A reaction-diffusion model of branching morphogenesis

It is a well known fact that in the course of their evolution higher organisms rapidly grow to a size where passive diffusion is not enough to provide the tissues with oxygen, nutrients, water and information. Nature has found a way out to this problem by the invention of complex-shaped organs of a filamentary structure, which thoroughly expand into the corresponding organic matrix. The vascular system of vertebrates is a paradigmatic example of such a situation.

While the underlying genetic mechanisms have begun to be elucidated only recently, mathematical models of reaction - diffusion type, accounting for some of the qualitative features observed, were proposed as early as in the 70s. Here we shall describe one such model, which was introduced by Hans Meinhardt. Some of the main features of the corresponding solutions, as for instance the local structure of the expanding filaments and the branching phenomena taking place there, will be presented and discussed by means of matched asymptotic expansions techniques.