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The Langevin or Kramers approach to biological modeling. (English) Zbl 1057.92012

Summary: In the Langevin or Ornstein-Uhlenbeck approach to diffusion, stochastic increments are applied to the velocity rather than to the space variable. The density of this process satisfies a linear partial differential equation of the general form of a transport equation which is hyperbolic with respect to the space variable but parabolic with respect to the velocity variable, the Klein-Kramers or simply Kramers equation [see H. A. Kramers, Physica 7, 284-304 (1940; Zbl 0061.46405)]. This modeling approach allows for a more detailed description of individual movement and orientation dependent interaction than the frequently used reaction diffusion framework.
For the Kramers equation, moments are computed, the infinite system of moment equations is closed at several levels, and telegraph and diffusion equations are derived as approximations. Then nonlinearities are introduced such that the semi-linear reaction Kramers equation describes particles which move and interact on the same time-scale. Also for these nonlinear problems a moment approach is feasible and yields nonlinear damped wave equations as limiting cases.
We apply the moment method to the Kramers equation for chemotactic movement and obtain the classical Patlak-Keller-Segel model [C. S. Pattlak, Bull Math. Biophys. 15, 311–338 (1953); E. Keller and L. Segel, J. Theor. Biol. 26, 399–415 (1970)]. We discuss similarities between chemotactic movement of bacteria and gravitational movement of physical particles.

MSC:

92C17 Cell movement (chemotaxis, etc.)
92B05 General biology and biomathematics
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
35K57 Reaction-diffusion equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences

Citations:

Zbl 0061.46405
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References:

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