Bernstein duality revisited: frequency-dependent selection, coordinated mutation and opposing environments

Abstract: This paper investigates the long-term behavior of a class of $\Lambda$-Wright--Fisher processes incorporating frequency-dependent selection, coordinated (bidirectional) selection, as well as individual and coordinated mutation. Our primary analytical tool is Bernstein duality, a generalization of moment duality. We introduce the corresponding dual process and establish the relevant duality relation. Without mutation, this work complements earlier studies that employed moment duality, Siegmund duality or other methods to classify the long-term behavior of similar processes. Notably, the current analysis encompasses parameter regimes that model bidirectional selection, a scenario that has proven challenging to analyze using moment duality. In the presence of mutation, we establish the ergodic properties of the process.