Dynamics of a kinetic model describing protein transfers in a cell population

We consider a cell population structured by a positive real number $x\in R_{+}$ , which represents the number of P-glycoproteins carried by the cell. These proteins combine two interesting properties: they are involved in the resistance of cancer cells to chemotherapy drugs, and the cells undergo frequent transfers of those proteins. In this article, we introduce a kinetic model to describe the dynamics of the cell population. We then consider an asymptotic limit of this equation: if transfers are frequent, the population can be described through a system of two coupled ordinary differential equations. Finally, we show that the solutions of the kinetic model converge to a unique steady-state in large times. The main idea of this manuscript is to combine Wasserstein distance estimates on the kinetic operator with more classical estimates on the macroscopic quantities.