Adaptive branching in evolution and epigenesis

We describe one of the simplest models that exhibit an adaptive branching behaviour. It is analysed both experimentally and formally, and its successive bifurcations provide a good model of what R. Thom called 'generalized catastrophes'. Two theorems on the stochastic adaptivity of the algorithm to very general shapes of target are given. The model further displays the phenomenon of abortive branching: each macroscopic branching appears after a burst of microscopic branchings that stop growing after a very short time. The mathematical analysis of the model explains why and how this behaviour occurs. Possible applications of these models to Evolution (natural and artificial) and Epigenesis are briefly mentioned, and a higher dimensional version is applied to growing a tree in a space of shapes in the context of a database of medical images.