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.

翻译：诸如大肠杆菌、精子或某些海藻等运动微生物，通常通过遵循游走-翻滚过程的自主运动粒子进行建模。基于个体的随机模型常被用于模拟边界上的聚集现象，这是一个吸引众多生物学家和生物物理学家关注的前沿研究领域。微观尺度下自主运动粒子行为复杂，而实际应用中更关注群体层面的特征，但后者又依赖于个体行为。描述运动微生物概率密度分布时间演化的动力学偏微分方程模型被广泛使用，然而如何施加考虑边界聚集现象的恰当边界条件却鲜有研究。本文针对在两块平行平板间通过游走-翻滚过程运动的自主运动粒子群，提出了二维约束游走-翻滚模型（CRTM）的边界条件。该模型满足相对熵不等式，因而具备长时间收敛性。我们建立了CRTM与文献[22]中研究的约束Fokker-Planck模型（CFPM）之间的关系。理论证明，当翻滚方向高度集中且频率足够高时，CRTM渐近收敛于CFPM。文中给出了带聚集效应的CRTM与CFPM的数值对比，并展示了确定性偏微分方程模型与基于个体的随机模拟的时间演化结果，两者吻合良好。