Introduction to the Modeling and Analysis of Complex Systems

284 CHAPTER 14. CONTINUOUS FIELD MODELS II: ANALYSISthan l c are going to shrink and disappear. This critical length scale tells us the characteristicdistance between aggregated points (or cities) spontaneously forming at the beginningof the process.We can confirm these analytical results with numerical simulations. Figure 14.4 showssimulation results with µ = 10 −4 , D = 10 −4 , f = 1, k = 1, and a eq = 1, while χ is variedas a control parameter. With these parameter values, inequality (14.76) predicts that thecritical value of χ above which aggregation occurs will beχ c = µka eq f = 10−4 , (14.79)which is confirmed in the simulation results perfectly.χ = 5.0 × 10 −5 χ = 1.0 × 10 −4 χ = 1.5 × 10 −4 χ = 3.0 × 10 −4Figure 14.4: Numerical simulation results of the Keller-Segel model with µ = 10 −4 ,D = 10 −4 , f = 1, k = 1, and a eq = 1. Cell densities are plotted in grayscale (darker =greater). The value of χ is given below each result.This concludes a guided tour of the linear stability analysis of continuous field models.It may have looked rather complicated, but the key ideas are simple and almost identicalto those of linear stability analysis of non-spatial models. Here is a summary of theprocedure:Linear stability analysis of continuous-field models1. Find a homogeneous equilibrium state of the system you are interested in.2. Represent the state of the system as a sum of the homogeneous equilibriumstate and a small perturbation function.3. Represent the small perturbation function as a product of a dynamic amplitude