Coalescent processes when the distribution of offspring number among individuals is highly skewed.

We report a complex set of scaling relationships between mutation and reproduction in a simple model of a population. These follow from a consideration of patterns of genetic diversity in a sample of DNA sequences. Five different possible limit processes, each with a different scaled mutation parameter, can be used to describe genetic diversity in a large population. Only one of these corresponds to the usual population genetic model, and the others make drastically different predictions about genetic diversity. The complexity arises because individuals can potentially have very many offspring. To the extent that this occurs in a given species, our results imply that inferences from genetic data made under the usual assumptions are likely to be wrong. Our results also uncover a fundamental difference between populations in which generations are overlapping and those in which generations are discrete. We choose one of the five limit processes that appears to be appropriate for some marine organisms and use a sample of genetic data from a population of Pacific oysters to infer the parameters of the model. The data suggest the presence of rare reproduction events in which approximately 8% of the population is replaced by the offspring of a single individual.