Journal of Mathematical Biology

, Volume 60, Issue 5, pp 727–760 | Cite as

Single-crossover recombination in discrete time

  • Ute von Wangenheim
  • Ellen BaakeEmail author
  • Michael Baake


Modelling the process of recombination leads to a large coupled nonlinear dynamical system. Here, we consider a particular case of recombination in discrete time, allowing only for single crossovers. While the analogous dynamics in continuous time admits a closed solution (Baake and Baake in Can J Math 55:3–41, 2003), this no longer works for discrete time. A more general model (i.e. without the restriction to single crossovers) has been studied before (Bennett in Ann Hum Genet 18:311–317, 1954; Dawson in Theor Popul Biol 58:1–20, 2000; Linear Algebra Appl 348:115–137, 2002) and was solved algorithmically by means of Haldane linearisation. Using the special formalism introduced by Baake and Baake (Can J Math 55:3–41, 2003), we obtain further insight into the single-crossover dynamics and the particular difficulties that arise in discrete time. We then transform the equations to a solvable system in a two-step procedure: linearisation followed by diagonalisation. Still, the coefficients of the second step must be determined in a recursive manner, but once this is done for a given system, they allow for an explicit solution valid for all times.


Population genetics Recombination dynamics Möbius linearisation Diagonalisation Linkage disequilibria 

Mathematics Subject Classification (2000)

92D10 37N30 06A07 60J05 


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Copyright information

© The Authors 2009

Authors and Affiliations

  • Ute von Wangenheim
    • 1
  • Ellen Baake
    • 1
    Email author
  • Michael Baake
    • 2
  1. 1.Technische FakultätUniversität BielefeldBielefeldGermany
  2. 2.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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