On the Singular Points of the Differential Equations in the Problem of Natural Selection

Abstract

Let (P1, P2,…, Pn) be an ensemble of n genotypes of the same species which differ among themselves for m independent characters (α1, α2,… αm), each of which is represented by a pair of alleles. Assuming that all of the possible combinations of these characters are effectively realized, one will have, as in my previous note (1), a genetic system composed of n = 3m different genotypes. The dynamics of this system can be represented by the system of differential equations (2):$$ p_{\rm i} '\, = \,\sum\limits_{h = 1}^n {\sum\limits_{k = 1}^n {\,n_{hk} \lambda ^i _{hk} } p_h \omega _{hk} \, - \,m_i p_i \, - \,p_i \,\sum\limits_{k = 1}^n {\,\mu _{hk} p_k } \,(i\, = \,1,2,...,n).} \, $$ (1)