Author:

Keywords:

Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, Asymptotic variance reduction, bacterial chemotaxis, velocity-jump process, DIFFUSION LIMIT, EQUATIONS, math.NA, 0102 Applied Mathematics, 0103 Numerical and Computational Mathematics, Applied Mathematics, 4901 Applied mathematics

Abstract:

We discuss variance reduced simulations for an individual-based model of chemotaxis of bacteria with internal dynamics. The variance reduction is achieved via a coupling of this model with a simpler process in which the internal dynamics has been replaced by a direct gradient sensing of the chemoattractants concentrations. In the companion paper ["Individual-based models for bacterial chemotaxis in the diffusion limit" (to appear in Math. Models Methods Appl. Sci., DOI: 10.1142/S0218202513500243)], we have rigorously shown, using a pathwise probabilistic technique, that both processes converge towards the same advection-diffusion process in the diffusive asymptotics. In this work, a direct coupling is achieved between paths of individual bacteria simulated by both models, by using the same sets of random numbers in both simulations. This coupling is used to construct a hybrid scheme with reduced variance. We first compute a deterministic solution of the kinetic density description of the direct gradient sensing model; the deviations due to the presence of internal dynamics are then evaluated via the coupled individual-based simulations. We show that the resulting variance reduction is asymptotic, in the sense that, in the diffusive asymptotics, the difference between the two processes has a variance which vanishes according to the small parameter. © 2013 World Scientific Publishing Company.