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Congestion-driven dendritic growth. (English) Zbl 1280.35065

Summary: In order to observe growth phenomena in biology where dendritic shapes appear, we propose a simple model where a given population evolves feeded by a diffusing nutriment, but is subject to a density constraint. The particles (e.g., cells) of the population spontaneously stay passive at rest, and only move in order to satisfy the constraint \(\rho \leq 1\), by choosing the minimal correction velocity so as to prevent overcongestion. We treat this constraint by means of projections in the space of densities endowed with the Wasserstein distance \(W_2\), defined through optimal transport. This allows to provide an existence result and suggests some numerical computations, in the same spirit of what the authors did for crowd motion (but with extra difficulties, essentially due to the fact that the total mass may increase). The numerical simulations show, according to the values of the parameter and in particular of the diffusion coefficient of the nutriment, the formation of dendritic patterns in the space occupied by cells.

MSC:

35K57 Reaction-diffusion equations
49J45 Methods involving semicontinuity and convergence; relaxation
49M25 Discrete approximations in optimal control
92B05 General biology and biomathematics
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