Acta Mathematicae Applicatae Sinica

, Volume 22, Issue 2, pp 313–324 | Cite as

Dynamic Behavior of a Logistic Type Equation with Infinite Delay

Original Papers

Abstract

A non-autonomous Logistic type equation with infinite delay is investigated. For general nonautonomous case, sufficient conditions which guarantee the uniform persistence and globally attractivity of the system are obtained; For almost periodic case, by means of a suitable Lyapunov functional, sufficient conditions are derived for the existence and uniqueness of almost periodic solution of the system. Some new results are obtained.

Keywords

Logistic type equation infinite delay Lyapunov functional existence uniqueness almost periodic solution 

2000 MR Subject Classification

34C25 92D25 34D20 34D40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barbălat I. Systems d’equations differential d’oscillations nonlinearies. Rev. Roumaine Math. Pure Appl., 4(2): 267–270 (1959)Google Scholar
  2. 2.
    Cao, Y.L., Thomas, C.G. Ultimate bounds and global asymptotic stability for differential delay equations. Rocky Mountain Journal of Mathematics, 25: 119–131 (1995)MATHMathSciNetGoogle Scholar
  3. 3.
    Chen, F.D., Shi, J.L. Periodicity in a Logistic type system with several delays. Computer and Mathematics with Applications, 48(1-2): 35–44 (2004)MATHMathSciNetGoogle Scholar
  4. 4.
    Chen, F.D. Positive periodic solutions of neutral Lotka-Volterra system with feedback control. Applied Mathematics and Computation, 162(3): 1279–1302 (2005)MATHMathSciNetGoogle Scholar
  5. 5.
    Chen, F.D., Sun, D.X., Shi, J.L. Periodicity in a food-limited population model with toxicants and state dependent delays. J. Math. Anal. Appl., 288(1): 132–142 (2003)CrossRefGoogle Scholar
  6. 6.
    Chen, F.D., Lin, F.X., Chen, X.X. Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control. Applied Mathematics and Computation, 158(1): 45–68 (2004)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chen, F.D. Periodic solution of nonlinear Integral-differential equations with infinite delay. Acta Mathematicae Applicatae Sinica, 26(1): 141–148 (2003) (in Chinese).MATHMathSciNetGoogle Scholar
  8. 8.
    Chen, F.D., Sun, D.X., Shi, J.L. On the existence and uniqueness of periodic solutions of a kind of Integro-differential equations. Acta Math. Sinica, 47(5): 973–984 (2004) (in Chinese)MathSciNetGoogle Scholar
  9. 9.
    Chen, F.D. On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay. Journal of Computational and Applied Mathematics, 180(1): 33–49 (2005)MATHMathSciNetGoogle Scholar
  10. 10.
    Chen, Y.M. Periodic solution of a delayed periodic Logistic equation. Applied Mathematics Letters, 16: 1047–1051 (2003)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Fan, G., Ouyang, Z. Existence of positive periodic solution for a single species model with state dependent delay. Journal of Biomathematics, 17(2): 173–178 (2002)Google Scholar
  12. 12.
    Fan, M., Ye, D., Wong, P.J.Y., Agarwal, R.P. Periodicity in a class of non-autonomous scalar equations with deviating arguments and applications to population models. Dynamical Systems: An International Journal, 19(3): 279–301 (2004)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Feng, C.H. On the existence and uniqueness of almost periodic solutions for delay Logistic equations. Applied Mathematics and Computation, 136: 487–494 (2003)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Gopalsamy, K. Stability and oscillation in delay differential equations of population dynamics, Mathematics and its Applications 74. Kluwer Academic Publishers Group, Dordrecht, 1992Google Scholar
  15. 15.
    Gopalsamy, K., He, X.Z., Wen, L. Global attractivity and level-crossing in a periodic logistic integro differential equation. Math. Nachr., 156: 25–44 (1992)MATHMathSciNetGoogle Scholar
  16. 16.
    Gopalsamy, K., He, X.Z., Wen, L. Global attractivity and Oscillations in an almost periodic delay logistic equation. Nonlinear Times and Digest, 1: 9–24 (1994)MATHMathSciNetGoogle Scholar
  17. 17.
    Gui, Z., Chen, L. Persistence and periodic solutions of a periodic Logistic equation with time delays. Journal of Mathematical Research and Exposition, 23(1): 109–114 (2003) (in Chinese)MATHMathSciNetGoogle Scholar
  18. 18.
    He, X.Z. Applications of delay differential equations to population dynamics in variable environments, Ph. D thesis. The Flinders University of South Australia, 1995Google Scholar
  19. 19.
    Hino, Y. Almost periodic solutions of functional differential equations with infinite retardation. Tohoku. Math. J., 32: 525–530 (1980)MATHMathSciNetGoogle Scholar
  20. 20.
    Li, Y., Kuang, Y. Periodic solutions in periodic state-dependent delay equations and population models. Proceeding of the American Mathematical Society, 130(5): 1345–1353 (2002)MATHMathSciNetGoogle Scholar
  21. 21.
    Li, Y.K. On a periodic Logistic equation with several delay. Advances in Mathematics, 28: 135–142 (1999)MATHGoogle Scholar
  22. 22.
    Sawano, K. Exponential asymptotic stability for functional differential equations with infinite retardation. Tohoku. Math. J., 31: 363–382 (1979)MATHMathSciNetGoogle Scholar
  23. 23.
    Seifert, G. Almost periodic solutions for delay Logistic equations with almost periodic time dependence. Differential and Integral Equations, 9(2): 335–342 (1996)MATHMathSciNetGoogle Scholar
  24. 24.
    Teng, Z.D. Permanence and stability in non-autonomous logistic systems with infinite delay. Dyn. Syst., 17(3): 187–202 (2002)MATHMathSciNetGoogle Scholar
  25. 25.
    Xu, J.H. The existence theorem of almost periodic solution for the Logistic equation with infinite delays. Ann. Math., 23(A): 307–310 (2002) (in Chinese)MATHGoogle Scholar
  26. 26.
    Xu, J.H., Wang, Z.C. Persistence for logistic equations with infinite delays. Math. Sci. Res. Hot-Line, 2(4): 31–35 (1998)MATHMathSciNetGoogle Scholar
  27. 27.
    Yan, J.R., Feng, Q.X. Global existence and oscillation in a nonlinear delay equation. Nonlinear Analysis, 43: 101–108 (2001)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Yang, X.T. The existence and asymptotic behavior of almost periodic solution for the Logistic equations with infinite delay. J. Sys. Sci. & Math. Scis., 21(4): 405–408 (2001) (in Chinese)MATHGoogle Scholar
  29. 29.
    Zhang, B.G., Gopalsamy, K. Global attractivity and oscillations in a periodic delay Logistic equation. J. Math. Anal. Appl., 150: 274–283 (1990)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.College of Mathematics and Computer ScienceFuzhou UniversityFuzhou 350002China

Personalised recommendations