| We build two models of population genetics with continuous geography. Rather than defining the models directly in terms of the dynamics of populations undergoing random change, we instead define associated coalescing processes that model lineages traced backwards in time. We then construct forwards-in-time models that have our coalescing processes as their duals. At each location in the "geography space" E (the circle of unit circumference), there is a population whose individuals have types from a "type space" K. Each population undergoes changes due to random mating, reproduction, and migration to other locations.;One model has as its dual a system of coalescing particles on E built from sticky Levy flows; the other has as its dual a system of particles on E performing independent copies of some Levy process and coalescing according to their local times together. Both processes (call them X and Xˆ respectively) live on a state space xi consisting of probability measure-valued functions; for each time t and for each point e ∈ E, Xt( e) and Xˆt(e) are measures on K describing the composition of the population at the point e at time t. For both models, we are able to deduce continuity in the time variable t of sample paths with respect to an appropriate topology on xi.;We then examine the case where the underlying Levy process of the sticky Levy flow and of the independent coalescing particles is a standard Brownian motion. We are able to analyze how the generator---in the sense of a martingale problem---of Xˆ behaves when applied to the algebra of functions generated by linear functionals xi → R of the form nm deye ne dkck , where psi is a C2 function on E, chi is a bounded function on K, and m is Lebesgue measure on E. We use this to show that Xˆ has a representative that is jointly continuous in both the time variable t and the geography variable e. Finally, we discover that the domain of the generator of X includes the above linear functionals, but does not include all of the algebra they generate. |
