IS32: Stochastic Eco-Evolutionary Models

Organizer: Nicolas Champagnat (Inria Center at Université de Lorraine)

Convergence of a general structured individual-based model with possibly unbounded growth, birth and death rates

Virgile BRODU

First, we introduce càdlàg measure-valued processes, with biological motivations. We focus on a construction with Poisson point measures, with useful martingale properties. Then, we present a general convergence result for these measure-valued processes. In the literature, such a convergence result is proven with bounded growth, birth and death rates (\([1]\),\([2]\),\([3]\),\([4]\),\([5]\)). We present a more general framework with unbounded rates, where the convergence still holds true in a weighted space of measures. This is joint work with Nicolas Champagnat and Coralie Fritsch.

Bibliography

\([1]\) Nicolas Fournier and Sylvie Méléard. "A microscopic probabilistic description of a locally regulated population and macroscopic approximations.” The Annals of Applied Probability, vol. 14, no. 4, 2004, pp. 1880-1919.

\([2]\) Nicolas Champagnat, Régis Ferrière and Sylvie Méléard. "Individual-based probabilistic models of adaptive evolution and various scaling approximations.” Seminar on Stochastic Analysis, Random Fields and Applications V: Centro Stefano Franscini, Ascona, May 2005, 2008, pp. 75-113.

\([3]\) Viet Chi Tran. "Large population limit and time behaviour of a stochastic particle model describing an age-structured population." ESAIM: PS, vol. 12, 2008, pp. 345-386.

\([4]\) Fabien Campillo, Nicolas Champagnat and Coralie Fritsch. "Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models." Journal of Mathematical Biology, vol. 73, no. 6-7, 2016, pp. 1781-1821.

\([5]\) Josué Tchouanti. "Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat." Stochastics and Partial Differential Equations: Analysis and Computations, vol .12, no. 1, 2024, pp. 466-524.