The Study Of Turing Instability Of The Solutions For A Class Of Compartmental Models

Aimed at the phenomena of the spatiotemporal pattern formations observed in the chemical reactions,in this thesis,we consider a class of compartmental models with cross-diffusion-like coupling.By using the center manifold theory,the normal form method,the local Hopf bifurcation theory and regular perturbation theory,we study the Turing instability of the spatially homogeneous Hopf bifurcating periodic solutions for this compartmental model.The specific contents are as follows:By using the local Hopf bifurcation theorem,we show the existence of the Hopf bifurcating periodic solution for the ordinary differential equations.In particular,by using the center manifold theory,the normal form methods,we are able to obtain the formula to determine the bifurcation direction and the stability of the local Hopf bifurcating periodic solutions.By utilizing regular perturbation theory,we show the existence of the periodic solutions for the perturbed ordinary differential equations,and we study the property of the period of periodic solutions.Then,we study the Turing instability of the spatially homogeneous periodic solutions for the compartmental model.We derive the precise conditions on diffusion rates so that under these conditions,the spatially homogeneous Hopf bifurcating periodic solutions can undergo Turing instability.The theoretical results of this thesis,reveal the different effects of the different diffusion rates on the Turing instability of the spatially homogeneous periodic solutions,and they also provide theoretical way for us to understand the mechanism of the spatiotemporal pattern formations for this compartmental model.