Estimates for solutions in one predator–prey model with two delay parameters
https://doi.org/10.25587/2411-9326-2025-4-92-100
Abstract
We consider a system of differential equations with two delay parameters describing the interaction of predators and prey. Under non-negative initial conditions, we prove the non-negativity and boundedness of solutions. We specify conditions for the coefficients of the system, under which the components of the solution stabilize to zero at infinity. Using Lyapunov–Krasovskii functionals we establish estimates of the stabilization rate.
Keywords
delay differential equations,
predator–prey model,
estimates for solutions,
Lyapunov–Krasovskii functional
Supporting agencies: The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project no. FWNF-2022-0008).
About the Authors
M. A. Skvortsova
Sobolev Institute of Mathematics; Novosibirsk State University
Russian Federation
Russian Federation
Maria A. Skvortsova
4 Koptyug Avenue, 630090 Novosibirsk
Pirogova st., 1, 630090 Novosibirsk
Yu. Mu
Novosibirsk State University
Russian Federation
Russian Federation
Yuanhai Mu
Pirogova st., 1, 630090 Novosibirsk
References
1. Lotka A.J., The Elements of Physical Biology, Baltimore, Williams & Wilkins Co.; London, Bailliere, Tindall & Cox, 1925.
2. Volterra V., “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi,” Memoria della Reale Accademia Nazionale dei Lincei, 2, 31–113 (1926).
3. Volterra V., Lecons sur la th´eorie math´ematique de la lutte pour la vie, Gauthier-Villars, Paris (1931).
4. Nedorezov L. V. and Utyupin Yu. V., Continuous-Discrete Models of Population Dynamics, State Public Scientific and Technological Library of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk (2011).
5. Brauer F. and Castillo-Chavez C., Mathematical Models in Population Biology and Epidemiology, Springer, New York (2001).
6. Ruan S., “On nonlinear dynamics of predator-prey models with discrete delay,” Math. Model. Nat. Phenom., 4, No. 2, 140–188 (2009).
7. Krasovskii N. N., Stability of Motion, Applications of Lyapunov?s Second Method to Differential Systems and Equations with Delay, Stanford Univ. Press, Stanford (1963).
8. Demidenko G. V. and Matveeva I. I., “Asymptotic properties of solutions to delay differential equations [in Russian],” Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform., 5, No. 3, 20–28 (2005).
9. Demidenko G. V. and Matveeva I. I., “The second Lyapunov method for time-delay systems,” in: Functional Differential Equations and Applications (A. Domoshnitsky, A. Rasin, S. Padhi, eds.), Springer Nature, Singapore, 2021, pp. 145–167 (Springer Proc. Math. Stat.; vol. 379).
10. Skvortsova M. A., “On estimates of solutions in a predator–prey model with two delays [in Russian],” Sib. Elektron. Mat. Izv., 15, 1697–1718 (2018).
11. Skvortsova M. A. and Yskak T., “Asymptotic behavior of solutions in one predator–prey model with delay,” Sib. Math. J., 62, No. 2, 324–336 (2021).
12. El′sgol′ts L. E. and Norkin S. B., Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Acad. Press, New York; London (1973) (Math. Sci. Eng.; vol. 105).
Review
For citations:
Skvortsova M.A., Mu Yu. Estimates for solutions in one predator–prey model with two delay parameters. Mathematical notes of NEFU. 2025;32(4):92-100. (In Russ.) https://doi.org/10.25587/2411-9326-2025-4-92-100
Views:
66
JATS XML
Email this article 