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Asymptotic Behavior of Solutions to a Delay Integro-Differential Equation Arising in Models of Living Systems


We consider a mathematical model describing the production of the components of some living system under the influence of positive and negative feedback. The model is presented in the form of the Cauchy problem for a nonlinear delay integro-differential equation. A theorem of the existence, uniqueness, and nonnegativity of the solutions to the model on the half-axis is proved for nonnegative initial data. The questions of the asymptotic behavior of the solutions and the stability of the equilibria of the model are investigated. Sufficient conditions for the asymptotic stability are obtained for nontrivial equilibria and the boundaries of their attraction domains are estimated. Examples illustrating the application of the obtained theoretical results are given.

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  1. 1

    N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Linear Functional Differential Equations (Nauka, Moscow, 1991; World Federation Publishers Company, Atlanta, GA, 1995).

    MATH  Google Scholar 

  2. 2

    N. V. Azbelev and V. V. Malygina, “On stability of trivial solution of nonlinear equations with aftereffect,” Izv. Vyssh. Uchebn. Zaved., Mat. No. 6(385), 20 (1994) [Russ. Math. 38:6, 18 (1994)].

    MathSciNet  MATH  Google Scholar 

  3. 3

    A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (Academic Press, New York, 1979).

    MATH  Google Scholar 

  4. 4

    L. Collatz, Functional Analysis and Numerical Mathematics (Mir, Moscow, 1969) [in Russian].

    Google Scholar 

  5. 5

    G. V. Demidenko and I. A. Mel’nik, “On a method of approximation of solutions to delay differential equations,” Sib. Mat. Zh. 51, 528 (2010) [Sib. Math. J. 51, 419 (2010)].

    MathSciNet  Article  Google Scholar 

  6. 6

    B. P. Demidovich, Lectures on Mathematical Stability Theory (Nauka, Moscow, 1967) [in Russian].

    MATH  Google Scholar 

  7. 7

    L. E. El’sgol’ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments (Nauka, Moscow, 1971; Academic Press, New York–London, 1973).

    MATH  Google Scholar 

  8. 8

    L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1984; Pergamon Press, Oxford etc., 1982).

    MATH  Google Scholar 

  9. 9

    V. B. Kolmanovskĭ and V. R. Nosov, Stability and Periodic Regimes of Control Systems with Aftereffect (Nauka, Moscow, 1981) [in Russian].

    Google Scholar 

  10. 10

    M. A. Krasnosel’skiĭ, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskiĭ, and V. Ya. Stetsenko, Approximate Solution of Operator Equations (Nauka, Moscow, 1969; Wolters–Noordhoff Publishing, Groningen, 1972).

    MATH  Google Scholar 

  11. 11

    M. R. S. Kulenovic, G. Ladas, and Y. G. Sficas, “Global attractivity in population dynamics,” Comput. Math. Appl. 18, 925 (1989).

    MathSciNet  Article  Google Scholar 

  12. 12

    G. Liu, J. Yan, and F. Zhang, “Existence and global attractivity of unique positive periodic solution for a model of hematopoiesis,” J. Math. Anal. Appl. 334, 157 (2007).

    MathSciNet  Article  Google Scholar 

  13. 13

    M. C. Mackey and L. Glass, “Oscillations and chaos in physiological control systems,” Science 197, 287 (1977).

    Article  Google Scholar 

  14. 14

    H. S. Morris, E. E. Ryan, and R. K. Dodd, “Periodic solutions and chaos in a delay-differential equation modelling haematopoesis,” Nonlinear Anal., Theory Methods Appl. (TMA) 7, 623 (1983).

    MathSciNet  Article  Google Scholar 

  15. 15

    N. V. Pertsev, “Two-sided estimates on solutions to an integro-differential equation which describes the process of hematogenic process,” Izv. Vyssh. Uchebn. Zaved., Mat. No. 6, 58 (2001) [Russ. Math. 45:6, 55 (2001)].

    MATH  Google Scholar 

  16. 16

    N. V. Pertsev, “Application of the monotone method and M-matrices to studying the behavior of solutions to some models of biological processes,” Sib. Zh. Ind. Mat. 5:4, 110 (2002) [in Russian].

    MathSciNet  MATH  Google Scholar 

  17. 17

    N. V. Pertsev, “Application of M-matrices in construction of exponential estimates for solutions to the Cauchy problem for systems of linear difference and differential equations,” Mat. Tr. 16:2, 111 (2013) [Sib. Adv. Math. 24, 240 (2014)].

    Article  Google Scholar 

  18. 18

    N. V. Pertsev, “Two-sided estimates for solutions to the Cauchy problem for Wazewski linear differential systems with delay,” Sib. Mat. Zh. 54, 1368 (2013) [Sib. Math. J. 54, 1088 (2013)].

    MathSciNet  Article  Google Scholar 

  19. 19

    Ya. B. Pesin, “Behavior of solutions to a strongly nonlinear differential equation with delay argument,” Differ. Uravn. 10, 1025 (1974) [Differ. Equations 10, 789 (1975)].

    MATH  Google Scholar 

  20. 20

    V. V. Voevodin and Yu. A. Kuznetsov, Matrices and Computations (Nauka, Moscow, 1984) [in Russian].

    MATH  Google Scholar 

Download references


The authors were financially supported by the Russian Foundation for Basic Research (project no. 18–29–10086).

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Correspondence to K. K. Loginov or N. V. Pertsev.

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Loginov, K.K., Pertsev, N.V. Asymptotic Behavior of Solutions to a Delay Integro-Differential Equation Arising in Models of Living Systems. Sib. Adv. Math. 31, 131–146 (2021).

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  • delay integro-differential equation
  • boundedness of solutions
  • asymptotic behavior of solutions
  • stability of equilibria
  • mathematical model of a living system