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arXiv · 2203.10809

Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat

Josué Tchouanti
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Abstract

We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from the modeling of bacterial growth in a chemostat. This system is derived, in a large population approximation, from a stochastic individual-based model where each individual is characterized by a non-negative real valued trait described by a diffusion. Two uniqueness results are highlighted. They differ in their hypotheses related to the influence of the resource on individual trait dynamics, the main difficulty being the non-linearity due to this dependence and the degeneracy of the diffusion coefficient. Further we show that the semi-group of the stochastic trait dynamics admits a density by probabilistic arguments, that allows the measure solution of the diffusiongrowth-fragmentation equation to be a function with a certain Besov regularity.

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Josué Tchouanti. 2023-03-21. Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat. https://doi.org/10.1007/s40072-023-00288-8

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