Introduction to the Modeling and Analysis of Complex Systems

14.4. LINEAR STABILITY ANALYSIS OF REACTION-DIFFUSION SYSTEMS 285and a static shape of the perturbation, which is chosen to be an eigenfunction ofthe spatial linear operator remaining in the equation (most likely just sine wavesfor ∇ 2 ).4. Eliminate the spatial linear operator and ignore higher-order terms of small perturbationsto simplify the equation into a linear non-spatial form.5. Calculate the eigenvalues of the resulting coefficient matrix.6. If the real part of the dominant eigenvalue is:• Greater than 0 ⇒ The homogeneous equilibrium state is unstable.• Less than 0 ⇒ The homogeneous equilibrium state is stable.• Equal to 0 ⇒ The homogeneous equilibrium state may be neutral (Lyapunovstable).7. In addition, if there are complex conjugate eigenvalues involved, oscillatory dynamicsare going on around the homogeneous equilibrium state.Finally, we should also note that this eigenfunction-based linear stability analysis ofcontinuous-field models works only if the spatial dynamics are represented by a local lineardifferential operator (e.g., ∂f/∂x, Laplacian, etc.). Unfortunately, the same approachwouldn’t be able to handle global or nonlinear operators, which often arise in mathematicalmodels of real-world phenomena. But this is beyond the scope of this textbook.Exercise 14.5 Add terms for population growth and decay to the first equation ofthe Keller-Segel model. Obtain a homogeneous equilibrium state of the revisedmodel, and then conduct a linear stability analysis. Find out the condition for whichspontaneous pattern formation occurs. Interpret the result and discuss its implications.14.4 Linear Stability Analysis of Reaction-Diffusion SystemsYou may have found that the linear stability analysis of continuous field models isn’t aseasy as that of non-spatial models. For the latter, we have a very convenient tool called