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

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 2^n - 1. Here a proof is provided. The upper bound of 3^{n-1} obtained by Lyubich et al. (2001) using Bezout's Theorem (1779) is reduced here to 2^n through a change of representation that reduces the third-order polynomials to second order. A further reduction to 2^n - 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 2^n - 1 fixed points given any n, which shows that 2^n - 1 is the sharpest possible upper bound that can be found for the general space of selection and transmission coefficients.