Unequal Crossover Dynamics in Discrete and Continuous Time

Details Unequal Crossover Dynamics in Discrete and Continuous Time Unequal Crossover Dynamics in Discrete and Continuous Time

2004-02-22

arxiv.org/abs/math/0402351v1

J. Math. Biol. 49 (2004), 201-226

Mathematics

Dynamical Systems

25 pages, 1 figure; to appear in J. Math. Biol

Scientific paper

10.1007/s00285-004-0273-7

We analyze a class of models for unequal crossover (UC) of sequences containing sections with repeated units that may differ in length. In these, the probability of an `imperfect' alignment, in which the shorter sequence has d units without a partner in the longer one, scales like q^d as compared to `perfect' alignments where all these copies are paired. The class is parameterized by this penalty factor q. An effectively infinite population size and thus deterministic dynamics is assumed. For the extreme cases q=0 and q=1, and any initial distribution whose moments satisfy certain conditions, we prove the convergence to one of the known fixed points, uniquely determined by the mean copy number, in both discrete and continuous time. For the intermediate parameter values, the existence of fixed points is shown.

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