  We study the long-time behaviour of a run and tumble model which is a
kinetic-transport equation describing bacterial movement under the effect of a
chemical stimulus. The experiments suggest that the non-uniform tumbling
kernels are physically relevant ones as opposed to the uniform tumbling kernel
which is widely considered in the literature to reduce the complexity of the
mathematical analysis. We consider two cases: (i) the tumbling kernel depends
on the angle between pre- and post-tumbling velocities, (ii) the velocity space
is unbounded and the post-tumbling velocities follow the Maxwellian velocity
distribution. We prove that the probability density distribution of bacteria
converges to an equilibrium distribution with explicit (exponential for (i) and
algebraic for (ii)) convergence rates, for any probability measure initial
data. To the best of our knowledge, our results are the first results
concerning the long-time behaviour of run and tumble equations with non-uniform
tumbling kernels.
