Extinction Probabilities for a Distylous Plant Population Modeled by an Inhomogeneous Random Walk on the Positive Quadrant

In this paper, we study a flower population in which self-reproduction is not permitted. Individuals are diploid, that is, each cell contains two sets of chromosomes, and distylous, that is, two alleles, A and a, can be found at the considered locus S. Pollen and ovules of flowers with the same genotype at locus S cannot mate. This prevents the pollen of a given flower from fecundating its own stigmata. Only genotypes AA and Aa can be maintained in the population, so that the latter can be described by a random walk in the positive quadrant whose components are the number of individuals of each genotype. This random walk is not homogeneous, and its transitions depend on the location of the process. We are interested in the computation of the extinction probabilities, as extinction happens when one of the axes is reached by the process. These extinction probabilities, which depend on the initial condition, satisfy a doubly indexed recurrence equation that cannot be solved directly. Our contribution is twofold: on the one hand, we obtain an explicit, though intricate, solution through the study of the PDE solved by the associated generating function. On the other hand, we provide numerical results comparing stochastic and deterministic approximations of the extinction probabilities.