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A link between microscopic and macroscopic models of self-organized aggregation. (English) Zbl 1262.35202

Summary: In some species, one of the roles of pheromones is to influence aggregation behavior. We first propose a macroscopic cross-diffusion model for the self-organized aggregation of German cockroaches that includes directed movement due to an aggregation pheromone. We then propose a microscopic particle model which is set into context with the macroscopic model. Our goal is to link the macroscopic and microscopic descriptions by using the singular and the hydrodynamic limit procedures. A hybrid model related to the macroscopic and microscopic models is also proposed as a cockroach aggregation model. This hybrid model assumes that each individual responds to pheromone concentration and moves by two-mode simple symmetric random walks. It shows that even though the movement of individuals is not directed, two-mode simple symmetric random walks and effect of the pheromone result in self-organized aggregation.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K40 Second-order parabolic systems
35K57 Reaction-diffusion equations
60K35 Interacting random processes; statistical mechanics type models; percolation theory
92D25 Population dynamics (general)
92C17 Cell movement (chemotaxis, etc.)
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