Mathematical Theory of Phenotypical Selection

Abstract

A general concept of phenotypical structure over a genotypical structure is developed. The direct decompositions of multilocus phenotypical structures are considered. Some aspects of phenotypical heredity are described in terms of graph theory. The acyclic phenotypical structures are introduced and studied on this base. The evolutionary equations are adjusted to the phenotypical selection. It is proved that if a phenotypical structure is acyclic then the set of fixed points of the corresponding evolutionary operator is finite except for a proper algebraic subset of the operator space. Some applications of this theorem are given.

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

“…Theorem 3 proves the conjecture of Feldman and Karlin on the maximum number of isolated fixed points in a system of selection and recombination, and extends it to arbitrary transmission processes, of which recombination and mutation represent special cases. This substantially sharpens the previous upper bound of 3 n−1 (Lyubich et al, 2001) on the number of isolated fixed points of an evolutionary system with selection and arbitrary transmission.…”

“…No attempt has been made to characterize the conditions on T ijk and w jk that produce only isolated and nondegenerate fixed points. More on this issue can be found in Lyubich et al (2001). One may mention, however, that such conditions are generic, in that for 'almost all' sets of n algebraic hypersurfaces of degree n, the intersection consists of isolated non-degenerate fixed points (Shafarevich 1994, p. 223, Garcia andLi 1980); and sets of T ijk values that produce degenerate fixed points are nowhere dense (Lyubich, 1992, Theorem 8.1.3) in the space of T ijk values.…”

“…(Also see Lyubich 1992, Theorem 9.1.1.) Lyubich (1992) shows that with arbitrary transmission but with no selection, 2 n−1 is an upper bound on the number of isolated fixed points of g. Lyubich et al (2001) show that with arbitrary transmission and selection, 3 n−1 is an upper bound on the number of isolated fixed points of g.…”

“…The approach taken in Lyubich (1992) and Lyubich et al (2001) to obtain the upper bounds relies on Bézout's Theorem (1779a).…”