Multiple merger gene genealogies in two species: Monophyly, paraphyly, and polyphyly for two examples of Lambda coalescents

Probabilities of monophyly, paraphyly, and polyphyly of two-species gene genealogies are computed for modest sample sizes and compared for two different Λ coalescent processes. Coalescent processes belonging to the Λ coalescent family admit asynchronous multiple mergers of active ancestral lineages. Assigning a timescale to the time of divergence becomes a central issue when different populations have different coalescent processes running on different timescales. Clade probabilities in single populations are also computed, which can be useful for testing for taxonomic distinctiveness of an observed set of monophyletic lineages. The coalescence rates of multiple merger coalescent processes are functions of coalescent parameters. The effect of coalescent parameters on the probabilities studied depends on the coalescent process, and if the population is ancestral or derived. The probability of reciprocal monophyly tends to be somewhat lower, when associated with a Λ coalescent, under the null hypothesis that two groups come from the same population. However, even for fairly recent divergence times, the probability of monophyly tends to be higher as a function of the number of generations for coalescent processes that admit multiple mergers, and is sensitive to the parameter of one of the example processes.