Reaction–diffusion equations
Reaction–diffusion equations combine local biological change with spatial movement.
General form
\[\frac{\partial u}{\partial t}=D\nabla^2u+f(u).\]The term \(f(u)\) describes local processes such as birth, death, infection or chemical reaction, while \(D\nabla^2u\) describes diffusion.
Multiple components
For interacting quantities \(u\) and \(v\),
\[\frac{\partial u}{\partial t}=D_u\nabla^2u+f(u,v),\]\[\frac{\partial v}{\partial t}=D_v\nabla^2v+g(u,v).\]Different diffusion rates and nonlinear interactions can generate spatially complex behaviour.
Key idea. Reaction–diffusion models couple local biological dynamics with movement through space.