On the nonidentifiability of migration time estimates in isolation with migration models

In recent years many studies have found evidence of gene flow between diverging populations by analyzing genetic data under an Isolation with Migration (IM) model (Pinho & Hey 2010). Given evidence of gene exchange, investigators often then wish to inquire of the time when gene flow occurred (e.g. Won & Hey 2005; Becquet & Przeworski 2009). For example, a model of divergence with gene flow would be suggested if gene flow occurred early or throughout the divergence process, whereas secondary contact would be the likely interpretation if gene flow was found to only have occurred after divergence had been ongoing for some time. Recently Strasburg & Rieseberg (2011) assessed the quality of estimates for the time of migration events using the method currently implemented in the IMA2 program (Hey 2010). They found that the credible intervals of estimated times were so wide as to make the method unsuitable for the question. These results suggest that some conclusions of previous studies that draw upon the posterior distribution for times of migration should be discounted (e.g. Won & Hey 2005; Strasburg et al 2008; Niemiller et al 2008; Nadachowska & Babik 2009). The Strasburg & Rieseberg (2011) study reports results from simulations. Here we examine, using the theory underlying the method implemented in the IMA2 program, the possible bases for their observations. We demonstrate that gene migration times are not fully identifiable using the general coalescent for genealogies in an IM model, as implemented in IMA2 and similar programs. In many respects the findings are general to methods that rely upon calculating the probabilities of genealogies under the coalescent, and so are of broader interest than any particular program. We note that the method implemented in IMA2 is the same as that in the IMA program (Hey & Nielsen 2007), and hereafter we refer simply to IMA.