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June 2009 A non-standard evolution problem arising in population genetics
Fabio A.C.C. Chalub, Max O. Souza
Commun. Math. Sci. 7(2): 489-502 (June 2009).


We study the evolution of the probability density of an asexual, one locus population under natural selection and random evolution. This evolution is governed by a Fokker-Planck equation with degenerate coefficients on the boundaries, supplemented by a pair of conservation laws. It is readily shown that no classical or standard weak solution definition yields solvability of the problem. We provide an appropriate definition of weak solution for the problem, for which we show existence and uniqueness. The solution displays a very distinctive structure and, for large time, we show convergence to a unique stationary solution that turns out to be a singular measure supported at the endpoints. An exponential rate of convergence to this steady state is also proved.


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Fabio A.C.C. Chalub. Max O. Souza. "A non-standard evolution problem arising in population genetics." Commun. Math. Sci. 7 (2) 489 - 502, June 2009.


Published: June 2009
First available in Project Euclid: 27 May 2009

zbMATH: 1176.92038
MathSciNet: MR2536449

Primary: 95D15
Secondary: 35K65

Keywords: boundary-coupled weak solutions , Degenerate parabolic equations , evolutionary dynamics , Gene fixation

Rights: Copyright © 2009 International Press of Boston

Vol.7 • No. 2 • June 2009
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