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Proof of the Feldman–Karlin conjecture on the maximum number of equilibria in an evolutionary system

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  • Altenberg, Lee

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.

Suggested Citation

  • Altenberg, Lee, 2010. "Proof of the Feldman–Karlin conjecture on the maximum number of equilibria in an evolutionary system," Theoretical Population Biology, Elsevier, vol. 77(4), pages 263-269.
    • Handle: RePEc:eee:thpobi:v:77:y:2010:i:4:p:263-269
    DOI: 10.1016/j.tpb.2010.02.007
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    1. Feldman, Marcus W., 2009. "Sam Karlin and multi-locus population genetics," Theoretical Population Biology, Elsevier, vol. 75(4), pages 233-235.
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    Cited by:

    1. Chaitanya Gokhale & Arne Traulsen, 2014. "Evolutionary Multiplayer Games," Dynamic Games and Applications, Springer, vol. 4(4), pages 468-488, December.
    2. Manh Hong Duong & The Anh Han, 2016. "On the Expected Number of Equilibria in a Multi-player Multi-strategy Evolutionary Game," Dynamic Games and Applications, Springer, vol. 6(3), pages 324-346, September.
    3. Han, The Anh & Traulsen, Arne & Gokhale, Chaitanya S., 2012. "On equilibrium properties of evolutionary multi-player games with random payoff matrices," Theoretical Population Biology, Elsevier, vol. 81(4), pages 264-272.
    4. Manh Hong Duong & Hoang Minh Tran & The Anh Han, 2019. "On the Expected Number of Internal Equilibria in Random Evolutionary Games with Correlated Payoff Matrix," Dynamic Games and Applications, Springer, vol. 9(2), pages 458-485, June.
    5. Chen, Luoer & Deng, Churou & Duong, Manh Hong & Han, The Anh, 2024. "On the number of equilibria of the replicator-mutator dynamics for noisy social dilemmas," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).

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