Application of stochastic computer simulation to a solution of a nonlinear Fokker-Planck equation governing bacteria chemotaxis

A stochastic computer simulation technique has been used to solve a one-dimensional time-dependent Fokker-Planck (F-P) equation governing spacetime distribution of bacteria in a substrate (food) gradient. Because of consumption of the substrate by the bacteria the F-P equation becomes nonlinear due to coupling of the distribution with the self-generated substrate gradient. The simulation is efficient and numerically accurate for generating transient solutions. We are able not only to produce a transient solution displaying approach to the steady state solitary wave solution known from an analytical result of Keller and Segal but also to show an interesting dependence of the solitary wave propagation speed on the concentration dependence of the substrate consumption rate.