  The motile micro-organisms such as E. coli, sperm, or some seaweed are
usually modelled by self-propelled particles that move with the run-and-tumble
process. Individual-based stochastic models are usually employed to model the
aggregation phenomenon at the boundary, which is an active research field that
has attracted a lot of biologists and biophysicists. Self-propelled particles
at the microscale have complex behaviors, while characteristics at the
population level are more important for practical applications but rely on
individual behaviors. Kinetic PDE models that describe the time evolution of
the probability density distribution of the motile micro-organisms are widely
used. However, how to impose the appropriate boundary conditions that take into
account the boundary aggregation phenomena is rarely studied. In this paper, we
propose the boundary conditions for a 2D confined run-and-tumble model (CRTM)
for self-propelled particle populations moving between two parallel plates with
a run-and-tumble process. The proposed model satisfies the relative entropy
inequality and thus long-time convergence. We establish the relation between
CRTM and the confined Fokker-Planck model (CFPM) studied in [22]. We prove
theoretically that when the tumble is highly forward peaked and frequent
enough, CRTM converges asymptotically to the CFPM. A numerical comparison of
the CRTM with aggregation and CFPM is given. The time evolution of both the
deterministic PDE model and individual-based stochastic simulations are
displayed, which match each other well.
