Kinetic modeling of congestion constraints: application to bacterial collective dynamics
Résumé
In this paper, we study a kinetic model describing congestion effects. We introduce a first-order model coupled with hard congestion complementarity conditions. Several properties of this model are analyzed and a numerical method is proposed for its resolution. As an application, we extend this framework to the biological context of the collective motion of the social and predatory soil bacterium \textit{Myxococcus xanthus}. We focus on the rippling phenomenon, characterized by localized back-and-forth movements of the bacteria that generate spatio-temporal wave trains at the macroscopic level. In our model, a congestion-generated pressure defines a new non-local signal perceived by the bacteria. Combined with a modulation of the bacteria's refractory period, these ingredients allow periodic patterns to emerge. Several properties of the resulting model are investigated through numerical experiments.
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