On Dynamical Systems of Quadratic Stochastic Operators Constructed for Bisexual Populations

For two classes of bisexual populations, we give a constructive description of quadratic stochastic operators which act to the Cartesian product of standard simplices. We 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, in a similar way, the male types are indexed by $\{1,2,\dots,\nu\}$. Quadratic stochastic operators are constructed for the bisexual population for the cases $n=\nu=2$ and $n=\nu=4$. In both cases, we study the dynamical systems generated by the quadratic operators of the bisexual population. We find all fixed points and limit points of the dynamical systems. Moreover, we give some biological interpretations of our results.