Environmental changes threaten many species and ecosystems. To assess their impacts, we use a mathematical approach based on reaction dispersion models. The first aim of my lecture is to derive relevant mathematical models with biological interpretation at different spatial and ecological scales (from individuals to metapopulations). Secondly, I will present classical and recent mathematical tools to quantify the ecological and evolutionary adaptation of species to environmental changes.
In the first part, I will present a recent approach based on the interior dynamics of stationary states, to understand the effect of migration on survival and genetic diversity in a source-sink model. In the second part, I will investigate the effect of spatial propagation on survival and genetic diversity. First, I will derive the reaction dispersion model from the individual based model of movement. Then I will present classical and recent results on spreading phenomena in the framework of reaction-dispersion equations. I will then extend the interior dynamics approach to these models. Finally, I will discuss evolutionary adaptation of a population structured by a phenotypic trait under a changing environment. I will derive a PDE model from a stochastic model and, using the Hamilton-Jacobi approach and large deviation techniques, I will present some approximations of these models. Then, I will present a new approach to track ancestral lineages in quantitative genetic model.