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Abstract Over the past 15 years, the D-statistic, a four-taxon test for organismal admixture (hybridization, or introgression) which incorporates single nucleotide polymorphism data with allelic patterns ABBA and BABA, has seen considerable use. This statistic seeks to discern significant deviation from either a given species tree assumption, or from the balanced incomplete lineage sorting that could otherwise defy this species tree. However, while the D-statistic can successfully discriminate admixture from incomplete lineage sorting, it is not a simple matter to determine the directionality of admixture using only four-leaf tree models. As such, methods have been developed that use 5 leaves to evaluate admixture. Among these, the DFOIL method, which tests allelic patterns on the “symmetric” tree S = (((1,2),(3,4)),5), succeeds in finding admixture direction for many five-taxon examples. However, DFOIL does not make full use of all symmetry, nor can DFOIL function properly when ancient samples are included because of the reliance on singleton patterns (such as BAAAA and ABAAA). Here, we take inspiration from DFOIL to develop a new and completely general family of five-leaf admixture tests, dubbed Δ-statistics, that can either incorporate or exclude the singleton allelic patterns depending on individual taxon and age sampling choices. We describe two new shapes that are also fully testable, namely the “asymmetric” tree A = ((((1,2),3),4),5) and the “quasisymmetric” tree Q = (((1,2),3),(4,5)), which can considerably supplement the “symmetric“ S = (((1,2),(3,4)),5) model used by DFOIL. We demonstrate the consistency of Δ-statistics under various simulated scenarios, and provide empirical examples using data from black, brown and polar bears, the latter also including two ancient polar bear samples from previous studies. Recently DFOIL and one of these ancient samples was used to argue for a dominant polar bear → brown bear introgression direction. However, we find, using both this ancient polar bear and our own, that by far the strongest signal using both DFOIL and Δ-statistics on tree S is actually bidirectional gene flow of indistinguishable direction. Further experiments on trees A and Q instead highlight what were likely two phases of admixture: one with stronger brown bear → polar bear introgression in ancient times, and a more recent phase with predominant polar bear → brown bear directionality. Code and documentation available at https://github.com/KalleLeppala/Delta-statistics. 

The polar bear ( Ursus maritimus ) has become a symbol of the threat to biodiversity from climate change. Understanding polar bear evolutionary history may provide insights into apex carnivore responses and prospects during periods of extreme environmental perturbations. In recent years, genomic studies have examined bear speciation and population history, including evidence for ancient admixture between polar bears and brown bears ( Ursus arctos ). Here, we extend our earlier studies of a 130,000- to 115,000-y-old polar bear from the Svalbard Archipelago using a 10× coverage genome sequence and 10 new genomes of polar and brown bears from contemporary zones of overlap in northern Alaska. We demonstrate a dramatic decline in effective population size for this ancient polar bear’s lineage, followed by a modest increase just before its demise. A slightly higher genetic diversity in the ancient polar bear suggests a severe genetic erosion over a prolonged bottleneck in modern polar bears. Statistical fitting of data to alternative admixture graph scenarios favors at least one ancient introgression event from brown bears into the ancestor of polar bears, possibly dating back over 150,000 y. Gene flow was likely bidirectional, but allelic transfer from brown into polar bear is the strongest detected signal, which contrasts with other published work. These findings may have implications for our understanding of climate change impacts: Polar bears, a specialist Arctic lineage, may not only have undergone severe genetic bottlenecks but also been the recipient of generalist, boreal genetic variants from brown bears during critical phases of Northern Hemisphere glacial oscillations.