Time depending dynamics of chains of evolution algebras

Abstract

The project is devoted to investigations of various types of evolution algebras (genetic, dibaric, algebras of free and bisexual population, Bernstein, etc.), development of the theory of non-associative varieties of algebras and their applications to mathematical biology and nonlinear dynamical systems. In history, mathematicians and geneticists once used non-associative algebras to study Mendelian genetics. Mendel first exploited symbols that are quite algebraically suggestive to express his genetic laws. Apparently, Serebrowsky was the first to give an algebraic interpretation of the sign "x", which indicated sexual reproduction, and to give a mathematical formulation of Mendel's laws. Glivenkov introduced the so-called Mendelian algebras for diploid populations with one locus or two unlinked loci. Independently, Kostitzin also introduced a symbolic multiplication to express Mendel's laws. The systematic study of algebras occurring in genetics can be attributed to I. M. H. Etherington. In his series of papers, he succeeded in giving a precise mathematical formulation of Mendel's laws in terms of non-associative algebras. Besides Etherington, fundamental contributions have been made by Gonshor, Schafer, Holgate, Hench, Reiser, Abraham, Lyubich, and Wörz-Busekros. The general genetic algebras could be developed into a field of independent mathematical interest, because these algebras are in general not-associative and do not belong to any of the well-known classes of non-associative algebras such as Lie algebras, alternative algebras, or Jordan algebras. Until 1980s, the most comprehensive reference in this area was Wörz-Busekros's book. More recent results, such as genetic evolution in genetic algebras, can be found in Lyubich's book. A good survey is Reed's article.