Introduction to the Modeling and Analysis of Complex Systems

246 CHAPTER 13. CONTINUOUS FIELD MODELS I: MODELINGEvelyn Keller and Lee Segel, to describe a completely different biological phenomenon[48, 49]. The Keller-Segel model was developed to describe chemotaxis—movementof organisms toward (or away from) certain chemical signals—of a cellular slime moldspecies Dictyostelium discoideum. These slime mold cells normally behave as individualamoebae and feed on bacteria, but when the food supply becomes low, they first spreadover the space and then aggregate to form “slugs” for long-distance migration (Fig. 13.11).Keller and Segel developed a mathematical model of this aggregation process, wherechemotaxis plays a critical role in cellular self-organization.Figure 13.11: Dictyostelium showing aggregation behavior. Image from WikimediaCommons (“Dictyostelium Aggregation” by Bruno in Columbus – Own work. Licensedunder Public domain via Wikimedia Commons – http://commons.wikimedia.org/wiki/File:Dictyostelium_Aggregation.JPG)Keller and Segel made the following assumptions based on what was already knownabout these interesting cellular organisms:• Cells are initially distributed over a space more or less homogeneously (with somerandom fluctuations).• Cells show chemotaxis toward a chemical signal called cAMP (cyclic adenosinemonophosphate).• Cells produce cAMP molecules.• Both cells and cAMP molecules diffuse over the space.