Reaction-Diffusion and Lattice Gas Models

Spatial and temporal multivariate RF models, where interactions between components act through chemical reactions, are introduced.These models give rise to morphogenesis. A first part gives a general description of Reaction-Diffusion models at two different scales:on a macroscopic scale, chemical concentration variables are solution of a system of non linear parabolic partial differential equations. On a microscopic scale, discrete versions of these models are obtained as Markov jump processes (for the chemical reaction) and random walks (for the diffusion).A second part studies the linear Reaction-Diffusion model, well suited to a particular kind of chemical reaction ( $$ {{X_{i}} \rightleftharpoons {X_{j}}}$$ X i ⇌ X j ) or to a damping effect. Models obtained from a stationary RF as initial conditions and from a spatial-temporal random source are considered, with in particular a specific Dilution RF. Then examples of simulations of non-linear Reaction-Diffusion RF, generating time oscillating or chaotic behaviors,are illustrated. In a third part is introduced a discrete implementation based on lattice gas simulations, to simulate complex flows in random media, random aggregates, and multi species chemical reactions.