A spatially homogeneous equilibrium of a reaction-diffusion system is a constant state satisfying . Since its spatial derivatives vanish, its stability to homogeneous perturbations is determined by the Jacobian matrix ; diffusion can additionally destabilize nonconstant modes.
A morphogen reaction-diffusion equation models a spatial concentration by diffusion and local production or decay. Linearization at a homogeneous equilibrium replaces by .
If a fast inhibitor satisfies , then its spatial Fourier transform obeysSubstitution produces a nonlocal scalar equation for the activator.
For activator diffusion , local linear decay , and coupling , eliminating the fast inhibitor givesIts pattern-forming threshold is
A Turing, or diffusion-driven, instability occurs when a spatially homogeneous equilibrium is stable to homogeneous perturbations but unstable to a nonzero spatial Fourier mode after unequal diffusion is introduced.
Let a stable two-species reaction equilibrium have Jacobianand diffusion matrix . A mode with has determinantSince its trace decreases with , diffusion-driven instability occurs exactly when
When , every spatial Fourier mode replaces the reaction Jacobian by . Each eigenvalue is shifted left by , so diffusion cannot destabilize a stable homogeneous equilibrium.
For reaction Jacobianthe critical diffusivity ratio isIt approaches one as the stable reaction system approaches marginality.
The Brusselator is a two-species autocatalytic reaction model. In the parametrizationits positive homogeneous equilibrium is .
For the spatially homogeneous Brusselator with parameters , set . The quadrilateralis a trapping region. The four inward-pointing tests use on , on , on , and on the sloping edge.
The spatially homogeneous Brusselator has its unique positive equilibrium at . Its Jacobian there has determinant and trace . If , the equilibrium is a repeller inside the Brusselator trapping region; the Poincare-Bendixson theorem then supplies a periodic orbit.
When and have diffusivities and , respectively, and the homogeneous Brusselator equilibrium is stable, a Turing instability occurs precisely whenAt onset,
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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.