The Effects of Diffusion and Advection on the Evolution of Competing Species: a Survey on the Lotka-Volterra Competition Model 戴佳原

This thesis is a rather complete survey concerning an ecologically meaningful problem: how would two competing species evolve in a given spatially heterogeneous and isolated environment? A special kind of the Lotka-Volterra competition model is derived by assuming that the mechanisms of redistribution consist of mutual competition, random diffusion, and advective motion. The main task is to analyze the evolutionary results of the competing species in the long run, or equivalently, to determine the stability of equilibria of the model. The mathematical methods such as maximum principles, calculus of variation, and the theory of monotone dynamical systems are utilized as the standard procedure. The main conclusion is that both random diffusion and advective motion decide the evolutionary results; thus different combinations of diffusion rates and advective tendencies may influence the evolutionary results. Accordingly, a preliminary bifurcation diagram can be established to provide certain theoretically reliable predictions. ∗