Introduction to the Modeling and Analysis of Complex Systems

13.4. MODELING SPATIAL MOVEMENT 247• Cells neither die nor divide.Did you notice the similarity between these assumptions and those we used for thepopulation-economy model? Indeed, they are identical, if you replace “cells” with “people”and “cAMP” with “money.” We could use the word “moneytaxis” for people’s migrationtoward economically active areas!The actual Keller-Segel equations look like this:∂a∂t = µ∇2 a − χ∇ · (a∇c) (13.27)∂c∂t = D∇2 c + fa − kc (13.28)In fact, these equations are simplified ones given in [49], as the original equations wererather complicated with more biological details. a and c are the state variables for celldensity and cAMP concentration, respectively. µ is the parameter for cell mobility, χ isthe parameter for cellular chemotaxis, D is the diffusion constant of cAMP, f is the rate ofcAMP secretion by the cells, and k is the rate of cAMP decay. Compare these equationswith Eqs. (13.25) and (13.26). They are the same! It is intriguing to see that completelydifferent phenomena at vastly distant spatio-temporal scales could be modeled in an identicalmathematical formulation.It is known that, for certain parameter settings (which will be discussed in the nextchapter), this model shows spontaneous pattern formation from almost homogeneousinitial conditions. A sample simulation result is shown in Fig. 13.12, where we see spots ofaggregated cell clusters forming spontaneously. You will learn how to conduct simulationsof continuous field models in the next section.In Fig. 13.12, we can also observe that there is a characteristic distance betweennearby spots, which is determined by model parameters (especially diffusion constants).The same observation applies to the formation of cities and towns at geographical scales.When you see a map, you will probably notice that there is a typical distance betweenmajor cities, which was probably determined by the human mobility centuries ago, amongother factors.Exercise 13.13 Consider introducing to the Keller-Segel model a new variable bthat represents the concentration of a toxic waste chemical. Make the followingassumptions:• cAMP gradually turns into the waste chemical (in addition to natural decay).• The waste chemical diffuses over space.