On the Karlin-Kimura approaches to the Wright-Fisher diffusion with fluctuating selection

The goal of this manuscript is a comparative study of two WrightFisher-like diffusion processes on the interval, one due to Karlin and the other one due to Kimura. Each model accounts for the evolution of one two-locus colony undergoing random mating, under the additional action of selection in random environment. In other words, we study the effect of disorder on the usual Wright-Fisher model with fixed (nonrandom) selection. There is a drastic qualitative difference between the two models and between the random and nonrandom selection hypotheses. We first present a series of elementary stochastic models and tools that are needed to undergo this study in the context of diffusion processes theory, including: Kolmogorov backward and forward equations, scale and speed functions, classification of boundaries, Doob-transformation of sample paths using additive functionals. In this spirit, we briefly revisit the neutral Wright-Fisher diffusion and the Wright-Fisher diffusion with nonrandom selection. With these tools at hand, we first deal with the Karlin approach to the Wright-Fisher diffusion model with randomized selection differentials. The specificity of this model is that in the large population case, the boundaries of the state-space are natural, hence inaccessible and so quasi-absorbing only. We supply some limiting properties pertaining to hitting times of points close to the boundaries. Next, we study the Kimura approach to the Wright-Fisher model with randomized selection, which may be viewed as a modification of the Karlin model, using an appropriate Doob transform which we describe. This model also has natural boundaries but they turn out to be much more attracting and sticky than in its Karlin’s version. This leads to a faster approach to the quasi-absorbing states, a larger time needed to move from the vicinity of one boundary to the other and to a local critical behavior of the branching diffusion obtained after the relevant Doob transformation. Running title: Karlin-Kimura Models with Fluctuating Selection.