On the Karlin–Kimura approaches to the Wright–Fisher diffusion with fluctuatingselection

Abstract

The goal of this work is a comparative study of two Wright–Fisher-like diffusionprocesses on the interval, one due to Karlin and the other one due to Kimura. Eachmodel accounts for the evolution of one two-locus colony undergoing randommating, under the additional action of selection in a random environment. In otherwords, we study the effect of disorder on the usual Wright–Fisher model withfixed (nonrandom) selection. There is a drastic qualitative difference betweenthe two models and between the random and nonrandom selection hypotheses.We first present a series of elementary stochastic models and tools that are needed toconduct this study in the context of diffusion process theory, including Kolmogorovbackward and forward equations, scale and speed functions, classification of boundaries,and Doob transformation of sample paths using additive functionals. In this spirit, webriefly revisit the neutral Wright–Fisher diffusion and the Wright–Fisher diffusion withnonrandom selection.With these tools at hand, we first deal with the Karlin approach to the Wright–Fisherdiffusion model with randomized selection differentials. The specificity of thismodel is that in the large population case, the boundaries of the state space arenatural and hence inaccessible, and so quasi-absorbing only. We supply somelimiting properties pertaining to times of hitting of points close to the boundaries.Next, we study the Kimura approach to the Wright–Fisher model with randomizedselection, which may be viewed as a modification of the Karlin model, using an appropriateDoob transform which we describe. This model also has natural boundaries, but they turnout to be much more attracting and sticky than in Karlin’s version. This leads to a fasterapproach to the quasi-absorbing states, to a larger time needed to move from the vicinity ofone boundary to the other and to a local critical behavior of the branching diffusionobtained after the relevant Doob transformation.