Introduction to the Modeling and Analysis of Complex Systems

14.3. LINEAR STABILITY ANALYSIS OF CONTINUOUS FIELD MODELS 283– the cells’ chemotaxis (or people’s “moneytaxis”) is stronger (χ),– there are more cells (or people) in the system (a eq ), and/or– the cells (or people) produce cAMP molecules (or economic values) at a fasterpace (f).• µ, D, and k on the right hand side indicate that the aggregation of cells (or theconcentration of population in major cities) is more likely to be suppressed if– the cells and cAMP molecules (or people and economic values) diffuse faster(µ and D), and/or– the cAMP molecules (or economic values) decay more quickly (k).It is quite intriguing that such an abstract mathematical model can provide such a detailedset of insights into the problem of urbanization (shift of population and economy from ruralto urban areas), one of the critical socio-economical issues our modern society is facingtoday. Isn’t it?Moreover, solving inequality (14.74) in terms of ω 2 gives us the critical condition betweenhomogenization and aggregation:χa eq f − µkµD> ω 2 (14.75)Note that ω can be any real number but ω 2 has to be non-negative, so aggregation occursif and only ifχa eq f > µk. (14.76)And if it does, the spatial frequencies of perturbations that are going to grow will be givenby√ω2π < 1 χa eq f − µk. (14.77)2π µDThe wave length is the inverse of the spatial frequency, so we can estimate the length ofthe growing perturbations as follows:√l = 2π ω > 2π µDχa eq f − µk = l c (14.78)This result means that spatial perturbations whose spatial length scales are greater thanl c are going to grow and become visible, while perturbations with length scales smaller