Proceedings of the American Mathematical Society

Author(s): Sebastian J. Schreiber
Journal: Proc. Amer. Math. Soc. 127 (1999), 993-1002.
MSC (1991): Primary 34D30, 58F12; Secondary 92D40, 92D25
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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

Whitney topology by a successionally stable vector field.

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

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 34D30, 58F12, 92D40, 92D25

Sebastian J. Schreiber
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225
Email: sschreib@cc.wwu.edu

DOI: 10.1090/S0002-9939-99-05169-2
PII: S 0002-9939(99)05169-2
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
Copyright of article: Copyright 1999, American Mathematical Society

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