 Reaction–diffusion system

A reaction-diffusion system couples local changes in several fields with their spatial spreading. For concentrations $u=(u_1,\dots ,u_m)$, a common model isThe diffusion coefficients $D_i$ describe transport, while the reaction rates $f_i$ describe local conversion, production or loss. Writing $D=\mathrm{diag}(D_1,\dots ,D_m)$ gives the vector equation $u_t=D\Delta u+f(u)$. Chemical concentrations and rates must be modeled consistently; a reaction term can exchange material with an external supply.

A spatially homogeneous equilibrium $u_{\ast }$ satisfies $f(u_{\ast })=0$. Linearizing a spatial mode with $-\Delta \varphi =q\varphi$ givesThus reaction stability tests $J$, while spatial stability tests all boundary-compatible values of $q$. Unequal diffusion can create a Turing instability even when homogeneous perturbations decay.^[ref] The Brusselator and quadratic activator-inhibitor model give explicit two-species examples. Initial and boundary conditions are part of the system; the kinetic equations alone do not select its spatial modes.

A reaction-diffusion system is a mathematical framework used to describe the behavior of multiple interacting chemical species or biological entities that can diffuse through space and interact via reaction processes. These systems are characterized by two key components: 1. **Reaction Terms**: This aspect describes the interactions or reactions between the species. For example, it might include terms representing the formation or decay of one species as a result of the presence of others.