Inference in two dimensions: allele frequencies versus lengths of shared sequence blocks.

We outline two approaches to inference of neighbourhood size, N, and dispersal rate, σ(2), based on either allele frequencies or on the lengths of sequence blocks that are shared between genomes. Over intermediate timescales (10-100 generations, say), populations that live in two dimensions approach a quasi-equilibrium that is independent of both their local structure and their deeper history. Over such scales, the standardised covariance of allele frequencies (i.e. pairwise FST) falls with the logarithm of distance, and depends only on neighbourhood size, N, and a 'local scale', κ; the rate of gene flow, σ(2), cannot be inferred. We show how spatial correlations can be accounted for, assuming a Gaussian distribution of allele frequencies, giving maximum likelihood estimates of N and κ. Alternatively, inferences can be based on the distribution of the lengths of sequence that are identical between blocks of genomes: long blocks (>0.1 cM, say) tell us about intermediate timescales, over which we assume a quasi-equilibrium. For large neighbourhood size, the distribution of long blocks is given directly by the classical Wright-Malécot formula; this relationship can be used to infer both N and σ(2). With small neighbourhood size, there is an appreciable chance that recombinant lineages will coalesce back before escaping into the distant past. For this case, we show that if genomes are sampled from some distance apart, then the distribution of lengths of blocks that are identical in state is geometric, with a mean that depends on N and σ(2).