The Stability Of Ecosystem Model For A Five Species

In the field of ecology, the interaction between species and number of species variation over time are the main research themes of ecologists. In recent centuries, the studies of the interdependence and mutual restraint way in which the species live have been greatly developed, a large number of simulation experiments and theoretical analysis of this phenomenon have been done by ecologists and mathematicians. In the ecological research, the most basic mathematical model is the Lotka-Volterra equations. In recent decades, many mathematicians and biologists discussed the connection between the predator prey system and the game theory, studied the variation of the population by adding some restrictions.This paper is organized as follows:In the first chapter, we give a summary of background, development and status quo for predator prey models.In the second chapter we mainly deal with the predator prey model by differential equation system. Especially, we analyze the local and global stability of this system by the qualitative theory.The third chapter introduces a competitive circulatory predator prey model. We establish the ordinary differential equations based on the level to describe species population size variation. By calculation, we obtain the steady states of the five species ecosystem. We prove that some of these points are locally stable and some are unstable. Under certain conditions, we discuss the global stability of the steady states by constructing the corresponding Lyapunov functions and draw the system coexistence phenomenon on a stable state. And the first and the fifth species must exist, but the three existing species needs certain conditions, and only one or all for the three species exist.