Stability Analysis For Some Serious Classes Of Discrete Predator-prey Model With Functional Response

The change of ecological population mostly obey the discrete law in the ecological system. Using difference equation describing the population dynamics of the state is more in line with reality. This paper mainly studies three serious of predator-prey systems with functional response. Using analytical method of difference equation and coincidence degree theory, we obtain the persistence of the population and the existence condition of periodic solution. Three parts of main work are as follows:In the first part, for a class of the predator-prey model with functional response function, using the differential equations thinking with piecewise constant argument, we derive from the corresponding constant coefficient of difference equation with functional function. By using the method of qualitative analysis, we obtain a condition for the existence and uniqueness of positive equilibrium point about discrete model. Using the eigenvalue method in algebra, we can obtain the sufficient conditions for partial stability of positive equilibrium point. By constructing a suitable Liapunov functional, we discuss global stability of positive equilibrium point.In the second part, taking into account of the impact of cyclical change, which in many acts and properties of the population along with the time, we build a predator-prey model with ratio-dependent periodic constant. Using analytical and inequalities method, properly zooming the right function of the model, we obtain the sufficient condition of persistence. By constructing a appropriate Liapunov functional, we obtain the sufficient condition of the global stability of positive periodic solution.In the third part, considering the factor of time delay, we discuss Leslie predator-prey model with time delay. By applying the continuation theorem of coincidence degree, we obtain existence condition of periodic solution. Using analytical and inequalities method, properly zooming the right function of the model, we obtain the sufficient condition of persistence. The research about persistence and stability of ecology system can be used to guide people to make full use of the nature and remold the nature. Those will have very important extensive theoretical and practical significance to maintain the diversity of ecosystems and the sustainable development of the ecological environment.