Motile microorganisms such as Escherichia coli, sperm cells, and certain species of seaweed are commonly modeled as self-propelled particles undergoing a run-and-tumble process. Individual-based stochastic models are often employed to study boundary aggregation phenomena, an active research area that has attracted considerable attention from biologists and biophysicists.

Although self-propelled particles exhibit complex behaviors at the microscale, population-level characteristics are often more relevant for practical applications and depend fundamentally on individual dynamics. Kinetic PDE models, which describe the time evolution of the probability density distribution of motile microorganisms, are therefore widely used. However, the formulation of appropriate boundary conditions that account for boundary aggregation phenomena has received relatively little attention.

In this talk, we propose boundary conditions for a confined run-and-tumble model (CRTM) describing populations of self-propelled particles moving between two parallel plates. The proposed model satisfies a relative entropy inequality, which implies long-time convergence to equilibrium. We further establish connections between the CRTM, a confined Fokker–Planck model, and a confined Keller–Segel model. We prove that, 1) when tumbling is sufficiently frequent and highly forward-peaked, the CRTM converges asymptotically to the confined Fokker–Planck model; 2) when the tumbling is sufficiently frequent and the cells are hard to escape from the wall, the CRTM converges to the confined Keller–Segel model.

Finally, numerical simulations of the deterministic PDE model are compared with individual-based stochastic simulations. The results show excellent agreement between the two approaches.