We consider the genealogy of a sample of individuals taken from a spatially structured population when the variance of the offspring distribution is relatively large. The space is structured into discrete sites of a graph G. If the population size at each site is large, spatial coalescents with multiple mergers, so called spatial Λ-coalescents, for which ancestral lines migrate in space and coalesce according to some Λ-coalescent mechanism, are shown to be appropriate approximations to the genealogy of a sample of individuals. We then consider as the graph G the two dimensional torus with side length 2L+1 and show that as L tends to infinity, and time is rescaled appropriately, the partition structure of spatial Λ-coalescents of individuals sampled far enough apart converges to the partition structure of a non-spatial Kingman coalescent. From a biological point of view this means that in certain circumstances both the spatial structure as well as larger variances of the underlying offspring distribution are harder to detect from the sample. However, supplemental simulations show that for moderately large L the different structure is still evident.

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In this thesis we consider the genealogy of a spatial Cannings model. This is a population model in which individuals are distributed over a countable set of sites G. The reproduction of individuals at each site is panmictic (exchangeable) and preserves the local population size. The offspring then migrate to other sites in G, also in an exchangeable manner. We consider the spatial coalescent introduced by sampling n individuals at present time and tracking their ancestral lines back in time. The resulting process is the spatial Cannings coalescent. Our main result shows, that an appropriately time-rescaled spatial Cannings coalescent converges to a spatial Ξ-coalescent in the large population limit.The key feature of our result is that the spatial structure is preserved into the limit as opposed to a fast migration limit. The influence of the migration on the local population size can yield a time-inhomogeneous limit and, in case of sites with a small population size, our limiting process may not have a strongly continuous semigroup. I would like to thank my friends and colleagues Alexander Hartmann and Fabian Telschow, who made for great targets to bounce mathematical ideas and questions off and also for their input regarding some formal as well as formatting questions concerning this thesis. Last, but not least, I would like to thank my parents Jürgen and Gabriela Heuer for their great support over the years. The Backward Model 92. The Wright-Fisher model in which we choose pν I k;x k;x,i q iI k;x to be a vector of i.i.d. Poisson distributed random variables conditioned on their sum being equal |I k;x |. An alternative way of describing this distribution would be that the vector is multinomially distributed. More precisely we consider an urn containing one ball for each color i I k;x . Now we draw |I k;x | times with replacement from the urn and set νI k;x k;x,i to be the total number of draws of color i.