Dynamics of Propagation Phenomena in Biological Pattern Formation

Abstract

A large variety of complexspatio-temporal patterns emerge from the processes occurring inbiological systems, one of them being the result of propagatingphenomena. This wave-like structurescan be modelled via reaction-diffusion equations. If a solution ofa reaction-diffusion equation represents a travelling wave, theshape of the solution will be the same at all time and the speedof propagation of this shape will be a constant. Travelling wavesolutions of reaction-diffusion systems have been extensivelystudied by several authors from experimental, numerical andanalytical points-of-view.In this paper we focus on two reaction-diffusion modelsfor the dynamics of the travelling waves appearing during theprocess of the cells aggregation. Using singular perturbationmethods to study the structure of solutions, we can deriveanalytic formulae (like for the wave speed, for example) in termsof the different biochemical constants that appear in the models.The goal is to point out if the models can describe inquantitative manner the experimental observations.