The Effects Of Diffusion And Spatial Variation In Competition Models

The dynamics of biological models have received intensive study and it has been an important aspect in the field of nonlinear partial differential equations. Qualitative properties of competition systems are studied by mathematicians and ecologists in recent years. In this paper, we will illustrate the combined effects of dispersal and spatial variations on the outcome of the competition by studying the stability of the steady state.In Chapter 1, the background and history about the related work are introduced.In Chapter 2, we examine a three species Lotka-Volterra competitive model with Neumann boundary condition in heterogenous environment:with the total resources being fixed at exactly the same level, we first study the stable of semi-trivial steady state and its limiting behaviors; In the second part, by using the upper and lower solution method, the existence of coexistence steady state is discussed and two special cases are also given; Finally, we proved the uniform boundedness of solutions in the space of Cx((0,∞), W21([0,1])) by the G-N inequality and comparison argument.In Chapter 3, a competitive model with nonlinear cross-diffusion V/1+U,U/1+V under the het-erogeneous Neumann boundary condition is also studied. We first show that a heterogeneous distribution of resources is usually superior to its homogeneous counterpart by discussing the stability of steady state. Then the slower diffuser always prevails when the two species are in the same spatial inhomogeneity. Finally, we study the more general case when both species have heterogeneous carrying capacities, the sufficient condition of the stable semi-trivial state is given and consequently, the coexistence becomes a much stronger possibility.