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Estimates of Solutions to Differential Equations with Distributed Delay Describing the Competition of Several Types of Microorganisms

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Abstract

We consider a model of \( n \)-species competition in a chemostat. This model is a system of \( n+1 \) differential equations with infinite distributed delay. The system consists of one equation for the nutrient concentration dynamics and \( n \) equations for the species population dynamics. The transformation of the nutrient into viable cells does not occur instantly and requires some time, which is taken into account by the presence of a delay. Under the condition that the concentration of the input nutrient is below a certain level, Lyapunov–Krasovskii functionals are constructed with the help of which estimates are obtained for all components of the solutions. The estimates characterize the extinction rates of all species in the chemostat and the rate of stabilization of the nutrient concentration to a constant concentration value.

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ACKNOWLEDGMENTS

The authors are grateful to G.V. Demidenko for attention to the work.

Funding

This work was supported financially by the Mathematical Center in Akademgorodok (agreement no. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    M. A. Skvortsova & T. Yskak

Authors
  1. M. A. Skvortsova
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  2. T. Yskak
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Correspondence to M. A. Skvortsova or T. Yskak.

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Translated by V. Potapchouck

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Skvortsova, M.A., Yskak, T. Estimates of Solutions to Differential Equations with Distributed Delay Describing the Competition of Several Types of Microorganisms. J. Appl. Ind. Math. 16, 800–808 (2022). https://doi.org/10.1134/S1990478922040196

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  • DOI: https://doi.org/10.1134/S1990478922040196

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