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Pré-Publication, Document De Travail Année : 2021

On the asymptotic behaviour of a run and tumble equation for bacterial chemotaxis

Résumé

We study the asymptotic behaviour of the run and tumble model for bacteria movement. Experiments show that under the effect of a chemical stimulus, the movement of bacteria is a combination of a transport with a constant velocity, "run", and a random change in the direction of the movement, "tumble". This so-called velocity jump process can be described by a kinetic-transport equation. We focus on the situation for bacteria called E. Coli where the tumbling rate depends on a chemical stimulus but the post tumbling velocities do not. In this paper, we show that the linear run and tumble equation converges to a unique steady state solution with an exponential rate in a weighted total variation distance in dimension d ≥ 1. We provide a constructive and quantitative proof by using Harris's Theorem from ergodic theory of Markov processes. The result is an improvement of a recent paper by Mischler and Weng [40], since we are able to remove the radial symmetry assumption on the chemoattractant concentration. We also consider a weakly non-linear run and tumble equation by coupling it with a nonlocal equation on the chemoattractant concentration. We construct a unique stationary solution for the weakly nonlinear equation and show the exponential convergence towards it. The novelty of our paper consist in our generalisation of the spectral gap result to dimension d ≥ 1 under relaxed assumptions and the methods we used in the linear setting; and, all the results in the non-linear setting. Contents
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hal-03186374 , version 1 (02-04-2021)

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  • HAL Id : hal-03186374 , version 1

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Josephine Evans, Havva Yoldaş. On the asymptotic behaviour of a run and tumble equation for bacterial chemotaxis. 2021. ⟨hal-03186374⟩
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