Single-crossover recombination in discrete time.

*(English)*Zbl 1208.92050Summary: Modelling the process of recombination leads to a large coupled nonlinear dynamical system. 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 [M. Baake and E. Baake, Can. J. Math. 55, No. 1, 3–41 (2003; Zbl 1056.92040)], this no longer works for discrete time. A more general model (i.e., without the restriction to single crossovers) has been studied before [J. H. Bennett, Ann. Hum. Genet. 18, 311–317 (1954); K. J. Dawson, Theor. Popul. Biol. 58, No. 1, 1–20 (2000; Zbl 1011.92038); Linear Algebra Appl. 348, No. 1–3, 115–137 (2002; Zbl 1003.92023)], and was solved algorithmically by means of Haldane linearisation. Using the special formalism introduced by Baake and Baake, 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.

##### MSC:

92D10 | Genetics and epigenetics |

39A60 | Applications of difference equations |

37N25 | Dynamical systems in biology |

60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |

06A07 | Combinatorics of partially ordered sets |

##### Keywords:

Population genetics; Recombination dynamics; Möbius linearisation; Diagonalisation; Linkage disequilibria##### References:

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