# Multilocus multiallele population with mutation and migration but without selection

A well-known mathematical model of population genetics was studied. It is assumed that the population size is infinite and that each individuum belongs to one of the finite class of available genotypes. In each generation the population state is described by a distribution of the genotypes. In the next generation the distribution is transformed by the Mendelian laws, the transformation, mathematically, being an essentially nonlinear operator. Particular cases of this model had been studied by Fisher. In the sixties Yu.I.Liubich, using an approach sketched by O.Reiersol, suggested a general method for solution of this problem ( without selection ) and applied it for the case of autosomal loci. In next articles this technique is developed and solved the problem on the limit behaviour of trajectories (in the framework of the above model)of genetic systems with any form of sexual differentiation, any form of polyploidy with migrations and mutations (but still without selection). An approach for obtaining an explicit solution of volutionary equations was sketched and such solutions were obtained for the case of haploid gametes. (Theorems on the trajectory convergence were obtained independently and for a less general case by H.Kesten and E.Ellison).

Kirzhner V. (1973). **On behavior of trajectories for a certain class of genetic systems**, *Soviet Doklady*, **209**, 2, 287-290 (in Russian). (English translation: *Soviet Mathematical Doklady*, 209 (1973), 2, 378-382. Article.freesex).

Kirzhner V.M. (1972). **On behaviour of trajectorics of polyploid genetic systems with migrations,** Proc. Low Temp.Phys.Inst. *"Computing Math.& Computers"*, **III**, 141-143 (in Russian).

Kirzhner V., Lyubich Yu. (1974). **An evolutionary equation and a limit theorem for general genetic systems without selection**, *Soviet Doklady*, **215**, 4, 776-779 (in Russian). (English translation: *Soviet Mathematical Doklady* 15 (1974), 2, 582-586. Article.freeall)

*The goal of the present work is to explicitly solve the evolutionary equations for a single locus autosomal polyploid population.*

Lyubich Yu.I., Kirzhner V.M. and R. Tutunikov (2004) **Explicit solution of the evolutionary equations for single locus autosomal polyploid populations.**
Preprint series: *International Erwin Schrodinger Institute for Mathematical Physics ESI* (Vienn) preprints 1500. (http://www.esi.ac.at/preprints/ESI-Preprints.html)