Asymptotic Behaviour for a Class of Non-monotone Delay Differential Systems with Applications

Abstract

The paper concerns a class of n-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.

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References

  1. 1.

    Bélair, J., Mackey, M., Mahaffy, J.: Age-structured and two-delays models for erythropoiesis. Math. Biosci. 241, 109–119 (2006)

    MATH  Google Scholar 

  2. 2.

    Berezansky, L., Braverman, E.: Boundedness and persistence of delay differential equations with mixed nonlinearity. Appl. Math. Comput. 279, 154–169 (2016)

    MathSciNet  Google Scholar 

  3. 3.

    Berezansky, L., Braverman, E., Idels, L.: Nicholson’s blowflies differential equations revisited: main results and open problems. Appl. Math. Model. 34, 1405–1417 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Berezansky, L., Idels, L., Troib, L.: Global dynamics of Nicholson-type delay systems with applications. Nonlinear Anal. RWA 12, 436–445 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Coppel, W.A.: Dichotomies in Stability Theory. Lecture Notes in Mathematics, vol. 629. Springer, Berlin (1978)

    Google Scholar 

  6. 6.

    Cushing, J.M.: An Introduction to Structured Population Dynamics, Conference Series in Applied Mathematics, vol. 71. SIAM, Philadelphia (1998)

    Google Scholar 

  7. 7.

    Faria, T.: Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays. Nonlinear Anal. 74, 7033–7046 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Faria, T.: Global dynamics for Lotka-Volterra systems with infinite delay and patch structure. Appl. Math. Comput. 245, 575–590 (2014)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Faria, T.: Persistence and permanence for a class of functional differential equations with infinite delay. J. Dyn. Differ. Equ. 28, 1163–1186 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Faria, T., Röst, G.: Persistence, permanence and global stability of an \(n\)-dimensional Nicholson system. J. Dyn. Differ. Equ. 26, 723–744 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Fiedler, M.: Special Matrices and Their Applications in Numerical Mathematics. Martinus Nijhoff Publ. Kluwer, Dordrechit (1986)

    Google Scholar 

  12. 12.

    Fink, A.M.: Almost Periodic Differential Equations. Lecture Notes in Math, vol. 377. Springer, Berlin (1974)

    Google Scholar 

  13. 13.

    Gurney, W.S.C., Blythe, S.P., Nisbet, R.M.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1980)

    Article  Google Scholar 

  14. 14.

    Hale, J.K.: Ordinary Differential Equations. Wiley, New York (1969)

    Google Scholar 

  15. 15.

    Liu, B.: Global stability of a class of delay differential equations. J. Comput. Appl. Math. 233, 217–223 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Liu, B.: Global stability of a class of Nicholson’s blowflies model with patch structure and multiple time-varying delays. Nonlinear Anal. RWA 11, 2557–2562 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Liu, B.: The existence and uniqueness of positive periodic solutions of Nicholson-type delay systems. Nonlinear Anal. RWA 12, 3145–3451 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Liu, B.: Global dynamic behaviors for a delayed Nicholson’s blowflies model with a linear harvesting term. Electron. J. Qual. Theory Differ. Equ. 2013(45), 1–13 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Mackey, M.C., Glass, L.: Oscillations and chaos in physiological control systems. Science 197(4300), 287–289 (1997)

    Article  MATH  Google Scholar 

  20. 20.

    Metz, J.A.J., Diekmann, O.: The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomath, vol. 68. Springer, Berlin (1986)

    Google Scholar 

  21. 21.

    Novo, S., Obaya, R., Sanz, A.M.: Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows. Nonlinearity 26, 1–32 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Obaya, R., Sanz, A.M.: Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems. J. Differ. Equ. 261, 4135–4163 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Sacker, R.J., Sell, G.R.: A spectral theory for linear differential systems. J. Differ. Equ. 27, 320–358 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Smith, H.L.: Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs. Providence, American Mathematical Society (1995)

    Google Scholar 

  25. 25.

    Smith, H.L.: An Introduction to Delay Differential Equations with Applications to Life Sciences, Texts in Applied Mathematics, vol. 57. Springer, Berlin (2011)

    Google Scholar 

  26. 26.

    Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. American Mathematical Society, Providence (2011)

    Google Scholar 

  27. 27.

    Takeuchi, Y., Cui, J., Miyazaki, R., Saito, Y.: Permanence of delayed population model with dispersal loss. Math. Biosci. 201, 143–156 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Wang, L.: Almost periodic solution for Nicholson’s blowflies model with patch structure and linear harvesting terms. Appl. Math. Model. 37, 2153–2165 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Wang, W., Wang, L., Chen, W.: Existence and exponential stability of positive almost periodic solution for Nicholson-type delay systems. Nonlinear Anal. RWA 12, 1938–1949 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Zhang, A.: New results on almost periodic solutions for a Nicholson’s blowflies model with a linear harvesting term. Electron. J. Qual. Theory Differ. Equ. 2014(37), 1–14 (2014)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Zhou, Q.: The positive periodic solution for Nicholson-type delay system with linear harvesting terms. Appl. Math. Model. 37, 5581–5590 (2013)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

This work was partially supported by Fundação para a Ciência e a Tecnologia under project UID/MAT/04561/2013 (T. Faria) and by Ministerio de Economía y Competitividad under project MTM2015-66330, and the European Commission under project H2020-MSCA-ITN-2014 (R. Obaya and A. M. Sanz). The authors are very grateful to the referee, whose careful reading and valuable comments led to significant improvements of the manuscript.

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Correspondence to Teresa Faria.

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Faria, T., Obaya, R. & Sanz, A.M. Asymptotic Behaviour for a Class of Non-monotone Delay Differential Systems with Applications. J Dyn Diff Equat 30, 911–935 (2018). https://doi.org/10.1007/s10884-017-9572-8

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Keywords

  • Delay differential equation
  • Non-autonomous Nicholson system
  • Quasi-monotone condition
  • Persistence
  • Permanence
  • Global asymptotic stability

Mathematics Subject Classification

  • 34K25
  • 34K12
  • 34K27
  • 34K20
  • 92D25