DOI 10.15507/2079-6900.24.202202.175-184

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 34D20

On the stability of Lotka-Volterra model with a delay

J. K. Khusanov1, A. E. Kaxxorov2

1Jizzakh Polytechnic Institute (Jizakh, Uzbekistan)

2I. Karimov Tashkent State Technical University (Tashkent, Uzbekistan)

Abstract. The paper examines the stability problem of biological, economic and other processes modeled by the Lotka-Volterra equations with delay. The difference between studied equations and the known ones is that the adaptability functions and the coefficients of the relative change of the interacting subjects or objects are non-linear and take into account variable delay in the action of factors affecting the number of subjects or objects. Moreover, these functions admit the existence of equilibrium positions’ set that is finite in a bounded domain. The stability study of three types of equilibrium positions is carried out using direct analysis of perturbed equations and construction of Lyapunov functionals that satisfy conditions of well-known theorems. Corresponding sufficient conditions for asymptotic stability including global stability are derived, as well as instability and attraction conditions of these positions.

Key Words: Lotka-Volterra model, delay differential equations, equilibrium position, stability

For citation: J. K. Khusanov, A. E. Kaxxorov. On the stability of Lotka-Volterra model with a delay. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 24:2(2022), 175–184. DOI:

Submitted: 28.02.2022; Revised: 01.04.2022; Accepted: 25.05.2022

Information about the authors:

Jumanazar Kh. Khusanov, Professor, Jizzakh Polytechnic Institute (4 I. Karimov St., Jizakh 130100, Uzbekistan), Ph.D. (Phys.-Math.), ORCID:,

Azizbeck E. Kaxxorov, Graduate Student, Academic Lyceum I. Karimov Tashkent State Technical University (2 University St., Tashkent 100095, Uzbekistan), ORCID:,

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.