MUlti-Scale models of Eco-evolutionary popUlation dynaMics

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.We will develop new methods for the asymptotic analysis of models with nonlinear reproduction operators, in the regime of small variance.

In these models, the nonlinearity of the reproduction operator comes from sexual reproduction, involving two parents, as opposed to asexual reproduction, which only involves one parent and which can be modeled by linear operators. Both of these reproductive modes are relevant to many living organisms.

A recent theory, that encompasses Hamilton-Jacobi equations offers robust methods for the asymptotic analysis of integro-differential models with linear asexual reproduction.

However, these methods, relying on comparison principles and viscosity solution theory, prove inadequate in the case of nonlinear sexual reproduction. Nonlinear reproduction operators introduce new features, yet their analysis remains considerably underdeveloped.

Their treatment requires a conceptual leap.Addressing stochastic effects in small subpopulations, we will also analyse large population and long time limits of stochastic individual-based models in the small variance regime.

In this way, we will palliate the inadequacies of Hamilton-Jacobi approximations.The methods that we will develop through this project will offer theoretical biologists the opportunity to go beyond their current possibilities and explore new frontiers.