Introduction to the Modeling and Analysis of Complex Systems

14.3. LINEAR STABILITY ANALYSIS OF CONTINUOUS FIELD MODELS 281dynamical system. What we have done is to constrain the shape of the small perturbations(i.e., deviations from the homogeneous equilibrium state) to a certain eigenfunctionto eliminate spatial effects, and then ignore any higher-order terms to linearize the dynamics.Each point in this new (∆a, ∆c) phase space still represents a certain spatialconfiguration of the original model, as illustrated in Fig. 14.3. Studying the stability of theorigin (∆a, ∆c) = (0, 0) tells us if the Keller-Segel model can remain homogeneous or if itundergoes a spontaneous pattern formation.aDcaa eqcxa eqcxc eqxc eqxDaaaa eqcxa eqcxc eqxc eqxhomogeneousequilibrium stateFigure 14.3: Visual illustration of how to interpret each point in the (∆a, ∆c) phasespace. Each call-out shows the actual state of the system that corresponds to eachpoint in the phase space.The only thing we need to do is to calculate the eigenvalues of the matrix in Eq. (14.65)and check the signs of their real parts. This may be cumbersome, but let’s get it done.Here is the calculation process, where λ is the eigenvalue of the matrix: