Trend to Equilibrium for Run and Tumble Equations with Non-uniform Tumbling Kernels

Trend to Equilibrium for Run and Tumble Equations with Non-uniform Tumbling Kernels

Evans, Josephine (University of Warwick)
Yoldas, H. (TU Delft Mathematical Physics)

2024

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.

Run and tumble equation
Hypocoercivity
Kinetic equations
Harris’s theorem

http://resolver.tudelft.nl/uuid:770479d8-36c9-43c9-8d3b-8ae90766a85c

https://doi.org/10.1007/s10440-024-00657-y

1572-9036

Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications, 191

Institutional Repository

journal article

© 2024 Josephine Evans, H. Yoldas