A brief survey of persistence in dynamical systems

  • Paul Waltman
Survey Articles
Part of the Lecture Notes in Mathematics book series (LNM, volume 1475)


Global Attractor Rest Point Morse Decomposition Uniform Persistence Semidynamical System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Busenberg, S., Cooke, K., Thieme, H.: Demographic change and persistence of HIV/AIDS in a heterogeneous population. To appear, SIAM J. Appl. MathGoogle Scholar
  2. 2.
    Burton, T.A., Hutson, V. (1989): Repellers with infinite delay, J. Math. Anal. Appl. 137, 240–263MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Butler, G., Freedman, H.I., Waltman, P. (1986): Uniformly persistent systems. Proc. AMS 96, 425–430MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Butler, G., Waltman, P. (1986): Persistence in dynamical systems. JDE 63, 255–263MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Conley, C. (1978): Isolated Invariant Sets and the Morse Index. CBMS vol. 38, Amer. Math. Soc., ProvidenceGoogle Scholar
  6. 6.
    Dunbar, S.R., Rybakowski, K.P., Schmitt, K. (1986): Persistence in models of predator-prey populations with diffusion. JDE 65, 117–138MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fonda, A (1988): Uniformly persistent semi-dynamical systems. Proc. AMS 104, 111–116MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Freedman, H.I., Moson, P. (1990): Persistence definitions and their connections. Proc. AMS, 109, 1025–1033MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Freedman, H.I., Rai, B. (1987): Persistence in a predator-prey-competitor-mutualist model. Janos Bolyai Math. Soc., 73–79Google Scholar
  10. 10.
    Freedman, H.I., So, J.W.-H. (1987): Persistence in discrete models of a population which may not be subject to harvesting. Nat. Res. Modeling 1, 135–145MathSciNetzbMATHGoogle Scholar
  11. 11.
    Freedman, H.I., So, J.W.-H. (1989): Persistence in discrete semidynamical systems. SIAM J. Math. Anal. 20, 930–938MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Freedman, H.I., Waltman, P. (1977): Mathematical analysis of some three-species food chain models. Math. Biosc. 33, 257–276MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Freedman, H.I., Waltman, P. (1984): Persistence in a model of three interscting predator-prey populations. Math. Biosc. 68, 213–231MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Freedman, H.I., Waltman, P. (1985): Persistence in a model of three competitive populations. Math Biosc. 73, 89–101MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Garay, B.M. (1989): Uniform persistence and chain recurrence. J. Math. Anal. Appl. 139, 372–381MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gard, T.C. (1980): Persistence in food chains with general interactions. Math. Biosc. 51, 165–174MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gard, T.C. (1987): Uniform persistence in multispecies population models. Math. Biosc. 85, 93–104MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gard, T.C., Hallam, T.G. (1979): Persistence in food webs I: Lotka-Volterra food chains. Bull. Math. Biol. 41, 477–491MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gard, T.C., Hallam, T.G., Svoboda, L.J. (1979): Persistence and extinction in three species Lotka-Volterra competitive systems. Math. Biosc. 46, 117–124MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gatica, J.A., So, J.W.-H. (1988): Predator-prey models with periodic coefficients. Applic. Anal. 27, 143–152MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hale, J.K. (1988): Asymptotic Behavior of Dissipative Systems. Amer. Math. Soc., ProvidencezbMATHGoogle Scholar
  22. 22.
    Harrison, G. (1979): Response of a population to stress: resistance and other stability concepts. Am. Nat. 113, 659–669CrossRefGoogle Scholar
  23. 23.
    Hallam, T.G., Ma, Z. (1986): Persistence in population models with demographic fluctuations. J. Math. Bio. 24, 327–340MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hale, J.K., Waltman, P. (1989): Persistence in infinite dimensional systems. SIAM J. Math. Anal. 20, 388–395MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hofbauer, J. (1980): A general cooperation theorem for hypercycles. Monat. Math. 91, 233–240MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hofbauer, J.: A unified approach to persistence. PreprintGoogle Scholar
  27. 27.
    Hofbauer, J., Hutson, V., Jansen, W. (1987): Coexistence for systems governed by difference equations of Lotka-Volterra type. J. Math Bio. 25, 553–570MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hofbauer, J., Sigmund, K. (1988): Dynamical Systems and the Theory of Evolution. Cambridge University PressGoogle Scholar
  29. 29.
    Hofbauer, J., So, J.W.-H. (1989): Uniform persistence and repellors for maps. Proc. AMS 107, 1137–1142MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hutson, V. (1984): A theorem on average Liapunov functions. Monats. Math. 98, 267–275MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Hutson, V., Law, R. (1985): Permanent coexistence in general models of three interacting species. J. Math. Biol. 21, 285–298MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hutson, V., Moran, W. (1982): Persistence of species obeying difference equations. J. Math. Bio. 15, 203–213MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hutson, V., Schmitt, K.: Permanence in dynamical systems. PreprintGoogle Scholar
  34. 34.
    Hutson, V., Vickers, G.T. (1983): A criterion for permanent coexistence of species with an application to a two-prey one-predator system. Math. Biosc. 63, 253–269MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Jansen, W. (1987): A permanence theorem for replicator and Lotka-Volterra systems. J. Math. Biol. 25, 411–422MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kirlinger, G. (1986): Permanence in Lotka-Volterra equations: linked predator-prey systems. Math. Biosc. 82, 165–191MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kirlinger, G. (1988): Permanence of some ecological systems with several predator and one prey species. J. Math. Biol. 26, 217–232MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Li, J., Hallam, T.G. (1988): Survival in continuous structured population models. J. Math. Bio. 26, 421–433MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    May, R.M., Leonard, W.J. (1975): Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29, 243–253MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Schuster, P. Sigmund, K. Wolf, R. (1979): Dynamical systems under constant organization III. Cooperative and competitive behaviour of hypercycles. JDE 32, 357–386CrossRefzbMATHGoogle Scholar
  41. 41.
    Schuster, P. Sigmund, K. Wolf, R. (1979): On ω-limits for competition between three species. SIAM J. Appl. Math. 37, 49–54MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    So, J.W.-H. (1990): Persistence and extinction in a predator-prey model consisting of nine prey genotypes. J. Australian Math. Soc. Ser.B 31, 347–365MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Thieme, H.R., Castillo-Chavez, C. (1989): On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic. In Mathematical and Statistical Approaches to AIDS Epidemiology (C. Castillo-Chavez, ed.), Lecture Notes in Biomathematics 83, Springer Verlag, Berlin-Heidelberg-New YorkCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Paul Waltman

There are no affiliations available

Personalised recommendations