Geodesic
Evolutionary Behavior in a Two-Locus System
Russian journal of nonlinear dynamics, Tome 19 (2023) no. 3, pp. 297-302.

Voir la notice de l'article provenant de la source Math-Net.Ru

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: gamete, dynamical system, fixed point, trajectory, limit point.
Mots-clés : loci 
@article{ND_2023_19_3_a0,
     author = {A. M. Diyorov and U. A. Rozikov},
     title = {Evolutionary {Behavior} in a {Two-Locus} {System}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {297--302},
     publisher = {mathdoc},
     volume = {19},
     number = {3},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ND_2023_19_3_a0/}
}
TY  - JOUR
AU  - A. M. Diyorov
AU  - U. A. Rozikov
TI  - Evolutionary Behavior in a Two-Locus System
JO  - Russian journal of nonlinear dynamics
PY  - 2023
SP  - 297
EP  - 302
VL  - 19
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ND_2023_19_3_a0/
LA  - en
ID  - ND_2023_19_3_a0
ER  - 
%0 Journal Article
%A A. M. Diyorov
%A U. A. Rozikov
%T Evolutionary Behavior in a Two-Locus System
%J Russian journal of nonlinear dynamics
%D 2023
%P 297-302
%V 19
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ND_2023_19_3_a0/
%G en
%F ND_2023_19_3_a0
A. M. Diyorov; U. A. Rozikov. Evolutionary Behavior in a Two-Locus System. Russian journal of nonlinear dynamics, Tome 19 (2023) no. 3, pp. 297-302. http://geodesic.mathdoc.fr/item/ND_2023_19_3_a0/

[1] Mat. Zametki, 111:5 (2022), 663–675 (Russian) | DOI | DOI | MR | Zbl

[2] Devaney, R. L., An Introduction to Chaotic Dynamical Systems, 3rd ed., CRC, Boca Raton, Fla., 2022, xiii, 419 pp. | MR | Zbl

[3] Ewens, W. J., Mathematical Population Genetics: 1. Theoretical Introduction, Interdiscipl. Appl. Math., 27, 2nd ed., Springer, New York, 2004, xx+417 pp. | DOI | MR

[4] Ganikhodjaev, N. N., Pah, C. H., and Rozikov, U., “Dynamics of Quadratic Stochastic Operators Generated by China's Five Element Philosophy”, J. Difference Equ. Appl., 27:8 (2021), 1173–1192 | DOI | MR | Zbl

[5] Lyubich, Yu. I., Mathematical Structures in Population Genetics, Biomathematics, 22, Springer, Berlin, 1992, x+373 pp. | MR | Zbl

[6] Rozikov, U. A., Population Dynamics: Algebraic and Probabilistic Approach, World Sci. Publ., Hackensack, N.J., 2020, xiv+443 pp. | MR

[7] Rozikov, U. A. and Xudayarov, S. S., “Quadratic Non-Stochastic Operators: Examples of Splitted Chaos”, Ann. Funct. Anal., 13:1 (2022), 17, 17 pp. | DOI | MR | Zbl