Multilevel mutation-selection systems and set-valued duals.

A class of measure-valued processes which model multilevel multitype populations undergoing mutation, selection, genetic drift and spatial migration is considered. We investigate the qualitative behaviour of models with multilevel selection and the interaction between the different levels of selection. The basic tools in our analysis include the martingale problem formulation for measure-valued processes and a generalization of the function-valued and set-valued dual representations introduced in Dawson-Greven (Spatial Fleming-Viot models with selection and mutation. Lecture notes in mathematics, vol 2092. Springer, Cham, 2014). The dual is a powerful tool for the analysis of the long-time behaviour of these processes and the study of evolutionary systems which model phenomena including altruism, the emergence of cooperation and more complex interactions.