I plan to explain the interactions between diffusion and the spatial inhomogeneity in mathematical ecology. Main examples used for illustration include the classic logistic equation and the Lotka-Volterra competition systems. Starting from eigenvalue problems for indefinite weights, we will study the various interesting phenomena associated with the (single) logistic equation as well as the competition systems in a systematic way. More realistic models, such as directed movements and taxis, will be discussed.

In this presentation we introduce the so called Cell Boundary Element (CBE) methods for partial differential equations. It can be interpreted as a hy-bridized DG method. The CBE method was introduced by the speaker and
his colleagues. The method is base on 1)a local solution decomposition 2)flux continuity on intercell boundary. Therefore, the method is defined on the skeleton of a mesh generation, which will reduce degrees of freedom a lot. Moreover, the method naturally satisfies local flux conservation property.
We apply our method for the following PDEs:

• 2nd order elliptic equations
• Stokes equations
• multiscale elliptic equations