Mathematical Modeling of Biological Pattern Formation

Summary

In this chapter we have presented mathematical modeling approaches to biological pattern formation. While reaction-diffusion models are appropriate to describe the spatio-temporal dynamics of (morphogenetic) signaling molecules or large cell populations, microscopic models at the cellular or subcellular level have to be chosen if one is interested in the dynamics of small populations. Interest in such “individual-based” approaches has recently grown significantly as more and more (genetic and proteomic) data are available. Important questions arise with respect to the mathematical analysis of microscopic individual-based models and the precise links to macroscopic approaches. While in physical processes typically the macroscopic equation is already known, the master equations in biological pattern formation are far from clear. In the remainder of the book we focus on cellular automata, which can be interpreted as an individual-based modeling approach.