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We show that the unstructured ancestral selection graph applies to part of the history of a sample from a population structured by restricted migration among subpopulations, or demes. The result holds in the limit as the number of demes tends to infinity with proportionately weak selection, and we have also made the assumptions of island-type migration and that demes are equivalent in size. After an instantaneous sample-size adjustment, this structured ancestral selection graph converges to an unstructured ancestral selection graph with a mutation parameter that depends inversely on the migration rate. In contrast, the selection parameter for the population is independent of the migration rate and is identical to the selection parameter in an unstructured population. We show analytically that estimators of the migration rate, based on pairwise sequence differences, derived under the assumption of neutrality should perform equally well in the presence of weak selection. We also modify an algorithm for simulating genealogies conditional on the frequencies of two selected alleles in a sample. This permits efficient simulation of stronger selection than was previously possible. Using this new algorithm, we simulate gene genealogies under the manydemes ancestral selection graph and identify some situations in which migration has a strong effect on the time to the most recent common ancestor of the sample. We find that a similar effect also increases the sensitivity of the genealogy to selection. K IMURA (1983) strongly promoted the idea that the Hudson (1983), and Tajima (1983). The reason for abundant genetic variation seen in nearly every spethis is clear: it is much simpler to predict the frequencies cies studied must be neutral. Ohta and Kimura (1971) of selected alleles using diffusion theory than using the suggested a less strict version of this in which polymorbackward-time approach of the coalescent. For neutral phisms are explained by the constant input of nearly loci, the coalescent provides a simple and useful descripneutral mutations, i.e., variants with selective advantages tion of the genetic ancestry of a sample of genetic dataor disadvantages smaller than the reciprocal of the popthe genealogy for short-from a large well-mixed, or panulation size. After a great deal of debate and many analymictic, population of constant size through time. Due to ses, summarized in Golding (1994), there are now few the close connection between genealogies and genetic adherents to the strict neutral mutation hypothesis. Modata, and to the ease with which genealogies can be simlecular techniques currently allow huge numbers of ulated, a growing set of computational tools use the coalespolymorphisms to be assayed with relative ease, and the cent to make inferences about population history from resulting genomic data can be used to estimate the DNA sequences; see Stephens (2001) and Tavaré (2004) strength of selection associated with genetic differences for reviews. (Sawyer and Hartl 1992;Bustamante et al. 2002).Th...

An ancestral influence graph is derived, an analogue of the coalescent and a composite of Griffiths' (1991) two-locus ancestral graph and Krone and Neuhauser's (1997) ancestral selection graph. This generalizes their use of branching-coalescing random graphs so as to incorporate both selection and recombination into gene genealogies. Qualitative understanding of a 'hitch-hiking' effect on genealogies is pursued via diagrammatic representation of the genealogical process in a two-locus, two-allele haploid model. Extending the simulation technique of Griffiths and Tavare (1996), computational estimation of expected times to the most recent common ancestor of samples of n genes under recombination and selection in two-locus, two-allele haploid and diploid models are presented. Such times are conditional on sample configuration. Monte Carlo simulations show that 'hitch-hiking' is a subtle effect that alters the conditional expected depth of the genealogy at the linked neutral locus depending on a mutation-selection-recombination balance.

Inequalities for spatial competition verify the pair approximation of statistical mechanics introduced to theoretical ecology by Matsuda, Satō and Iwasa, among others. Spatially continuous moment equations were introduced by Bolker and Pacala and use a similar assumption in derivation. In the present article, I prove upper bounds for the kth central moment of occupied sites in the contact process of a single spatial dimension. This result shows why such moment closures are effective in spatial ecology.2010 Mathematics subject classification: primary 92D40; secondary 60K35.