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Modeling chemotaxis from \(L^2\)-closure moments in kinetic theory of active particles. (English) Zbl 1280.35160

The paper considers the derivation of macroscopic equations describing a spread of active particles from microscopic kinetics in absence of biological/biochemical interactions as well as in the case of a presence of reagents inducing chemotaxis process. The principal idea consists of the introduction of an additional microscopic state called activity and moment expansions. The first part of the work includes velocity-jump processes with an integral term (like a master-equation transition operator) on the right-hand side of Euler equations governing linear transport processes. The final macroscopic model is obtained by the usage of variational approach using the moment closure up to the second moment. The second part of the work considers a chemosensitive movement of a binary mixture. In this case, the velocity-dependent turning operator depends additionally on the concentration of the external signal, and on its gradient. The Keller-Segel model is derived as a partial example via two steps: i) closure of the three moment equations to get hyperbolic approximations and ii) passing to the parabolic limit for fast speeds and large turning rates.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C17 Cell movement (chemotaxis, etc.)
35K57 Reaction-diffusion equations
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