Particles approximations of Vlasov equations with singular forces : Propagation of chaos

We obtain the mean field limit and the propagation of chaos for a system of particles interacting with a singular interaction force of the type $1/|x|^\alpha$, with $\alpha <1$ in dimension $d \geq 3$. We also provide results for forces with singularity up to $\alpha < d-1$ but with large enough cut-off. This last result thus almost includes the most interesting case of Coulombian or gravitational interaction, but it is also interesting when the strength of the singularity $\alpha$ is larger but close to one, in which case it allows for very small cut-off.Comment: 47 pages. This version: improvement of the presentation, in particular of the deterministic result

Cellulose Biodegradation Models; An Example of Cooperative Interactions in Structured Populations

We introduce various models for cellulose bio-degradation by micro-organisms. Those models rely on complex chemical mechanisms, involve the structure of the cellulose chains and are allowed to depend on the phenotypical traits of the population of micro-organisms. We then use the corresponding models in the context of multiple-trait populations. This leads to classical, logistic type, reproduction rates limiting the growth of large populations but also, and more surprisingly, limiting the growth of populations which are too small in a manner similar to the effects seen in populations requiring cooperative interactions (or sexual reproduction). This study hence offers a striking example of how some mechanisms resembling cooperation can occur in structured biological populations, even in the absence of any actual cooperation.Comment: 37 pages, accepted to ESAIM: Mathematical Modelling and Numerical Analysis (2017