  Consider a bisexual population such that the set of females can be
partitioned into finitely many different types indexed by $\{1,2,\dots,n\}$
and, similarly, that the male types are indexed by $\{1,2,\dots,\nu \}$.
Recently an evolution algebra of bisexual population was introduced by
identifying the coefficients of inheritance of a bisexual population as the
structure constants of the algebra. In this paper we study constrained
evolution algebra of bisexual population in which type "1" of females and males
have preference. For such algebras sets of idempotent and absolute nilpotent
elements are known. We consider two particular cases of this algebra, giving
more constrains on the structural constants of the algebra. By the first our
constrain we obtain an $n+\nu$- dimensional algebra with a matrix of structural
constants containing only 0 and 1. In the second case we consider $n=\nu=2$ but
with general constrains. In both cases we study dynamical systems generated by
the quadratic evolution operators of corresponding constrained algebras. We
find all fixed points, limit points and some 2-periodic points of the dynamical
systems. Moreover we study several properties of the constrained algebras
connecting them to the dynamical systems. We give some biological
interpretation of our results.
