Analysis Of The Dynamic Behaviors Of A Reaction-Diffusion Model With Cross-Diffusion

In the field of theoretical biology,it is very important to study the mechanism of pattern formations.In recent decades,Turing instability of the solutions of the reaction-diffusion equations,as a typical problem of the research of the pattern formations,has been widely concerned in academic circles.This thesis studies a kind of reaction-diffusion predator-prey model with cross-diffusions.By using center manifold theorem,normal form theory,implicit function existence theorem,Floquet multiplier theory,and Poincare-Andronov-Hopf bifurcation theorem,we study the Turing instability of the Hopf bifurcating periodic solution for the reaction-diffusion predator-prey model with cross-diffusions.The specific studies are as follows:Firstly,we study the existence of the Hopf bifurcating periodic solutions of the ordinary differential system.In particular,we calculate the real parts and the imaginary parts of the first Lyapunov coefficient.The related theoretical results pave the way for our investigation of the Turing instability of the Hopf bifurcating periodic solutions.Secondly,through the Poincare-Andronov-Hopf bifurcation theorem,we proved the existence of periodic solutions for the perturbed auxiliary equations,and obtain the approximate expression of the periodic solutions.In particular,we characterize the first prime of the periodic function of the perturbed parameter.Finally,by using Floquet theory,we derive the conditions for Turing instability of the Hopf bifurcating periodic solutions for the reaction-diffusion predator-prey system with cross-diffusions.The aforementioned results obtained in the thesis enrich the dynamic behaviors of the reaction-diffusion predator-prey system with cross-diffusions,and provide a new idea for the study of the formation mechanism of Turing patterns.