Single-crossover recombination in discrete time. (English) Zbl 1208.92050

Summary: 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.


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
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