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Dynamics of competition

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1714)

Keywords

  • Periodic Solution
  • Periodic Orbit
  • Stable Manifold
  • Cooperative System
  • Order Interval

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References

  1. Ahmad, S., Lazer, A.C.: Asymptotic behavior of solutions of periodic competition-diffusion system, Nonlinear Analysis 13 (1993) 263–284.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Berman, A., Plemmons, R., Nonnegative matrices in the mathematical sciences, Academic Press, New York, 1979.

    MATH  Google Scholar 

  3. Capasso, V.: Mathematical structures of epidemic systems, Lecture Notes in Biomathematics, 97, Springer-Verlag, New York.

    Google Scholar 

  4. Conley, C.: The gradient structure of a flow: I, IBM Research, RC 3939 (17806) Yorktown Heights, NY, 1972. Also, Ergodic Theory and Dynamical Systems 8 (1988) 11–26.

    MATH  Google Scholar 

  5. Conley, C.: Isolated Invariant Sets and the Morse Index, CBMS 38, Amer. Math. Soc., Providence, R.I., 1978.

    MATH  Google Scholar 

  6. Conway, E., Hoff, D., Smoller, J.: Large-time behavior of solutions of systems of reaction-diffusion equations, Siam J. Appl. Math. 35 (1978) 1–16.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Coppel, W.A.: Stability and Asymptotic Behavior of Differential Equations, Health, Boston, 1965.

    MATH  Google Scholar 

  8. Dancer, E., Hess, P.: Stability of fixed points for order preserving discrete time dynamical systems, J. reine angew. Math. 419 (1991) 125–139.

    MathSciNet  MATH  Google Scholar 

  9. deMottoni, P.: Qualitative analysis for some quasi-linear parabolic systems, Inst. Math. Pol. Acad. Sci. Zam 190 (1979).

    Google Scholar 

  10. deMottoni, P., Schiaffino, A.: Competition systems with periodic coefficients: a geometric approach, J. Math. Biology 11 (1982) 319–335.

    CrossRef  MathSciNet  Google Scholar 

  11. Goldbeter, A., Segel, L.: Unified mechanism for relay and oscillation of cyclic AMP in Distyostelium discoideum, Proc. Nat. Acad. Sci. U.S.A. 74 (1977) 1543–1547.

    CrossRef  Google Scholar 

  12. Goldbeter, A.: Biochemical Oscillations and Cellular Rhythms, the molecular bases of periodic and chaotic behavior, Cambridge Univ. Press, London, 1996.

    CrossRef  MATH  Google Scholar 

  13. Hale, J., Somolinas, A.: Competition for fluctuating nutrient. J. Math. Biology 18 (1983), 255–280.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Hale, J., Waltman, P. Persistence in infinite-dimensional systems, SIAM J. Math. Anal. 20 (1989), 388–395.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Harrison, G.: Comparing predator-prey models to Luckinbill's experiment with Didinium and Paramecium, Ecology 76 (1995) 357–374.

    CrossRef  Google Scholar 

  16. Hess, P., Lazer, A.C.: On an abstract competition model and applications, Nonlinear Analysis T.M.A. 16 (1991) 917–940.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Hirsch, M.: Systems of differential equations which are competitive or cooperative 1: limit sets, SIAM J. Appl. Math. 13 (1982) 167–179.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Hirsch, M.: Systems of differential equations which are competitive or cooperative II: convergence almost everywhere, SIAM J. Math. Anal. 16 (1985), 423–439.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. Hirsch, M.: The dynamical systems approach to differential equations, Bull. A.M.S. 11 (1984) 1–64.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Hirsch, M.: Systems of differential equations which are competitive or cooperative III. Competing species. Nonlinearity 1 (1988a) 51–71.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. Hirsch, M.: Stability and Convergence in Strongly Monotone dynamical systems, J. reine angew. Math. 383 (1988b) 1–53.

    MathSciNet  MATH  Google Scholar 

  22. Hirsch, M.: Systems of differential equations that are competitive or cooperative. IV: Structural stability in three dimensional systems. SIAM J. Math. Anal. 21 (1990) 1225–1234.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. Hirsch, M.: Systems of differential equations that are competitive or cooperative. V: Convergence in 3-dimensional systems, J. Diff. Eqns. 80 (1989) 94–106.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. Hirsch, M.: Systems of differential equations that are competitive or cooperative. VI.: A local C r closing lemma for 3-dimensional systems, Ergod. Th. Dynamical Sys. 11 (1991) 443–454.

    MathSciNet  MATH  Google Scholar 

  25. Hofbauer, J. and So, J.W.-H.: Multiple limit cycles for three dimensional Lotka-Volterra equations, Appl. Math. Lett. 7 (1994) 65–70.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. Hsu, S.-B., Smith, H., Waltman, P.: Dynamics of competition in the unstirred chemostat, Canadian Applied Math. Quart. 2 (1994) 461–483.

    MathSciNet  MATH  Google Scholar 

  27. Hsu, S.-B., Smith, H., Waltman, P.: Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc. 348 (1996) 4083–4094.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. Kishimoto, K., Weinberger, H.: The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Diff. Eqns. 58 (1985) 15–21.

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. Leung, A.: Systems of Nonlinear Partial Differential Equations, Kluwer Academic Publishers, Boston, 1989.

    CrossRef  Google Scholar 

  30. Luckinbill, L.: Coexistence in laboratory populations of Paramecium aurelia and its predator Didinium nasutum, Ecology 54 (1973) 1320–1327.

    CrossRef  Google Scholar 

  31. Matano, H., Mimura, M.: Pattern formation in competition-diffusion systems in nonconvex domains, Pub. Res. Inst. Math. Sci. Kyoto Univ. 19 (1983) 1050–1079.

    MathSciNet  MATH  Google Scholar 

  32. McShane, E.J.: Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934) 837–842.

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. Mimura, M., Ei, S.-I., Fang, Q.: Effect of domain shape on coexistence problems in a competition-diffusion system, J. Math. Biol. 29 (1991) 219–237.

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. Morita, Y.: Reaction-Diffusion systems in nonconvex domains: invariant manifold and reduced form, J. Dyn. and Diff. Eqns. 2 (1990) 69–115.

    CrossRef  MathSciNet  MATH  Google Scholar 

  35. Nisbet, R.M., Gurney, W.S.C.: Modelling Flutuating Populations, New York, Wiley, 1982.

    MATH  Google Scholar 

  36. Pao, C.V.: Nonlinear parabolic and elliptic equations, Plenum Press, New York, 1992.

    MATH  Google Scholar 

  37. Pascual, M.: Periodic response to periodic forcing of the Droop equations for phytoplankton growth. J. Math. Biol. 32 (1994) 743–759.

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations, Prentice Hall, N.J., 1967.

    MATH  Google Scholar 

  39. Segel, L.: Modeling dynamic phenomena in molecular and cellular biology, Cambridge, London, 1984.

    Google Scholar 

  40. Smale, S.: On the differential equations of species in competition. J. Math. Biol. 3 (1976) 5–7.

    CrossRef  MathSciNet  MATH  Google Scholar 

  41. Smoller, J.: Shock Waves and Reaction Diffusion Equations, Springer-Verlag, New York, 1983.

    CrossRef  MATH  Google Scholar 

  42. Smith, H.L.: Periodic orbits of competitive and cooperative systems, J. Diff. Eqns. 65 (1986) 361–373.

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. Smith, H.L.: Monotone Dynamical Systems: An introduction to the Theory of Competitive and Cooperative Systems AMS Math. Surv. & Monographs 41, Providence, R.I, 1995.

    Google Scholar 

  44. Smith, H.L.: An application of monotone dynamical systems theory to a model of microbial competition, in Differential Equations and Control Theory, Proc. of Int. Conf. on Differential Equations and Control Theory, Wuhan, China, ed. Z. Deng et al, Marcel Dekker, Inc., New York, 1996.

    Google Scholar 

  45. Smith, H.L.: The periodically forced Droop model for phytoplankton growth in a chemostat J. Math. Biol. 35 (1997) 545–556.

    CrossRef  MathSciNet  MATH  Google Scholar 

  46. Smith, H.L.: Periodic competitive differential equations and the discrete dynamics of competitive maps, J. Diff. Eqns., 64 (1986) 165–194.

    CrossRef  MathSciNet  MATH  Google Scholar 

  47. Smith, H.L.: Periodic solutions of periodic competitive and cooperative systems, Siam J. Math. Anal. 17 (1986) 1289–1318.

    CrossRef  MathSciNet  MATH  Google Scholar 

  48. Smith, H.L.: Complicated dynamics for low-dimensional strongly monotone maps, to appear, Proc. WCNA, (1996).

    Google Scholar 

  49. Smith, H.L.: Planar Competitive and Cooperative Difference Equations, to appear, J. Difference Equations.

    Google Scholar 

  50. Smith, H., Waltman, P.: The Theory of the Chemostat, Cambridge Univ. Press, London, 1995.

    CrossRef  MATH  Google Scholar 

  51. Smith, H.L., Waltman, P.: Competition for a single limiting resource in continuous culture: the variable yield model, SIAM J. Appl. Math. 54 (1994) 1113–1131.

    CrossRef  MathSciNet  MATH  Google Scholar 

  52. Smith, H., Waltman, P.: Competition in an unstirred multidimensional chemostat, in Differential Equations and Applications to Biology and Industry, Proceedings of Claremont International Conference Dedicated to the Memory of Stavros Busenberg, (M. Martelli et al, eds.), World Scientific, Singapore, 1996.

    Google Scholar 

  53. Smith, H., Waltman, P.: A classification theorem for three dimensional competitive systems, J. Diff. Eq. 70 (1987) 325–332.

    CrossRef  MathSciNet  MATH  Google Scholar 

  54. Thieme, H.R.: Persistence under relaxed point-dissipativity (with application to an epidemic model), SIAM J. Math Anal. 24 (1993) 407–435.

    CrossRef  MathSciNet  MATH  Google Scholar 

  55. Tereščák, I.: Dynamics of C 1 smooth strongly monotone discrete-time dynamical systems, preprint.

    Google Scholar 

  56. Xiao, D., Li, W.: Limit cycles for competitive three-dimensional Lotka-Volterra system, preprint.

    Google Scholar 

  57. Zhu, H.-R.: The Existence of Stable Periodic Orbits for Systems of Three Dimensional Differential Equations that are Competitive, Ph.D. thesis, Arizona State University, 1991.

    Google Scholar 

  58. Zhu, H.-R., Smith, H.L.: Stable periodic orbits for a class of three dimensional competitive systems, J. Diff. Eqns. 110 (1994) 143–156.

    CrossRef  MathSciNet  MATH  Google Scholar 

  59. Zeeman, M.L.: Hopf bifurcation in competitive three-dimensional Lotka-Volterra systems, Dynamics and Stability of Systems 8, (1993) 189–217.

    CrossRef  MathSciNet  MATH  Google Scholar 

  60. Zhou, L., Pao, C.V.: Asymptotic behavior of a competitive-diffusion system in population dynamics, J. Nonlinear Analysis 6 (1982) 1163–1184.

    CrossRef  MathSciNet  MATH  Google Scholar 

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Smith, H.L. (1999). Dynamics of competition. In: Capasso, V. (eds) Mathematics Inspired by Biology. Lecture Notes in Mathematics, vol 1714. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0092378

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  • DOI: https://doi.org/10.1007/BFb0092378

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