# Equilibria points multilocus multiallele systems (finiteness and infinity)

**Brief introduction.** (brief.eq)

*A general haploid selection model with arbitrary number of multiallelic loci and arbitrary linkage distribution is considered. The population is supposed to be panmictic. A dynamically equivalent diploid selection model is introduced. There is a position effect in this model if the original haploid selection is not multiplicative. If haploid selection is additive then the fundamental theorem is established even with an estimate for the change in the mean fitness. On this basis exponential convergence to an equilibrium is proved. As rule, the limit states are single-gamete ones. If, moreover, linkage is tight, then the single-gamete state with maximal fitness attracts the population for almost all initial states.*

Kirzhner V., Lyubich Yu. (1997). **Multilocus dynamics under haploid selection.** *J. Math. Biol*., **35**, 391-408. (Article.gaplsel)

*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.*

Yuri Lyubich, Valery Kirzhner, Anna Ryndin (2001)
**Mathematical Theory of Phenotypical Selection**, *Advances in Applied Mathematics*, **26**, 4, pp. 330-352. (Article.phen)

*A mathematical approach to interactions between genotypes and phenotypes in a multilocus multiallele population is developed. No a priori information on a fitness function is required. In particular, some structural definitions of epistasis and the position effect are given in terms of a decomposition of phenotypical structures. On this base a distance to the additive non-epistasis is introduced and an explicit formula for it is obtained. A class of phenotypical structures including multilocus dominance is described in terms of directed graphs. The evolutionary equations are adjusted to a fitness function compatible with a phenotypical structure. Some results on the finiteness of the equilibria set are presented.*

Yuri Lyubich, Valery Kirzhner (2003) **Mathematical frameworks for phenotypical selection and epistasis** *J. Theoretical Biol.*, **221**, 625-638.(Article.selep)

*It is proven that the set of equilibria under multilocus Mendel dominance in a population with any fixed-recombination coefficients is finite generically if the selection is nonepistatic in Karlin's sense.*

Yuri Lyubich, Valery Kirzhner. (2003) **Finiteness of Equilibria Set for a Nonepistatic Selection under Multilocus Mendel Dominance.** *Appl. Math. Letters*, **16**, 421-424. (Article.multipl)

*The concept of monomial selection is introduced as a natural generalization of multiplicative selection. It is proven that the equilibrium set of the multilocus multiallele population under monomial selection is generically finite. The result is new even in the multiplicative case. An upper bound for the number of equilibria is given.*

Valery Kirzhner and Yuri Lyubich. (2003) **The multilocus multiplicative selection equilibria.** *Appl. Math. Letters*, **16** (6),853-856. (Article.epis)