| It is well known that the interactions between diffusion and spatial heterogeneity could create very interesting phenomena. In this thesis, using the classical Lotka-Volterra competition system, we will illustrate the combined effects of dispersal and spatial variation on the outcome of the competition.;We first show that, with the total resources being fixed at exactly the same level, a heterogeneous distribution of resources is usually superior to its homogeneous counterpart in the presence of diffusion. Then we study the more general case when both species have heterogeneous carrying capacities, but still with the same amount of total resources. Limiting behaviors of co-existence steady states as the dispersal rates tend to 0 or infinity are also obtained.;In the end, we investigate the much broader situations - including different strengths and distributions of the resources, and with different competition abilities. Stability properties of semi-trivial and co-existence steady states are characterized under various circumstances. |
