Proof of the Feldman-Karlin Conjecture on the Maximum Number of Equilibria in an Evolutionary System

Lee Altenberg

2010. Theoretical Population Biology 77: 263269.

Feldman and Karlin conjectured that the number of 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 Bezout'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 fixed point `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.


Altenberg introduces the transformation matrix [Ti,j,k], which gives the probability of a genotype jk producing a gamete of type i. This matrix can encompass various genetic distortions such as recombination, mutation, gene conversions, etc. Hence, the proof by Altenberg goes much further than the original constraints of the Feldman and Karlin conjecture and also beyond the scope of our present game theoretic formulation.

— Hana, The Anh, Arne Traulsen, Chaitanya S. Gokhale. 2012. On equilibrium properties of evolutionary multi-player games with random payoff matrices. Theoretical Population Biology 81: 264-272

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