Summary: The population formalism of “adaptive evolution” has been developed in the last twenty years along ideas presented in other chapters in this volume. This mathematical formalism addresses the question of explaining how selection of a favorable phenotypical trait in a population occurs. In the language of Metz’s Chapter, it refers to meso-evolution. It uses models based, usually, on integro-differential equations for the population structured by a phenotypical trait. A self-contained mathematical formulation of adaptive evolution also contains the description of mutations and leads to partial differential equations. Then the complete evolution picture follows from the model ingredients mostly driven by the changing adaptive landscape.
It is possible to introduce scaling parameters and perform asymptotic analysis. Then highly concentrated population densities (well-separated Dirac masses) arise that can undergo branching patterns. This phenomenon is interpreted as the speciation process.
The process in which concentrated solutions occur and a continuous set of traits cannot be present is subtle and numerical methods can induce artifacts if not correctly shaped. Simulations on Monte-Carlo methods can be compared to deterministic numerical methods as finite differences.
For the entire collection see [Zbl 1220.92045].