In this lecture we give an overview of some mathematical developments that have been motivated by and contributed to the study of evolving population systems such as thosethat arise in molecular and evolutionary biology, evolutionary economics, and evolutionary computation. Mathematical formulations of these systems have led to classes of stochastic processes that are sufficiently rich to describe spatially and hierarchically structured interacting populations and their genealogical structures. These processes are formulated in terms of interacting particle systems and measure-valued Markov processes on structured spaces. Basic tools for the study of particle systems and measure-valued processes have been developed over the past 25 years. These include martingale problem methods, dual processes, countable particle representations, canonical representations, Palm measures, coupling methods, multi-scale asymptotics, large deviations, etc. Using these methods considerable progress has been achieved in understanding some universality classes of behaviors of spatially and hierarchically structured systems in different fitness landscapes and space and time scales. In recent years progress has also been made on some interesting classes of more complex interactions but the development and analysis of populations with complex interactions remains an active and challenging area of research.
Group for Research in Decision Analysis