On modelling pattern formation by activator-inhibitor systems
SummaryThe formation of spatially patterned structures in biological organisms has been modelled in recent years by various mechanisms, including pairs of reaction-diffusion equations 
$$u_t = D_{\text{1}} \nabla ^{\text{2}} u + f(u,v)$$
, 
$$v_t = D_{\text{2}} \nabla ^{\text{2}} v + g(u,v)$$
.Their analysis has been by computer simulation. In some cases, u can be interpreted as an activator and v an inhibitor. The following problem is treated: given a “pattern” u = ϕ(x), v = Ψ(x), find a system which has it as a stable stationary solution (stability is used in various senses in the paper). This inverse problem is shown to have solutions for reasonable ϕ and Ψ. The solutions constructed are of activator-inhibitor type with D2>D1.