Sous-titre : PDE analysis of eco-evolutionary dynamics with quantitative traits in patchy and changing environments.
Ecosystem fragmentation is ubiquitous and presents mounting challenges to species' adaptation, especially when coupled with our current changing climate. To be able to predict how evolutionary trajectories are shaped by the tension between adaptation to local selection pressures and migration to reach more favourable conditions is increasingly pressing. This has been the subject of a long sustained interest from theoretical evolutionary biology and more recently from the mathematical community. I will first try to give an overview of the analytical frameworks that have been developed, before presenting how two of my contributions fit in their continuity. Both of these follow the same PDE framework aimed at studying how population dynamics of adaptation to a two-patch environment are influenced by sexual reproduction. In that context, the transmission of the quantitative trait by sexual reproduction can be modelled by a non-local collisional operator. In a small variance regime, this operator constrains the local distributions to be close to Gaussian distributions with fixed variance. I will show how this can lead to a separation between ecological and evolutionary time scales thanks to a slow-fast analysis. When the environment is stable, this allows to reduce the analysis to a phase-line study that identifies a particularly relevant evolutionary equilibrium describing a specialist species . Next, I will highlight how, when the environment is changing, the previous phase-line study can be greatly leveraged to reveal how specialist species undergo sharp dynamics of habitat switch corresponding to evolutionary tipping points