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We derive a central limit theorem for a spatial $\Lambda$-Fleming-Viot model with fluctuating population size. At each reproduction, a proportion of the population dies and is replaced by a not necessarily equal mass of new individuals. The mass depends on the local population size and a function thereof. Additionally, as new individuals have a single parental type, with growing population size, events become more frequent and of smaller impact, modelling the successful reproduction of a higher number of individuals. From the central limit theorem we derive a Wright-Mal\'ecot formula quantifying the asymptotic probability of identity by descent and thus isolation by distance. The formula reflects that ancestral lineages are attracted by centres of population mass and coalesce with a rate inversely proportional to the population size. Notably, we obtain this information despite the varying population size rendering the dual process intractable., Comment: 54 pages, 2 figures

We propose an extension of the classical $\Lambda$-Fleming-Viot model to intrinsically varying population sizes. During events, instead of replacing a proportion of the population, a random mass dies and a, possibly different, random mass of new individuals is added. The model can also incorporate a drift term, representing infinitesimally small, but frequent events. We investigate elementary properties of the model, analyse its relation to the $\Lambda$-Fleming-Viot model and describe a duality relationship. Through the lookdown framework, we provide a forward-in-time analysis of fixation and coming down from infinity. Furthermore, we present a new duality argument allowing one to deduce well-posedness of the measure-valued process without the necessity of proving uniqueness of the associated lookdown martingale problem.

In this paper, we uncover new asymptotic isolation by distance patterns occurring under long-range dispersal of offspring. We extend a recent work of the first author, in which this information was obtained from forwards-in-time dynamics using a novel stochastic partial differential equations approach for spatial $\Lambda$-Fleming-Viot models. The latter were introduced by Barton, Etheridge and V\'eber as a framework to model the evolution of the genetic composition of a spatially structured population. Reproduction takes place through extinction-recolonisation events driven by a Poisson point process. During an event, in certain ball-shaped areas, a parent is sampled and a proportion of the population is replaced. We generalize the previous approach of the first author by allowing the area from which a parent is sampled during events to differ from the area in which offspring are dispersed, and the radii of these regions follow power-law distributions. In particular, while in previous works the motion of ancestral lineages and coalescence behaviour were closely linked, we demonstrate that local and non-local coalescence is possible for ancestral lineages governed by both fractional and standard Laplacians., Comment: 46 pages, 3 Figures