Travelling Wave Solutions of Competition-Diffusion Systems

We divide the thesis into two parts to investigate the travelling wave of two types partial differential
equations coming from ecology. In Part 1, we consider the 3-species Lotka-Volterra competition-diffusion systems. In
Part 2, we consider a free boundary problem for a two-species competitive model.

For the 3-species Lotka-Volterra competition-diffusion system, a travelling wave solution can be considered as a heteroclinic orbit of a vector field in R^6. Under suitable assumptions on the parameters of the equations, we apply a bifurcation theory of heteroclinic orbits to show that a 3-species travelling wave can bifurcate from two 2-species waves which connect to a common equilibrium. The three components of the 3-species wave obtained are positive and have the profiles that one component connects a positive state to zero, one component connects zero to a positive state, and the third component is a pulse between the previous two with a long middle part close to a positive constant. As concrete examples of application of our result,
we find several explicit regions of the parameters of the equations where the bifurcations of 3-species travelling waves occur.

The free boundary problem for a two-species competitive model in ecology was proposed by Mimura, Yamada and Yotsutani. Motivated by the
spreading-vanishing dichotomy obtained by Du and Lin, we
suppose the spreading speed of the free boundary tends to a constant as time tends to infinity and consider the corresponding travelling wave problem. We establish the existence and uniqueness of a travelling wave solution for
this free boundary problem.