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The Qualitative Analysis of a Fast-Slow Predator-Prey System

Author: ZouHuiJuan
Tutor: XiaoDongMei
School: Shanghai Jiaotong University
Course: Applied Mathematics
Keywords: Predator - prey system - Fast System Singular perturbation Slow Manifold Saddle - node bifurcation 3D Hopf bifurcation
CLC: O175
Type: Master's thesis
Year: 2012
Downloads: 22
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Abstract


The study of the actual ecological problems , the evolution of the ecological environment and the evolution of biological species is a problem that can not be ignored . Literature of Cortez et al proposed a class of predator - based biometric speed of evolution of three-dimensional power system to describe the process of the interactions of predator and its prey asked , considering the biological characteristics of the predator species , the evolution and ecology of species interactions has a different time scale of observation data show that the evolution of the predator species biometric important influence on the dynamic behavior of the ecological food chain system . c. rtez et al in the the 2D classic predator- prey model based on the introduction of a description of the evolution of the ecological characteristics of the predator species , to obtain a three-dimensional single fast variable speed of power system . c. through the numerical simulation rtez et al in the text ... the observed species population oscillation behavior of these two populations co-exist in a stable state , or may occur . The papers on the basis of their article , continue to study this model through the theoretical analysis of the stability of the coexistence equilibrium point and the branch condition may occur . Specifically , we do the following work : full parameters of a three-dimensional model of qualitative analysis , given the number of local stability of the boundary equilibrium and the positive equilibrium conclusion , the three-dimensional system conditions of the Hopf bifurcation , and through the two-dimensional center manifold theorem Hopf bifurcation and stability of periodic solution . This part of the work appears in the third chapter of the thesis . Second, by studying the three - dimensional system slow positive equilibrium point of the system and the existence and stability of the boundary equilibrium morphology analysis proved that the system has experienced saddle node bifurcation and Hopf bifurcation . And use of the singular perturbation theory to study the dynamic behavior of the system , the use of fast subsystem zero-order slow manifold expressions come to the existence of the slow manifold maintained by Fenichel theorem . Numerical simulation . This part of the work appear in the fourth chapter of this thesis .

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations
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