Kinetically cooperative models: boundary movement in optical resolution, phase transitions, and biological morphogenesis

Computer simulations are described for simple models of kinetically cooperative competition between two or more a priori equally matched antagonists. These might be optically enantiomeric molecules, crystal defects in two phases, differentiation states of biological cells, etc. The models give a perspective on the relationship between the reaction–diffusion theory of pattern formation, homeogenetic induction between biological cells, and the currently popular "cellular automaton" computer programmes. The models express self-activation without inhibition. Therefore, they do not have all that is needed for indefinitely long stabilization of pattern by reaction–diffusion, as first established in the Turing activator–inhibitor model and used later in most reaction–diffusion models. But by the same token, boundaries between disparate regions do not reach stable positions, but must move until they finally reach the edges of the system and disappear. Therefore, these models are convenient for studying aspects of boundary movement in the régime of small-number statistics. To see this without artefacts from array geometry, we use hexagonal arrays in place of the more popular square ones.We conclude that the behaviour of these models in one respect contrasts with and in another resembles the expected deterministic behaviour in large-number statistics. Regions totally surrounded by an antagonist often grow, while deterministically they must shrink. On the other hand, boundaries tend to straighten, as they would deterministically. In a system of overall rectangular habit, with or without periodic edge conditions making it a cylinder or a torus, boundaries tend to align with the shorter dimension, making a pattern of "square stripes".