On Volterra quadratic stochastic operators of a two-sex population on $S^1\times S^1$

TL;DR

This paper studies a family of Volterra quadratic stochastic operators modeling two-sex populations, analyzing fixed points, their stability, and convergence properties across different parameter subfamilies.

Contribution

It introduces a four-parameter family of operators, characterizes fixed points, and classifies subfamilies with guaranteed convergence to fixed points.

Findings

Each operator has at least two fixed points.

Some operators have infinitely many fixed points.

Certain subfamilies are proven to be regular with convergent trajectories.

Abstract

We consider a four-parametric $(a, b, \alpha, \beta)$ family of Volterra quadratic stochastic operators for a bisexual population (i.e., each organism of the population must belong either to the female sex or the male sex). We show that independently on parameters each such operator has at least two fixed points. Moreover, under some conditions on parameters the operator has infinitely many (continuum) fixed points. Choosing parameters, numerically we show that a fixed point may be any type: attracting, repelling, saddle and non-hyperbolic. We separate five subfamilies of quadratic operators and show that each operator of these subfamilies is regular, i.e. any trajectory constructed by the operator converges to a fixed point.