The Qualitative Analysis Of A Fast-Slow Predator-Prey System

In the literature [1], the author propose a class of simple Fast-Slow Predator-Prey System .Through numerical simulation, In the study of practical biologicalprobiems common predator-prey model has inherent limitations.First,because ofecological evolution and biological evolution has di?erent time scales.Second,theactual predator model is not a simple two-dimensional model systen,so the authorspropose the use of di?erent time scales to measure the model and introduce thehigh-power system model.In the literature [1], Through numerrical simulation,the author observed thegroup's action under the speed-based coexistence and species may occur in thepopulation oscillation.This paper is based on this model,and select the appropriatefunction to study the stability of the systen and give the possible branch conditions.This paper is divided into five parts:In Chapter 1 , we overview the development of predator-prey system and theway of building up and simplifying the model we study in this paper;In Chapter 2 , we introduce the qualitative theory, bifurcation theory of di?er-ential equations and singular perturbation theory used in this article,give relevantdefinitions, basic theorems and research methods;In Chapter 3 ,Through the qualtiative analysis of the three-dimensional model,giventhe local stability of the boundary equilibrium and the number of relevent balanceconclusions,get the conditions that the three-dimensional model appear hopf bi-furcation,and by the two-dimensional center manifold study the hopf bifurcationand the stability of periodic solutions;In Chapter 4 , By studying the slow system of the three-dimensional model,provedthe system has experienced saddle-node and hopf bifurcation.And the use of thedingular perturbation theory to study the dynamic behavior of the system,Usingthe zero-order fast system to get the slow manifold's expressions,through theFenichel theory get the existence of slow manifolds.At the same time,give a nu-merical simulation;In Chapter 5 , we provide conclusions of our findings and exhibit problems which haven't been solved.