Introduction to the Modeling and Analysis of Complex Systems

14.4. LINEAR STABILITY ANALYSIS OF REACTION-DIFFUSION SYSTEMS 293This means that the diffusion of v must be at least 4.5 times faster than u in order to causethe diffusion instability. In other words, u acts more locally, while the effects(of v reach) 1 −1over longer spatial ranges. If you look back at the original coefficient matrix,2 −1.5you will realize that u tends to increase both u and v, while v tends to suppress u andv. Therefore, this represents typical “short-range activation and long-range inhibition”dynamics that we discussed in Section 11.5, which is essential in many pattern formationprocesses.Figure 14.6 shows the numerical simulation results with the ratio of the diffusion constantssystematically varied. Indeed, a sharp transition of the results across ρ = 4.5 isactually observed! This kind of transition of a reaction-diffusion system’s behavior betweenhomogenization and pattern formation is called a Turing bifurcation, which Turinghimself showed in his monumental paper in the 1950s [44].Exercise 14.10Below is a variant of the Turing pattern formation model:∂u∂t = u(v − 1) − α + D u∇ 2 u (14.131)∂v∂t = β − uv + D v∇ 2 v (14.132)Here α and β are positive parameters. Let (α, β) = (12, 16) throughout this exercise.Do the following:1. Find its homogeneous equilibrium state.2. Examine the stability of the homogeneous equilibrium state without diffusionterms.3. With (D u , D v ) = (10 −4 , 10 −3 ), conduct a linear stability analysis of thismodel around the homogeneous equilibrium state to determine whether nonhomogeneouspatterns form spontaneously. If they do, estimate the lengthscale of the patterns.4. Determine the critical ratio of the two diffusion constants.5. Confirm your predictions with numerical simulations.