Advertisement

Journal of Dynamics and Differential Equations

, Volume 28, Issue 3–4, pp 1163–1186 | Cite as

Persistence and Permanence for a Class of Functional Differential Equations with Infinite Delay

  • Teresa Faria
Article

Abstract

The paper deals with a class of cooperative functional differential equations (FDEs) with infinite delay, for which sufficient conditions for persistence and permanence are established. Here, the persistence refers to all solutions with initial conditions that are positive, continuous and bounded. The present method applies to a very broad class of abstract systems of FDEs with infinite delay, both autonomous and non-autonomous, which include many important models used in mathematical biology. Moreover, the hypotheses imposed are in general very easy to check. The results are illustrated with some selected examples.

Keywords

Infinite delay Persistence Permanence Quasimonotone condition Lotka–Volterra model 

AMS Subject Classification

34K12 34K25 92D25 

Notes

Acknowledgments

Work supported by Fundação para a Ciência e a Tecnologia, under UID/MAT/04561/2013.

References

  1. 1.
    Ahmad, S., Lazer, A.C.: Average growth and total permanence in a competitive Lotka-Volterra system. Ann. Mat. Pura Appl. 185, S47–S67 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aiello, W.G., Freedman, H.I.: A time-delay model of single species growth with stage structure. Math. Biosci. 101, 139–153 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aiello, W.G., Freedman, H.I., Wu, J.: Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM J. Appl. Math. 52, 855–869 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Al-Omari, J.F.M., Al-Omari, S.K.Q.: Global stability in a structured population model with distributed maturation delay and harvesting. Nonlinear Anal. RWA 12, 1485–1499 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Al-Omari, J.F.M., Gourley, S.A.: Stability and traveling fronts in Lotka-Volterra competition models with stage structure. SIAM J. Appl. Math. 63, 2063–2086 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Arino, J., Wang, L., Wolkowicz, G.S.K.: An alternative formulation for a delayed logistic equation. J. Theor. Biol. 241, 109–119 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bastinec, J., Berezansky, L., Diblik, J., Smarda, Z.: On a delay population model with a quadratic nonlinearity without positive steady state. Appl. Math. Comput. 227, 622–629 (2014)MathSciNetGoogle Scholar
  8. 8.
    Faria, T.: Stability and extinction for Lotka-Volterra systems with infinite delay. J. Dyn. Differ. Equ. 22, 299–324 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Faria, T.: A note on permanence of nonautonomous cooperative scalar population models with delays. Appl. Math. Comput. 240, 82–90 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Faria, T.: Global dynamics for Lotka-Volterra systems with infinite delay and patch structure. Appl. Math. Comput. 245, 575–590 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Faria, T., Muroya, Y.: Global attractivity and extinction for Lotka-Volterra systems with infinite delay and feedback controls. Proc. R. Soc. Edinb. Sect. A 145, 301–330 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Faria, T., Oliveira, J.J.: General criteria for asymptotic and exponential stabilities of neural network models with unbounded delay. Appl. Math. Comput. 217, 9646–9658 (2011)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Freedman, H.I., Wu, J.H.: Persistence and global asymptotic stability of single species dispersal models with stage structure. Q. Appl. Math. 49, 351–371 (1991)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Haddock, J.R., Hornor, W.: Precompactness and convergence in norm of positive orbits in a certain fading memory space. Funkcial. Ekvac. 31, 349–361 (1988)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Haddock, J.R., Nkashama, M.N., Wu, J.H.: Asymptotic constancy for linear neutral Volterra integrodifferential equations. Tôhoku Math. J. 41, 689–710 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hale, J.K.: Functional differential equations with infinite delay. J. Math. Anal. Appl. 48, 276–283 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hale, J.K., Kato, J.: Phase space for retarded equations with infinite delay. Funkcial. Ekvac. 21, 11–41 (1978)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hino, Y., Murakami, S., Naito, T.: Functional Differential Equations with Infinite Delay. Springer, New-York (1993)zbMATHGoogle Scholar
  19. 19.
    Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)zbMATHGoogle Scholar
  20. 20.
    Kuang, Y.: Global stability in delay differential systems without dominating instantaneous negative feedbacks. J. Differ. Equ. 119, 503–532 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kuang, Y., Smith, H.L.: Global stability for infinite delay Lotka-Volterra type systems. J. Differ. Equ. 103, 221–246 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mallet-Paret, J., Sell, G.R.: Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions. J. Differ. Equ. 125, 385–440 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mallet-Paret, J., Sell, G.R.: The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay. J. Differ. Equ. 125, 441–489 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Meng, X., Chen, L.: Periodic solution and almost periodic solution for a nonautonomous Lotka Volterra dispersal system with infinite delay. J. Math. Anal. Appl. 339, 125–145 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Miller, R.K.: Asymptotic behavior of solutions of nonlinear Volterra equations. Bull. Am. Math. Soc. 72, 153–156 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Miller, R.K., Sell, G.R.: Existence, uniqueness and continuity of solutions of integral equations. Ann. Mat. Pura Appl. 80, 135–152 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Murakami, S., Naito, T.: Fading memory spaces and stability properties for functional differential equations with infinite delay. Funkcial. Ekval. 32, 91–105 (1989)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Saito, Y., Takeuchi, Y.: A time-delay for predator-prey growth with stage structure. Can. Appl. Math. Q. 13, 293–302 (2003)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Smith, H.L.: Invariant curves for mappings. SIAM J. Math. Anal. 15, 1053–1067 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Smith, H.L.: Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems. American Mathematical Society, Providence, RI (1995)zbMATHGoogle Scholar
  31. 31.
    Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. American Mathematical Society, Providence, RI (2011)zbMATHGoogle Scholar
  32. 32.
    Teng, Z., Chen, L.: Global stability of periodic Lotka-Volterra systems with delays. Nonlinear Anal. 45, 1081–1095 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Teng, Z., Rehim, M.: Persistence in nonautonomous predator-prey systems with infinite delay. J. Comput. Appl. Math. 197, 302–321 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wu, J.: Global dynamics of strongly monotone retarded equations with infinite delay. J. Integral Equ. Appl. 4, 273–307 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zhao, X.-Q., Jing, Z.-J.: Global asymptotic behavior in some cooperative systems of functional differential equations. Can. Appl. Math. Q. 4, 421–444 (1996)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Departamento de Matemática and CMAF-CIO, Faculdade de CiênciasUniversidade de LisboaLisboaPortugal

Personalised recommendations