Parameter Plane Of A Discrete-Time Predator-Prey Model

The predator-prey system is a part of the ecosystem, which has an important significance for the protection of the environment, the rational use of biological resources, and the maintenance of ecosystem balance. The research on the relationship between predator and prey has been one of the most important topics in the field of ecology and biology. In order to study the interaction between predators and preys, we consider a discrete-time predator-prey system of Leslie type with a generalized Holling type III functional response p(x)= Chapter 1 briefly describes the research background, summarizes the literatures and gives some basic concepts and theroems. In Chapter 2, we give a basic analysis and introduction for the predator-prey system. The discrete-time system has very fine bifurcation structure including rich periodic and quasiperiodic solutions and chaotic motions. In Chapter 3, by using C language and MATLAB software, we calculate the maximum Lyapunov exponent of the model under specific parameters and plot the two-dimensional parameter plane. Through the observation of the parameter plane, we judge that the rich solutions should be related to the Flip bifurcation, Hopf bifurcation and so on. In Chapter 4, when b< 0 and the integral step size is chosen as a bifurcation parameter, we investigate the stability and bifurcation of the discrete-time predator-prey system in a strict mathematical way. In Chapter 5, we analysis and identify the results of the fourth chapter by plotting bifurcation diagrams, maximum Lyapunov exponents diagrams, and phase portraits. Chapter 6 summarized this paper. The interaction between preys and predators is more complicated than those when 6 > 0 and the integral step size is a constant.