Turing Diffusion Instability Band

Worked Example: Unequal Diffusion and Instability

Consider a system with reaction Jacobian J = [[1, -2], [2, -3]] and diffusion matrix D = diag(1, 16). The homogenous state without diffusion is stable since its trace is -2 and determinant is 1. When diffusion is introduced and we test a Fourier mode with squared wavenumber q = 1/4, we evaluate the eigenvalues of J - qD.

Given D = 16, q = 1/4 Trace = -2 - (1 + D)q = -2 - (1 + 16)(1/4) = -2 - 17/4 = -6.25 Det = 1 + (3 - D)q + Dq^2 = 1 + (3 - 16)(1/4) + 16(1/4)^2 = 1 - 13/4 + 1 = -1.25 Growth (Max Eigenvalue) = (Trace + sqrt(Trace^2 - 4*Det)) / 2 = (-6.25 + sqrt(39.0625 - 4(-1.25))) / 2 = (-6.25 + sqrt(44.0625)) / 2 = 0.19397951183793838

Since the maximum growth rate is positive (approximately 0.194), this mode is growing, placing the system in an unstable band. The bounds of the unstable band can also be independently determined: q = (D - 3 ± sqrt((D - 3)² - 4D)) / (2D). For D = 16, the bounds are roughly 0.086 and 0.726.

Experiment: Changing Wavenumbers and Diffusivity

In this experiment, we explore how changing the diffusion ratio and the spatial mode impacts stability. Follow these steps in the interactive Turing Diffusion Instability Band module:

1. Set the Diffusivity ratio control to 16.
2. Set the Squared wavenumber control to 1/4. Observe the Mode stability indicator showing "Growing".
3. Change the Squared wavenumber to 0. The system behaves like the homogeneous case without diffusion, and the mode decays.
4. Change the Diffusivity ratio to 9 and the Squared wavenumber to 1/3. You will find that the system reaches a "Critical touch" with a maximum growth rate of exactly 0.
5. Finally, change the Diffusivity ratio to 1. With equal diffusion, the system cannot destabilize, and there is "No unstable band" regardless of the wavenumber.

Limitations

This linear stability analysis determines whether tiny spatial perturbations will initially grow. It is a local linear instability and does not predict the final nonlinear pattern saturation or guarantee that a finite domain will support a specific growing eigenmode. Additionally, the analysis applies to an infinite or periodic domain where Fourier modes are exact; bounded domains may restrict the available wavenumbers.

References

For a detailed mathematical treatment of reaction-diffusion systems and the Turing instability, see:

• UCSD Physics 221A Lectures on Nonlinear Dynamics, which outlines the conditions required for diffusion-driven instabilities, including unequal diffusion rates.