In this thesis I study a model of self propelled particles exhibiting run-and tumble
dynamics on lattice. This non-Brownian diffusion is characterised by
a random walk with a finite persistence length between changes of direction,
and is inspired by the motion of bacteria such as Escherichia coli. By defining
a class of models with multiple species of particle and transmutation between
species we can recreate such dynamics. These models admit exact analytical
results whilst also forming a counterpart to previous continuum models of run-and-
tumble dynamics. I solve the externally driven non-interacting and zero-range
versions of the model exactly and utilise a field theoretic approach to
derive the continuum fluctuating hydrodynamics for more general interactions. I
make contact with prior approaches to run-and-tumble dynamics of lattice and
determine the steady state and linear stability for a class of crowding interactions,
where the jump rate decreases as density increases.
In addition to its interest from the perspective of nonequilibrium statistical
mechanics, this lattice model constitutes an efficient tool to simulate a class
of interacting run-and-tumble models relevant to bacterial motion. Pattern
formation in bacterial colonies is confirmed to be able to stem solely from the
interplay between a diffusivity that depends on the local bacterial density and
regulated division of the cells, in particular without the need for any explicit
chemotaxis. This simple and generic mechanism thus provides a null hypothesis
for pattern formation in bacterial colonies which has to be falsified before
appealing to more elaborate alternatives. Most of the literature on bacterial
motility relies on models with instantaneous tumbles. As I show, however, the finite tumble duration can play a major role in the patterning process. Finally a
connection is made to some real experimental results and the population ecology
of multiple species of bacteria competing for the same resources is considered