Macroscopic Cross-Diffusion Models Derived from Spatially Discrete Continuous Time Microscopic Models

We formulate a continuous time, discrete in space model for two spatially interacting species. The spatial interaction is described in terms of a measure for the desire or ability of a population to move from one location into a neighboring site. This can depend on local densities of both populations in the current and the target site. Refining the spatial resolution and passing to a continuous in space model, one obtains a system of partial differential equations with cross diffusion terms. We show that certain cross-diffusion models that have been used in the literature to describe interacting species can be derived as special cases with our approach.