The goal of MUSEUM is to develop robust methods for the asymptotic analysis of unconventional multi-scale integro-differential equations arising in evolutionary biology, and to investigate their connection with underlying 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. The sexual reproduction can indeed be modeled by nonlocal and nonlinear terms of collision type. 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 explicitly incorporating demographic stochasticity. In particular, Hamilton-Jacobi equations have been derived from integro-differential equations under the assumption of small mutational effects. These integro-differential equations themselves arise as limits of stochastic, individual-based models considering large populations. However, the order in which these limiting procedures are applied can affect the outcome, since they do not always commute, potentially leading to inaccuracies in the resulting deterministic approximations. To address this issue, we will perform asymptotic analyses of stochastic processes that account for both limits simultaneously. This unified approach will allow us to identify and characterize new limiting objects, thereby overcoming the limitations of conventional Hamilton-Jacobi approximations.
In collaboration with biologists, we will propose practical tools to analyze the evolutionary dynamics of complex traits, ensuring that our advancements are effectively transferred to the biological community. Our research focuses, for example, on the evolutionary adaptation of age-structured pathogen traits during epidemics, as well as on the adaptive evolution of size-structured traits in plant populations.
This project is funded by the European Research Council.
PhD students