Singular solutions of the diffusion equation of population genetics

McKane, A.J and Waxman, D (2007) Singular solutions of the diffusion equation of population genetics. Journal of Theoretical Biology, 247 (4). pp. 849-858. ISSN 0022-5193

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Abstract

The forward diffusion equation for gene frequency dynamics is solved subject to the condition that the total probability is conserved at all times. This can lead to solutions developing singular spikes (Dirac delta functions) at the gene frequencies 0 and 1. When such spikes appear in solutions they signal gene loss or gene fixation, with the �weight� associated with the spikes corresponding to the probability of loss or fixation. The forward diffusion equation is thus solved for all gene frequencies, namely the absorbing frequencies of 0 and 1 along with the continuous range of gene frequencies on the interval (0; 1) that excludes the frequencies 0 and 1. Previously, the probabilities if the absorbing frequencies 0 and 1 were found by appeal to the backward diffusion equation, while those in the continuous range (0; 1) were found from the forward diffusion equation. Our uni fied approach does not require two separate equations for a complete dynamical treatment of all gene frequencies within a diffusion approximation framework. For cases involving mutation, migration and selection, it is shown that a property of the deterministic part of gene frequency dynamics determines when fixation and loss can occur. It is also shown how solution of the forward equation, at long times, leads to the standard result for the fixation probability.

Item Type: Article
Schools and Departments: School of Life Sciences > Biology and Environmental Science
Subjects: Q Science > QH Natural history > QH0301 Biology
Depositing User: Chris Keene
Date Deposited: 19 Feb 2008
Last Modified: 09 Mar 2017 06:21
URI: http://sro.sussex.ac.uk/id/eprint/1183
Google Scholar:14 Citations

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