On dynamical systems of quadratic stochastic operators constructed for bisexual populations

TL;DR

This paper constructs and analyzes quadratic stochastic operators for bisexual populations with multiple types, studying their fixed points and dynamics to understand population evolution.

Contribution

It provides a constructive description of quadratic stochastic operators for bisexual populations with multiple types and analyzes their dynamical behavior.

Findings

Identified all fixed points of the quadratic operators.

Determined limit points of the dynamical systems.

Provided biological interpretations of the mathematical results.

Abstract

For two classes of bisexual populations we give a constructive description of quadratic stochastic operators which act to the Cartesian product of standard simplexes. We consider a bisexual population such that the set of females can be partitioned into finitely many different types indexed by $\{1,2,...,n\}$ and, similarly, that the male types are indexed by $\{1,2,...,\nu\}$. Quadratic stochastic operators were constructed for the bisexual population for the cases $n=\nu=2$ and $n=\nu=4$. In both cases, we study 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.