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

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.