Travelling Wave Solutions Of Several Tumor-invasion Singularly Perturbed Models

By using geometric singular perturbation theory and canard theory,this the-sis is mainly concerned with the existence of travelling wave solutions in several singular perturbation models modelling tumor invasions.The models are reaction-diffusion-convection equations.The main ideas are as follows.Firstly,the PDEs are transformed into singular perturbation ODEs by introducing traveling coordinate.Then,we obtain the existence of shock-fronted wave solutions to PDEs as well as the corresponding parameter conditions by constructing the singular heteroclinic orbits of ODEs.The thesis is consisted of four chapters:Chapter 1 is the introduction.'This chapter reviews the background,signifi-cance and recent advance on the research.Also we introduce geometric singularly perturbed theory and canard theory as well as the main work of this thesis briefly.In chapter 2,by introducing a so-called generalized Allee function to describe the growth of tumors,we reformulate a tumor-invasion model which is described by reaction-diffusion-convection equation.A key differences between the generalized Allee and Allee functions is the higher power of nonlinearity generated by the former one.In fact,this tumor-invasion model can be viewed as a degenerated situation of the Allee case,which had been studied by Sewalt et al.[J.Theoretical Bio.,(394)2016,pp.77-92].By using geometric singular perturbation theory and canard theory,we pay our attention to make clear the influence of the higher power of nonlinearity——the parameter m,on the existence of shock-fronted waves in this model.We find t,hat:(1)Types ? and ? waves are impossible when the power of nonlinearity m?4,and hence,Types ? and ? waves can exist only when m = 2 and 3;(2)there can have at most two equilibriums(of the regular system)in the wall:one is always a saddle while the other can be a stable focus or a stable node—depending fully on the power of nonlinearity m;(3)it is nearly impossible for the second equilibrium to act as a node when the power of nonlinearity m?8,while it can be a node " easily" when m is large,e.g.m>9.If the second equilibrium is a node,there are totally two "holes" in the wall,which leads to several new shock-fronted wave solutions.In chapter 3,we add competition effect into the generalized Allee growth and reformulate a tumor-invasion model.Based on the results in chapter 2,we aim to figure out the influence of competition on the existence of shock-fronted waves in the generalized Allee model.We find that:there also can have at most two equilibriums of system in the wall:one is always a saddle while the other can be a focus or a node—depending fully on the power of nonlinearity m as well as the competition coefficient ?.It is observed by calculations and phase plane analysis that it is nearly impossible for the second equilibrium to act as a node when m = 1.It should be a unstable focus.Furthermore,by controlling 0<?<?max,then we can find there are Type ?,?,? and ? waves in this model,which are non-uniqueness—there are two or more singular heteroclinic orbits connecting the same two fixed equilibriums.When m>1,by using the phase plane analysis and the equilibriums analysis can determined that it is impossible for model possess Types ?and? waves,but we do not discuss the existence of other waves owing to the complexity.Chapter 4 is the summary and prospect on the research in this thesis.