Kinetic PDEs and inverse problems: an application in mathematical biology

Kinetic equations provide a robust mesoscopic framework for describing particle-based physical phenomena, such as the run-and-tumble motion of bacteria. In the presence of external stimuli, this motion becomes biased, leading to chemotaxis. We investigate the inverse problem of inferring the velocity-jump kernel, the fundamental driver of bacterial motion, under a fixed stimulus concentration. Experimental challenges suggest the use of macroscopic density measurements to infer the microscopic parameters. We bridge this scaling discrepancy through a strategically chosen experimental design, inspired from kinetic theory. This allows us to provide an analytical proof for the existence and uniqueness of the reconstruction. After relaxation to a more practical setting, numerical simulations demonstrate the efficacy of this design, recommending future laboratory implementations. Another line of research examines the multiscale behavior of the model in an inverse setting, where we show that the reconstruction remains consistent across the transition from the scaled kinetic model to its diffusion limit.

This is joint work with Christian Klingenberg (Wuerzburg, Germany), Qin Li (Madison, Wisc., USA) and Min Tang (Shanghai, China).