Alan Turing's work in Morphogenesis has received wide attention during the
past 60 years. The central idea behind his theory is that two chemically
interacting diffusible substances are able to generate stable spatial patterns,
provided certain conditions are met. Turing's proposal has already been
confirmed as a pattern formation mechanism in several chemical and biological
systems and, due to their wide applicability, there is a great deal of interest
in deciphering how to generate specific patterns under controlled conditions.
However, techniques allowing one to predict what kind of spatial structure will
emerge from Turing systems, as well as generalized reaction-diffusion systems,
remain unknown. Here, we consider a generalized reaction diffusion system on a
planar domain and provide an analytic criterion to determine whether spots or
stripes will be formed. It is motivated by the existence of an associated
energy function that allows bringing in the intuition provided by phase
transitions phenomena. This criterion is proved rigorously in some situations,
generalizing well known results for the scalar equation where the pattern
selection process can be understood in terms of a potential. In more complex
settings it is investigated numerically. Our criterion can be applied to
efficiently design Biotechnology and Developmental Biology experiments, or
simplify the analysis of hypothesized morphogenetic models.Comment: 19 pages, 10 figure