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

Abstract

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

On quantitative hypocoercivity estimates based on Harris-type theorems

“…A recent quantitative result concerning the hypocoercivity of this equation using Harris's theorem can be found in [35] where the authors could remove the radial symmetry assumption on S(x) of [50].…”

“…The main result of [35] is given by Theorem 3.7. Suppose that t → f t is the solution of the run and tumble equation (1.1)-(3.46) with the initial data f 0 ∈ P(R d × V).…”

“…All the constants appearing above are coming from the hypotheses. The computations can be found in Section 2.1 of [35] but here we discuss how to get an intuition about the shape of the weight function (3.49). We expect that Lyapunov functional will have exponential tails if we would like to have an exponential decay to equilibrium.…”

“…In [35], we consider another coupling for the run and tumble equation where S solves Then there exists some constant C depending on C 1 , C 2 , and α such that if η < C then there exists a unique steady state solution to the nonlinear run and tumble equation (1.1)- (3.46) where S solves (3.51). Moreover for any initial data f 0 ∈ P(R d × V) satisfying f 0 φ ≤ C with φ = e σ x where the constant C > 0 depends on ψ ′ ∞ , ∇ x N ∞ and some other constants coming from the previous hypotheses and Theorem 3.7, there exist some positive constants C and λ satisfying…”

“…This coupling plays the role of an intermediate step between the linear model (1.1)-(3.46) and the nonlinear model (1.1)-(3.46)-(3.50). The motivation behind (3.51) and how it is obtained are dicussed in the last section of[35]. The result concerning the nonlinear equation where S solves (3.51) is as follows: Suppose that t → f t is the solution of the run and tumble equation (1.1)-(3.46) where S solves (3.51) with the initial data f 0 ∈ P(R d × V).…”