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

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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 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 solution `at infinity'. While the original conjecture was made for systems of viability 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.