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Successional stability of vector fields
in dimension three

Author: Sebastian J. Schreiber
Journal: Proc. Amer. Math. Soc. 127 (1999), 993-1002
MSC (1991): Primary 34D30, 58F12; Secondary 92D40, 92D25
MathSciNet review: 1641101
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Abstract: A topological invariant, the community transition graph, is introduced for dissipative vector fields that preserve the skeleton of the positive orthant. A vector field is defined to be successionally stable if it lies in an open set of vector fields with the same community transition graph. In dimension three, it is shown that vector fields for which the origin is a connected component of the chain recurrent set can be approximated in the $C^1$ Whitney topology by a successionally stable vector field.

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  • 1. R. A. Armstrong and R. McGehee, Competitive exclusion, Amer. Natur. 116 (1980), 151-170. MR 82d:92029
  • 2. W. Brannath, Heteroclinic networks on the tetrahedron, Nonlinearity 7 (1994), 1367-1384. MR 95e:58128
  • 3. G. J. Butler, H. I. Freedman, and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96 (1986), 425-430. MR 87d:58119
  • 4. G. J. Butler and P. Waltman, Persistence in dynamical systems, J. Differential Equations 63 (1986), 255-263. MR 87k:54058
  • 5. T. J. Case, Invasion resistence, species build-up and community collapse in metapopulation models with interspecies competition, Bio. J. of the Linn. Soc. 42 (1991), 239-266.
  • 6. C. Conley, Isolated Invariant Sets and Morse Index, Amer. Math. Soc., CBMS 38 (1978). MR 80c:58009
  • 7. J. A. Drake, The mechanics of community assembly and succession, J. Theor. Bio. 147 (1990), 213-233.
  • 8. B. M. Garay, Uniform persistence and chain recurrence, J. Math. Anal. and Appl. 139 (1989), 372-382. MR 90h:54049
  • 9. A. Hastings and A. Klebanoff, Chaos in three-species food chains, J. Math. Biol. 32 (1994), 427-451. MR 95g:92019
  • 10. J. Hofbauer, Heteroclinic cycles in ecological differential equations, Tatra Mt. Math. Publ 4 (1994), 105-116. MR 95i:34083
  • 11. J. Hofbauer and K. Sigmund, The theory of evolution and dynamical systems: mathematical aspects of selection, Cambridge University Press, Cambridge, 1988. MR 91h:92019
  • 12. S. B. Hsu, S. P. Hubbell, and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of microorganisms, SIAM J. Appl. Math. 32 (1977), 366-383. MR 55:7458
  • 13. S. B. Hsu, H. L. Smith, and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc. 348 (1996), 4083-4094. MR 97d:92021
  • 14. V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci. 111 (1992), 1-71. MR 93d:92003
  • 15. M. Krupa, Robust heteroclinic cycles, J. Nonlinear Sci. 7 (1997), 129-176. MR 98g:58126
  • 16. M. Krupa and I. Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry, Ergod. Th. & Dynam. Sys. 15 (1995), 121-147. MR 96g:58101
  • 17. I. Kupka, Contribution á la theorie des champs generiques, Contrib. Diff. Eqs. 2 (1963), 457-484. MR 29:2818a
  • 18. R. Law and R. D. Morton, Alternative permanent states of ecological communities, Ecology 74 (1993), 1347-1361.
  • 19. -, Permanence and the assembly of ecological communities, Ecology 77 (1996), 762-775.
  • 20. R. M. May, Stability and complexity in model ecosystems, 2nd edn., Princeton University Press, Princeton, 1975.
  • 21. Z. Nitecki and M. Shub, Filtrations, decompositions and explosions, Amer. J. Math. 97 (1975), 1029-1047. MR 52:15561
  • 22. J. Palis and W. de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York, 1982. MR 84a:58004
  • 23. M. M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1966), 214-227. MR 35:499
  • 24. W. M. Post and S. L. Pimm, Community assembly and food web stability, Math. Biosci. 64 (1983), 169-192.
  • 25. C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math 89 (1967), 1010-1021. MR 37:2257
  • 26. S. J. Schreiber, Generalist and specialist predators that mediate permanence in ecological communities, J. Math. Bio. 36 (1997), 133-148. MR 98k:92017
  • 27. P. Schuster, K. Sigmund, and R. Wolff, Dynamical systems under constant organization 3: Cooperative and competitive behavior of hypercycles, J. Differential Equations 32 (1979), 357-368. MR 82b:34035b
  • 28. S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Normale Superiore Pisa 18 (1963), 97-116. MR 29:2818b
  • 29. H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: the variable-yield model, SIAM J. Appl. Math. 54 (1994), 1113-1131. MR 95b:92014
  • 30. F. W. Wilson, Smoothing derivatives of functions and applications, Trans. Amer. Math. Soc. 139 (1969), 413-428. MR 40:4974
  • 31. G. S. K. Wolkowicz, Successful invasion of a food web in a chemostat, Math. Biosci. 93 (1989), 249-268. MR 90f:92027

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Additional Information

Sebastian J. Schreiber
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225

Keywords: Generic properties of vector fields, ecological succession, population dynamics
Received by editor(s): January 28, 1997
Additional Notes: Part of this research was completed during a postdoctoral fellowship sponsored by Andrew P. Gutierrez at the University of California, Berkeley. For his support and encouragement, the author is grateful.
Communicated by: Hal L. Smith
Article copyright: © Copyright 1999 American Mathematical Society