Schnakenbergmodel

The Schnakenberg model is a two-species reaction-diffusion system used to study Turing-type pattern formation. It provides a minimal, analytically tractable framework for diffusion-driven instabilities in chemical or biological systems.

The model describes two interacting chemical concentrations u(x,t) and v(x,t) evolving in space and time according to

∂u/∂t = Du ∇^2 u + α − u + u^2 v

∂v/∂t = Dv ∇^2 v + β − u^2 v

where Du and Dv are the diffusion coefficients, and α and β are positive feed terms. The terms α − u + u^2 v and β − u^2 v come from a simple reaction network that, in the absence of diffusion, drives the system toward a homogeneous steady state.

Homogeneous steady state and linearization show a fixed point at

u* = α + β, v* = β/(α + β)^2.

The Jacobian of the reaction terms at this steady state has entries involving α and β, and the stability without diffusion is determined by its trace and determinant. Diffusion can destabilize this state when Du and Dv differ sufficiently, typically with the inhibitor (the field v) diffusing faster than the activator (u). This diffusion-driven instability can produce stationary spatial patterns.

In practice, patterns such as spots, stripes, or labyrinthine structures emerge on bounded domains, especially under no-flux (Neumann) boundary conditions. The specific pattern depends on domain size, boundary conditions, and parameter values, including α, β and the diffusivity ratio Dv/Du.

The Schnakenberg model remains a canonical example in the study of Turing patterns, offering analytic tractability and a clear illustration of how diffusion can generate spatial order from homogeneous equilibria.