Dynamics of Propagation Phenomena in Biological Pattern Formation

¹ Institute of Mathematics "O. Mayer", Romanian Academy, 700505 Iaşi, Romania
² Institute of Applied Mathematics and Mechanics Warsaw University, 02-097 Warsaw, Poland
³ Departamento de Matemática Aplicada, Facultad de Matemáticas Universidad Complutense, 28040 Madrid, Spain

Corresponding author: lorelai@mimuw.edu.pl

Abstract

A large variety of complex spatio-temporal patterns emerge from the processes occurring in biological systems, one of them being the result of propagating phenomena. This wave-like structures can be modelled via reaction-diffusion equations. If a solution of a reaction-diffusion equation represents a travelling wave, the shape of the solution will be the same at all time and the speed of propagation of this shape will be a constant. Travelling wave solutions of reaction-diffusion systems have been extensively studied by several authors from experimental, numerical and analytical points-of-view. In this paper we focus on two reaction-diffusion models for the dynamics of the travelling waves appearing during the process of the cells aggregation. Using singular perturbation methods to study the structure of solutions, we can derive analytic formulae (like for the wave speed, for example) in terms of the different biochemical constants that appear in the models. The goal is to point out if the models can describe in quantitative manner the experimental observations.