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| HAL: hal-00664536, version 1 |
| arXiv: 1201.6461 |
| Detailed view |  Export this paper |
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| Available versions: | v1 (2012-01-31) | v2 (2012-09-27) |
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| Extinction probabilities for a distylous plant population modeled by an inhomogeneous random walk on the positive quadrant |
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Pauline Lafitte-Godillon 1Kilian Raschel 2 |
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| (2012-01-30) |
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| 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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| 1: | Mathématiques Appliquées aux Systèmes - EA 4037 (MAS) |
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Ecole Centrale Paris |
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| 2: | Laboratoire de Mathématiques et Physique Théorique (LMPT) |
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CNRS : UMR6083 – Université François Rabelais - Tours |
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| 3: | Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP) |
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Polytechnique - X – CNRS : UMR7641 |
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| 4: | Laboratoire Paul Painlevé (LPP) |
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CNRS : UMR8524 – Université Lille I - Sciences et technologies |
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| Laboratoire MAS Laboratoire de Mathématiques et Physique Théorique LPP |
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| Subject | : | Mathematics/Probability Mathematics/Analysis of PDEs Environmental Sciences/Biodiversity and Ecology |
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| 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 |
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| Attached file list to this document: | |||||
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| hal-00664536, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00664536 | |
| oai:hal.archives-ouvertes.fr:hal-00664536 | |
| From: Viet Chi Tran | |
| Submitted on: Monday, 30 January 2012 22:22:52 | |
| Updated on: Tuesday, 31 January 2012 08:26:41 | |
