This article is cited in 1 paper

Nonlinear physics and mechanics

Evolutionary Behavior in a Two-Locus System

A. M. Diyorov^a, U. A. Rozikov<sup>bcd

a</sup> The Samarkand branch of Tashkent University of Information Technologies, st. Ibn Sino 2A, Samarkand, 140100 Uzbekistan
^b Central Asian University, st. Milliy Bog 264, Tashkent, 111221 Uzbekistan
^c National University of Uzbekistan, University st. 4, Tashkent, 100174 Uzbekistan
^d V. I. Romanovskiy Institute of Mathematics, University st. 9, Tashkent, 100174 Uzbekistan

Abstract: In this short note we study a dynamical system generated by a two-parametric quadratic operator mapping a 3-dimensional simplex to itself. This is an evolution operator of the frequen- cies of gametes in a two-locus system. We find the set of all (a continuum set of) fixed points and show that each fixed point is nonhyperbolic. We completely describe the set of all limit points of the dynamical system. Namely, for any initial point (taken from the 3-dimensional simplex) we find an invariant set containing the initial point and a unique fixed point of the operator, such that the trajectory of the initial point converges to this fixed point.

Keywords: loci, gamete, dynamical system, fixed point, trajectory, limit point.

MSC: 37N25, 92D10

Received: 25.12.2022
Accepted: 30.06.2023

Language: English

DOI: 10.20537/nd230701