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Global existence for a parabolic chemotaxis model with prevention of overcrowding. (English) Zbl 0998.92006

Summary: We study a version of the Keller-Segel model [E.F. Keller and L.A. Segel, J. Theor. Biol. 26, 399-415 (1970)] where the chemotactic cross-diffusion depends on both the external signal and the local population density. A parabolic quasi-linear strongly coupled system follows. By incorporation of a population-sensing (or “quorum-sensing”) mechanism, we assume that the chemotactic response is switched off at high cell densities. The response to high population densities prevents overcrowding, and we prove local and global existence in time of classical solutions. Numerical simulations show interesting phenomena of pattern formation and formation of stable aggregates. We discuss the results with respect to previous analytical results on the Keller-Segel model.

MSC:

92C17 Cell movement (chemotaxis, etc.)
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K25 Higher-order parabolic equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
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