Section: Research Program

PDE analysis and simulation

PDEs arise at several levels of our models. Parabolic equations  (B. Perthame, Parabolic equations in biology, Springer, 2015) can be used for large cell populations and also for intracellular spatio-temporal dynamics of proteins and their messenger RNAs in gene regulatory networks, transport equations  (B. Perthame, Transport equations in biology, Springer, 2007) are used for protein aggregation / fragmentation models and for the cell division cycle in age-structured models of proliferating cell populations. Existence, uniqueness and asymptotic behaviour of solutions have been studied  [65], [62]. Other equations, of the integro-differential type, dedicated to describing the Darwinian evolution of a cell population according to a phenotypic trait, allowing exchanges with the environment, genetic mutations and reversible epigenetic modifications, are also used  [81], [80], [79], [82], possibly enriched to classical PDEs by the adjunction of diffusion and advection terms  [63]. Through multiscale analysis, they can be related to stochastic and free boundary models used in cancer modelling.