Proof of the Feldman-Karlin Conjecture on the Maximum Number of Equilibria in an Evolutionary System - Quantitative Biology > Populations and EvolutionReport as inadecuate




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Abstract: Feldman and Karlin conjectured that the number of isolated fixed points fordeterministic models of viability selection and recombination among n possiblehaplotypes has an upper bound of 2^n - 1. Here a proof is provided. The upperbound of 3^{n-1} obtained by Lyubich et al. 2001 using Bezout-s Theorem1779 is reduced here to 2^n through a change of representation that reducesthe third-order polynomials to second order. A further reduction to 2^n - 1 isobtained using the homogeneous representation of the system, which yieldsalways one solution `at infinity-. While the original conjecture was made forsystems of viability selection and recombination, the results here generalizeto viability selection with any arbitrary system of bi-parental transmission,which includes recombination and mutation as special cases. An example isconstructed of a mutation-selection system that has 2^n - 1 fixed points givenany n, which shows that 2^n - 1 is the sharpest possible upper bound that canbe found for the general space of selection and transmission coefficients.



Author: Lee Altenberg

Source: https://arxiv.org/







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