A Study Of Unstable Traveling Wave Solution

The Fisher-KPP equation is a very important reaction-diffusion equation arising in biology.Biologists use it to describe the population growth model. The Fisher-KPP equation has a wave-like solution like hyperbolic equations do.This wave-like solution is called the traveling wave solution or the TW solution,which represents the spread of the advantageous species through space.In this thesis,we study the TW solution of the Fisher-KPP equation.Previous research showed that the stability of the TW solution can be significantly affected by the traveling speed c. There exists a critical speed c*.If c< c*, no stable TW solution exists;while for each c≥c*,there exists a unique solution.However,when one computes the TW solution numerically,the solution is found out to be stable only with c= c*.The numerical solution will converge to the solution with c= c* if a proper initial datum is given.Observing the large time behaviour of the TW solution through phase space method,we propose a transform. This special transform then changes the Fisher-KPP equation into a new different partial differential equation.It can be proved that the new equation derived from this transform also has a TW solution,which has a one-to-one match with the TW solution of the Fisher-KPP equation.This corresponding relation then inspire us to solve the TW solution of the Fisher-KPP equation by investigating the new equation.In order to solve the transformed equation,we use the iteration method with relaxation proposed by M. Tang and W. R. Sun. In this thesis,we manage to get the TW solution and the traveling speed of the Fisher-KPP equation numerically in two respective cases where c> c* and 0< c<c*...