Time-varying Ecosystem Solution To Study

Eighties of the last century, the former Soviet Union expert in differential equations Mironenko create the theory of reflective function which provided a new avenue to study the qualitative behavior of solutions of the differential equation x’=X(t,x). From now on, more and more experts began to investigate the reflective function and obtained many good results.On the basis of work which has been done, in this paper we research the periodic solution of time-varying ecological differential system. First, it introduces the background、present status and significance of this article in introduction. Then, to convenient, it gives the detailed definition and the basic qualities of the reflective function. We give a description of the relation between reflective function and the Poincare mapping. In this paper, we introduce the basic concepts which will be used from beginning to end.The aim of this paper is to combine the method of reflective function to investigate the qualitative behavior of solution of time-varying ecological-II functional response Kolmogrov model.Known by Mironenko, for the2n-dimensional differential system Suppose that F(t,x) satisfied Let us study the qualitative behavior of n-dimensional differential system x’=P(t,x,F) to discuss the qualitative behavior of2n-dimensional differential system.In this paper we use the results of Mironenko to investigate the qualitative behavior of solution of time-varying ecological-Ⅱ functional response Kolmogrov model. The first section discusses for the structure of reflective function of differential system (1.4) which satisfied conditions (1.2),(1.3).The second part studies the reflective function of the differential equation x’=P(t,x,F). It has shaped like F(t,x)=f0(t)+f1(t)x and F(t,x)=(f0(t)+f1(t)x)/(g0(t)+g1(t)x).In this part, sufficient conditions for this reflective function are obtained.The third section considers the reflective function F=(F1(t,x,y), F2(t,x,y))T of the differential system (1.4). Suppose that the first component of the reflective function Fx(t,x,y) nothing to do with y, we obtain the structure form of F2(t,x,y) which with respect to components of a vector y. In the same time, we give the sufficient conditions of the differential system with this reflective function and the qualitative behavior of periodic solutions of time-varying system.Finally, two simple examples are exploited to show the upper conclusions.In conclusion, this paper mainly using the method of reflective function for one-dimensional differential equation to investigate the qualitative behavior of periodic solutions of two-dimensional differential system, by using the theory of reflective function to study the qualitative behavior of solution of time-varying ecological (1.4).