Traveling wave fronts for a diffusive competition model with time delay

The main purpose of this dissertation is to study mono-stable traveling wave fronts/solutions for a class of monotone reaction-diffusion systems with time delay. First, I prove that the existence of an upper solution is sufficient for the existence of a traveling wave solution that connects an unstable equilibrium point and a stable equilibrium point (while in a classical approach one needs both upper and lower solutions). Then I apply this result to investigating the existence of traveling wave solutions for an important class of diffusive Lotka-Volterra competition model with time delayed effect. By a careful construction of upper solutions, I obtain the existence of traveling wave solutions. In particular, I find the precise and explicit formula for the minimum wave speed cm of traveling waves under certain conditions. That is, the competition model has a traveling wave connecting an unstable equilibrium point to a stable one if and only if the wave speed c ≥ cm. The explicit formula of the minimum wave speed has an important application to the theoretical ecology. The results obtained in this dissertation also extended the known results on the minimum wave speed for Lotka-Volterra competition model without time delay.