  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,
\emph{"run"}, and a random change in the direction of the movement,
\emph{"tumble"}. This so-called \emph{velocity jump process} can be described
by a kinetic-transport equation. We focus on the situation for bacteria called
\emph{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 $d\geq 1$. 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
\cite{MW17}, since we are able to remove the radial symmetry assumption on the
chemoattractant concentration. We also consider a \emph{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 non-linear equation and show the exponential convergence towards it.
The novelty of our paper consist in our generalisation of the spectral gap
result to dimension $d\geq 1$ under relaxed assumptions and the methods we used
in the linear setting; and, all the results in the non-linear setting.
