Activator–Inhibitor Models

Abstract

Let us consider the activator–inhibitor system. The general model equation is given by the reaction–diffusion system

$$\begin{cases}\frac{\partial u}{\partial t}=a\varDelta u+\gamma f(u,v)&\text{in}\ \varOmega\times(0,\infty), \\\noalign{\vspace{3pt}}\frac{\partial v}{\partial t}=b\varDelta v+\gamma g(u,v)&\text{in}\ \varOmega\times(0,\infty),\end{cases}$$

(9.1)

in a domain Ω⊂ℝ^d. Here, u denotes the concentration of the activator, and v the concentration of the inhibitor in Ω, respectively. The functions f(u,v) and g(u,v) denote the kinetics and are real smooth functions defined for u≥0 and v≥0. The activator and the inhibitor diffuse in Ω with diffusion rates a>0 and b>0, respectively. The constant γ>0 denotes the reaction rate. Species of the activator or inhibitor must be specified in each activator–inhibitor system under consideration. The form of kinetic functions f(u,v) and g(u,v) must also be fixed in an adequate manner.