Confined run-and-tumble model with boundary aggregation: long time behavior and convergence to the confined Fokker-Planck model

TL;DR

The paper develops a kinetic model for self-propelled particles confined between two parallel plates, introducing a bulk density $n(t,y,\theta)$ and boundary densities $n_{\pm}(t,\theta)$ with switching kernels. It proves mass conservation and a relative entropy inequality, ensuring the solution converges to a unique steady state $(m,m_{+},m_{-})$, and shows that under a forward-peaked tumbling scaling with parameter $\varepsilon$, the CRTM converges to the confined Fokker-Planck model ($CFPM$) studied previously, linking boundary-aggregation dynamics to a diffusion-limit description. Numerical PDE and stochastic simulations validate the theory and demonstrate boundary aggregation and agreement between CRTM and CFPM, enabling inference of effective diffusion coefficients from tumbling data. Overall, the work builds a rigorous bridge from microscopic run-and-tumble kinetics to a boundary-aware macroscopic description in confined geometries, with potential for extracting model coefficients directly from experimental tumbling rates.

Abstract

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.