ABSTRACT
A family of heterogeneous mean-field systems with jumps is analyzed. These systems are constructed as a Gibbs measure on block graphs. When the total number of particles goes to infinity, the law of large numbers is shown to hold in a multi-class context, resulting in the weak convergence of the empirical vector towards the solution of a McKean-Vlasov system of equations. We then investigate the local stability of the limiting McKean-Vlasov system through the construction of a local Lyapunov function. We first compute the limit of adequately scaled relative entropy functions associated with the explicit stationary distribution of the N-particles system. Using a Laplace principle for empirical vectors, we show that the limit takes an explicit form. Then we demonstrate that this limit satisfies a descent property, which, combined with some mild assumptions shows that it is indeed a local Lyapunov function.
ABSTRACT
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.
