Introduction to the Modeling and Analysis of Complex Systems

282 CHAPTER 14. CONTINUOUS FIELD MODELS II: ANALYSIS∣ −µω2 − λ χa eq ω 2f −Dω 2 − k − λ ∣ = 0 (14.66)(−µω 2 − λ)(−Dω 2 − k − λ) − χa eq ω 2 f = 0 (14.67)λ 2 + (µω 2 + Dω 2 + k)λ + µω 2 (Dω 2 + k) − χa eq ω 2 f = 0 (14.68)λ = 1 (−(µω 2 + Dω 2 + k) ±√(µω22 + Dω 2 + k) 2 − 4 ( µω 2 (Dω 2 + k) − χa eq ω 2 f ))(14.69)Now, the question is whether either eigenvalue’s real part could be positive. Since all theparameters are non-negative in this model, the term before “±” can’t be positive by itself.This means that the inside the radical must be positive and sufficiently large in order tomake the real part of the eigenvalue positive. Therefore, the condition for a positive realpart to arise is as follows:√(µω 2 + Dω 2 + k) < (µω 2 + Dω 2 + k) 2 − 4 ( µω 2 (Dω 2 + k) − χa eq ω 2 f ) (14.70)If this is the case, the first eigenvalue (with the “+” operator in the parentheses) is realand positive, indicating that the homogeneous equilibrium state is unstable. Let’s simplifythe inequality above to get a more human-readable result:(µω 2 + Dω 2 + k) 2 < (µω 2 + Dω 2 + k) 2 − 4 ( µω 2 (Dω 2 + k) − χa eq ω 2 f ) (14.71)0 < −4 ( µω 2 (Dω 2 + k) − χa eq ω 2 f ) (14.72)χa eq ω 2 f > µω 2 (Dω 2 + k) (14.73)χa eq f > µ(Dω 2 + k) (14.74)At last, we have obtained an elegant inequality that ties all the model parameters togetherin a very concise mathematical expression. If this inequality is true, the homogeneousequilibrium state of the Keller-Segel model is unstable, so it is expected that the systemwill show a spontaneous pattern formation. One of the benefits of this kind of mathematicalanalysis is that we can learn a lot about each model parameter’s effect on patternformation all at once. From inequality (14.74), for example, we can make the following predictionsabout the aggregation process of slime mold cells (and people too, if we considerthis a model of population-economy interaction):• χ, a eq , and f on the left hand side indicate that the aggregation of cells (or theconcentration of population in major cities) is more likely to occur if