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

Abstract

Abstract. 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.Key words. inhomogeneous random walk on the positive quadrant, boundary absorption, transport equation, method of characteristics, self-incompatibility in flower populations, extinction in diploid population with sexual reproduction AMS subject classifications. 60G50, 60J80, 35Q92, 92D25

Walks in the quarter plane: Analytic approach and applications

“…• finance, as the dynamics of certain limit order books may be approximated by random processes in the quarter plane, see [5]; • population biology, as the quarter plane is the natural space to parametrize any two-dimensional population, see [12] for an example; • probability theory, since random walks in cones (e.g., quantum random walks, non-colliding random walks) are a very actual topic; • queueing theory, as any two-dimensional queue (and many of them are important for the theory, see [4]) can be modeled by random walks in the quarter plane; • etc. The present article aims at summarizing part of our contributions to that domain obtained in the PhD thesis [15] and afterwards [8,11,12].…”

“…The present article aims at summarizing part of our contributions to that domain obtained in the PhD thesis [15] and afterwards [8,11,12]. We choose to focus here on two particular applications, that we consider representative of the variety of the problems related to (deterministic or random) walks in the quarter plane.…”

“…This survey is organized as follows: We shall first briefly present these two applications, in Sections 1.2 and 1.3. Next, in Section 1.4, we will sketch the main ideas of the analytic methods that we used in [8,11,12,15] to study these problems. Then, in the main Sections 2 and 3, we will return to the two applications, in more details; in particular, there we shall present our contributions.…”

“…In this survey we shall only consider one instance (taken from [12]) of such a two-dimensional population, but it is worth keeping in mind that many others could be similarly analyzed. The example we take here is the following: We consider a flower population of two types, say a and A, for which we 1 Of course, it is possible to count the numbers of trajectories; here, we ask ourselves if it is possible to find closed-form expressions for these numbers, or for their generating function.…”

“…The quantity q 0 then typically follows by a normalizing argument. We refer to [6,7,12,13,15] for more details.…”