Unequal Crossover Dynamics in Discrete and Continuous Time |
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Authors: | Oliver?Redner Email author" target="_blank">Michael?BaakeEmail author |
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Institution: | (1) Institut für Mathematik, Univ. Greifswald, Jahnstr. 15a, 17487 Greifswald, Germany;(2) Fakultät für Mathematik, Univ. Bielefeld, Pf. 100131, 33501 Bielefeld, Germany |
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Abstract: | 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 qd 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.Copyright 2004 by Michael BaakeRevised version: 3 December 2003 |
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Keywords: | Unequal crossover Recombination Quadratic operators Probability generating functions Lyapunov functions |
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