually have a certain effect on densiv constraints. In the third chapter the Hopf bifurcation and stability for a class of a single-speices model with interference and delays are investigated; Sufficient conditions of locally asymtotic stability of the modle are derived by using the characteristic value theory; Considering the delayτas a parameter, conditons of the existence of Hopf bifurcation and the bifurcation value and the stability of it are obtained;Finally fitting maps about the examples are presented by using Matlab and the effects of the parameters on bifurcation periodic solutions are disscused.The properties of the difference equation are concerning by many scholars. The oscillation and asympotic stability for a class of difference equation are investigated in the last chapter. Necessary and sufficient conditions of oscillation of all the solutions of the difference equation are derived by studying the roots of the cubic equation using cardano formula;Necessary and sufficient conditions of asympotic stability of all the solutions of it are obtained by using the Jury rules; Finally the feasibility of the theorys is shown by some examples.