Asymptotic properties of solutions in a model of interaction of populations with several delays

  • Skvortsova Maria A., sm-18-nsu@yandex.ru Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, Novosibirsk 630090, Russia; Novosibirsk State University, 1 Pirogov Street, Novosibirsk 630090, Russia
Keywords: model of interaction of populations, delay differential equations, estimates of solutions, modified Lyapunov-Krasovskii functional

Abstract

We consider a model describing the interaction of n species of microorganisms. The model is presented as a system of nonlinear differential equations with several constant delays. The components of solutions to the system are responsible for the biomass of the i-th species and for the concentration of the nutrient. The delay parameters denote the time required for the conversion of the nutrient to viable biomass for the i-th species. In the paper, we study asymptotic properties of the solutions to the considered system. Under certain conditions on the system parameters, we obtain estimates for all components of the solutions. The established estimates characterize the rate of decrease in the biomass of microorganisms and the rate of stabilization of the nutrient concentration to the initial value of the concentration. The estimates are constructive and all values characterizing the stabilization rate are indicated explicitly. The modified Lyapunov–Krasovskii functionals were used to obtain the estimates.

References


[1]
Wolkowicz G. S. K., Xia H., “Global asymptotic behavior of a chemostat model with discrete delays,” SIAM J. Appl. Math., 57, No. 4, 1019–1043 (1997).

[2]
Demidenko G. V. and Matveeva I. I., “Asymptotic properties of solutions to delay differential equations [in Russian],” Vestn. NGU. Ser. Mat. Mekh. Inform., 5, No. 3, 20–28 (2005).

[3]
Khusainov D. Ya., Ivanov A. F., Kozhametov A. T., “Convergence estimates for solutions of linear stationary systems of differential-difference equations with constant delay,” Differ. Equ., 41, No. 8, 1196–1200 (2005).

[4]
Mondie S. and Kharitonov V. L., “Exponential estimates for retarded time-delay systems: LMI approach,” IEEE Trans. Autom. Control., 50, No. 2, 268–273 (2005).

[5]
Demidenko G. V. and Matveeva I. I., “Stability of solutions to delay differential equations with periodic coefficients of linear terms,” Sib. Math. J., 48, No. 5, 824–836 (2007).

[6]
Demidenko G. V., “Stability of solutions to linear differential equations of neutral type,” J. Anal. Appl., 7, No. 3, 119–130 (2009).

[7]
Vodop’yanov E. S. and Demidenko G. V., “Asymptotic stability of the solutions of linear differential equations with retarded argument under the perturbation of the coefficients [in Russian],” Mat. Zamet. YAGU, 18, No. 2, 32–40 (2011).

[8]
Matveeva I. I. and Shcheglova A. A., “Estimates of the solutions of a class of nonlinear differential equations with retarded argument and parameters [in Russian],” Mat. Zamet. YAGU, 19, No. 1, 60–69 (2012).

[9]
Matveeva I. I., “Estimates of solutions to a class of systems of nonlinear delay differential equations,” J. Appl. Ind. Math., 7, No. 4, 557–566 (2013).

[10]
Demidenko G. V. and Matveeva I. I., “On estimates of solutions to systems of differentia equations of neutral type with periodic coefficients,” Sib. Math. J., 55, No. 5, 866–881 (2014).

[11]
Matveeva I. I., “On exponential stability of solutions to periodic neutral-type systems,” Sib. Math. J., 58, No. 2, 264–270 (2017).

[12]
Matveeva I. I., “Estimates of the exponential decay of solutions to linear systems of neutral type with periodic coefficients,” J. Appl. Ind. Math., 13, No. 3, 511–518 (2019).

[13]
Yskak T., “On the stability of systems of linear differential equations of neutral type with distributed delay,” J. Appl. Ind. Math., 13, No. 3, 575–583 (2019).

[14]
Skvortsova M. A., “Stability of solutions in the predator-prey model with delay [in Russian],” Mat. Zamet. SVFU, 23, No. 2, 108–120 (2016).

[15]
Skvortsova M. A., “Estimates for solutions in a predator-prey model with delay [in Russian],” Vestn. Irkutsk Gos. Univ., Ser. Mat., 25, 109–125 (2018).

[16]
Skvortsova M. A., “On estimates of solutions in a predator-prey model with two delays [in Russian],” Sib. Electron. Math. Rep., 15, 1697–1718 (2018).

[17]
Hartman Ph., Ordinary Differential Equations, John Wiley & Sons, New York; London; Sydney (1964).
How to Cite
Skvortsova, M. (2020) “Asymptotic properties of solutions in a model of interaction of populations with several delays”, Mathematical notes of NEFU, 26(4), pp. 63-72. doi: https://doi.org/10.25587/SVFU.2019.23.62.006.
Section
Mathematics