Extinction probabilities for a distylous plant population modeled by an inhomogeneous random walk on the positive quadrant

Details Extinction probabilities for a distylous plant population modeled by an inhomogeneous random walk on the positive quadrant Extinction probabilities for a distylous plant population modeled by an inhomogeneous random walk on the positive quadrant

2012-01-31

arxiv.org/abs/1201.6461v1

Mathematics

Probability

20 pages

Scientific paper

In this paper, we study a distylous flower population in which self-reproduction is not permitted. Individuals are diploid, and 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 to fecundate its stigmates. 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, where extinction happens when one of the axis 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. We consider the associated generating function and show that it satisfies a partial differential equation that is solved but whose solution is explicit, though hardly tractable. Numerical results comparing stochastic and deterministic approximations of the extinction probabilities are studied.

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