A non-standard evolution problem arising in population genetics

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Abstract

We study the evolution of the probability density of an a sexual, 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.
Original languageEnglish
Pages (from-to)489-502
JournalCommunications In Mathematical Sciences
Volume7
Issue number2
Publication statusPublished - 2009

Keywords

  • Gene fixation
  • evolutionary dynamics
  • degenerate parabolic equations
  • boundarycoupled weak solutions

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