 Proof of the Feldman–Karlin conjecture on the maximum number of equilibria in an evolutionary systemLee Altenberg Theoretical Population Biology, 2010, vol. 77, issue 4, 263-269 Abstract: Feldman and Karlin conjectured that the number of isolated fixed points for deterministic models of viability selection and recombination among n possible haplotypes has an upper bound of 2n−1. Here a proof is provided. The upper bound of 3n−1 obtained by Lyubich et al. (2001) using Bézout’s Theorem (1779) is reduced here to 2n through a change of representation that reduces the third-order polynomials to second order. A further reduction to 2n−1 is obtained using the homogeneous representation of the system, which yields always one solution ‘at infinity’. While the original conjecture was made for systems of selection and recombination, the results here generalize to viability selection with any arbitrary system of bi-parental transmission, which includes recombination and mutation as special cases. An example is constructed of a mutation-selection system that has 2n−1 fixed points given any n, which shows that 2n−1 is the sharpest possible upper bound that can be found for the general space of selection and transmission coefficients. Keywords: Feldman Karlin conjecture; Selection; Recombination; Transmission; Fixed points; Equilibria; Bézout’s Theorem; Homotopy method (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:thpobi:v:77:y:2010:i:4:p:263-269 DOI: 10.1016/j.tpb.2010.02.007 Access Statistics for this article Theoretical Population Biology is currently edited by Jeremy Van Cleve More articles in Theoretical Population Biology from Elsevier |
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