Analysis of a general reaction–diffusion model using Lie symmetries and conservation laws

Abstract

Turing’s model to explains the formation of patterns in morphogenesis considered a system of chemicals, termed morphogens, that react and diffuse through tissues. These reaction–diffusion systems can start homogeneously but later develop patterns due to instabilities triggered by random disturbances. Building on this foundation, Kepper realized the Chlorite-Iodide Malonic-Acid reaction, an example of an oscillatory reaction in a homogeneous solution that forms spatial patterns in a non-homogeneous environment. This work led to further studies, such as the Lengyel-Epstein reaction–diffusion model, which describes the dynamics of chemical concentrations of activator and inhibitor species. This paper extends these classical models by investigating a general reaction–diffusion system through the lens of Lie symmetries. We analyze the system using Lie point symmetry generators and Lie symmetry groups, enabling us to reduce the equations via these symmetries. Furthermore, we compute the conservation laws for the general reaction–diffusion model using the multipliers approach, involving dependent variables, independent variables, and their derivatives up to a certain order. By applying various symmetry groups, we derive new solutions from known ones, offering deeper insights into the dynamics of pattern formation in biological systems.