The goal of MUSEUM is to develop robust methods for the asymptotic analysis of unconventional multi-scale integro-differential equations from evolutionary biology and their connection with stochastic processes. These equations involve nonstandard nonlinear integral terms and, under specific conditions of small variance and long time, exhibit solutions that concentrate around one or multiple evolving points. A recent theory, involving Hamilton-Jacobi equations, offers robust methods for the asymptotic analysis of integro-differential models with linear asexual reproduction.
In MUSEUM, we will design new approaches for the asymptotic study of models with nonlinear reproduction operators in the regime of small variance. In such models, the nonlinearity arises from sexual reproduction. Existing methods based on Hamilton-Jacobi equations, which rely on comparison principles and viscosity solution theory, are not well suited to handle the complexities introduced by nonlinear sexual reproduction. Despite the rich structure brought by these nonlinear operators, their mathematical analysis remains significantly underdeveloped.
We will also develop new methods to improve deterministic approximations by accounting for demographic stochasticity. Hamilton-Jacobi equations have been derived from integro-differential equations under the assumption of small mutational effects, where these integro-differential equations themselves originate from stochastic individual-based models in the large population limit. However, these limiting procedures do not always commute, potentially resulting in inaccuracies in deterministic approximations. To address this, we will conduct asymptotic analyses of stochastic processes that consider both limits simultaneously and analyze the new limiting objects. This approach will help overcome the limitations of standard Hamilton-Jacobi approximations.
In collaboration with biologists, we will propose new tools to analyze the evolutionary dynamics of complex traits. Our research focuses, for example, on the evolutionary adaptation of age-dependent traits in pathogens during epidemics, as well as on the adaptive evolution of size-structured plant species.
This project is funded by the European Research Council.
PhD students