Turing models and general reaction–diffusion systems have been used to study mechanisms leading to emergent spatial patterns in Biology, Chemistry, and Physics. 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 models remain unknown, limiting scientists to run large sets of computationally intensive simulations to test parameter choices.

In response to this need, 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. Our criterion is motivated by the existence of an associated energy function that allows bringing in the intuition provided by phase transitions phenomena. We proved our selection criterion 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, we performed numerical investigations.

Our work constitutes a first step towards rigorous pattern prediction in arbitrary geometries/conditions. This theory is largely applicable to the efficient design of Biotechnology and Developmental Biology experiments, as well as in simplifying the analysis of morphogenetic models. In this talk, I will present the criterion itself [1], along with applications in which it is being used.

[1] Marquez-Lago T, Padilla P. A selection criterion for patterns in reaction-diffusion systems. Theor Biol Med Model. 2014;11:7