Successional stability of vector fields in dimension three

Schreiber, Sebastian

doi:10.1090/S0002-9939-99-05169-2

Successional stability of vector fields in dimension three
HTML articles powered by AMS MathViewer

by Sebastian J. Schreiber PDF

Proc. Amer. Math. Soc. 127 (1999), 993-1002 Request permission

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.

References

Robert A. Armstrong and Richard McGehee, Competitive exclusion, Amer. Natur. 115 (1980), no. 2, 151–170. MR 596657, DOI 10.1086/283553
Werner Brannath, Heteroclinic networks on the tetrahedron, Nonlinearity 7 (1994), no. 5, 1367–1384. MR 1294548, DOI 10.1088/0951-7715/7/5/006
Geoffrey Butler, H. I. Freedman, and Paul Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96 (1986), no. 3, 425–430. MR 822433, DOI 10.1090/S0002-9939-1986-0822433-4
Geoffrey Butler and Paul Waltman, Persistence in dynamical systems, J. Differential Equations 63 (1986), no. 2, 255–263. MR 848269, DOI 10.1016/0022-0396(86)90049-5
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.
Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133, DOI 10.1090/cbms/038
J. A. Drake, The mechanics of community assembly and succession, J. Theor. Bio. 147 (1990), 213–233.
Barnabas M. Garay, Uniform persistence and chain recurrence, J. Math. Anal. Appl. 139 (1989), no. 2, 372–381. MR 996964, DOI 10.1016/0022-247X(89)90114-5
Aaron Klebanoff and Alan Hastings, Chaos in three-species food chains, J. Math. Biol. 32 (1994), no. 5, 427–451. MR 1284166, DOI 10.1007/BF00160167
Josef Hofbauer, Heteroclinic cycles in ecological differential equations, Tatra Mt. Math. Publ. 4 (1994), 105–116. Equadiff 8 (Bratislava, 1993). MR 1298459
Josef Hofbauer and Karl Sigmund, The theory of evolution and dynamical systems, London Mathematical Society Student Texts, vol. 7, Cambridge University Press, Cambridge, 1988. Mathematical aspects of selection; Translated from the German. MR 1071180
S. B. Hsu, S. Hubbell, and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math. 32 (1977), no. 2, 366–383. MR 434492, DOI 10.1137/0132030
S. B. Hsu, H. L. Smith, and Paul Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc. 348 (1996), no. 10, 4083–4094. MR 1373638, DOI 10.1090/S0002-9947-96-01724-2
Vivian Hutson and Klaus Schmitt, Permanence and the dynamics of biological systems, Math. Biosci. 111 (1992), no. 1, 1–71. MR 1175114, DOI 10.1016/0025-5564(92)90078-B
M. Krupa, Robust heteroclinic cycles, J. Nonlinear Sci. 7 (1997), no. 2, 129–176. MR 1437986, DOI 10.1007/BF02677976
Martin Krupa and Ian Melbourne, Asymptotic stability of heteroclinic cycles in systems with symmetry, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 121–147. MR 1314972, DOI 10.1017/S0143385700008270
Ivan Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations 2 (1963), 457–484 (French). MR 165536
R. Law and R. D. Morton, Alternative permanent states of ecological communities, Ecology 74 (1993), 1347–1361.
—, Permanence and the assembly of ecological communities, Ecology 77 (1996), 762–775.
R. M. May, Stability and complexity in model ecosystems, 2nd edn., Princeton University Press, Princeton, 1975.
Z. Nitecki and M. Shub, Filtrations, decompositions, and explosions, Amer. J. Math. 97 (1975), no. 4, 1029–1047. MR 394762, DOI 10.2307/2373686
Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541, DOI 10.1007/978-1-4612-5703-5
M. M. Peixoto, On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1967), 214–227. MR 209602, DOI 10.1016/0022-0396(67)90026-5
W. M. Post and S. L. Pimm, Community assembly and food web stability, Math. Biosci. 64 (1983), 169–192.
Charles C. Pugh, An improved closing lemma and a general density theorem, Amer. J. Math. 89 (1967), 1010–1021. MR 226670, DOI 10.2307/2373414
Sebastian J. Schreiber, Generalist and specialist predators that mediate permanence in ecological communities, J. Math. Biol. 36 (1997), no. 2, 133–148. MR 1601772, DOI 10.1007/s002850050094
J. Hofbauer, P. Schuster, K. Sigmund, and R. Wolff, Dynamical systems under constant organization. II. Homogeneous growth functions of degree $p=2$, SIAM J. Appl. Math. 38 (1980), no. 2, 282–304. MR 564015, DOI 10.1137/0138025
Ivan Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations 2 (1963), 457–484 (French). MR 165536
Hal L. Smith and Paul Waltman, Competition for a single limiting resource in continuous culture: the variable-yield model, SIAM J. Appl. Math. 54 (1994), no. 4, 1113–1131. MR 1284704, DOI 10.1137/S0036139993245344
F. Wesley Wilson Jr., Smoothing derivatives of functions and applications, Trans. Amer. Math. Soc. 139 (1969), 413–428. MR 251747, DOI 10.1090/S0002-9947-1969-0251747-9
Gail S. K. Wolkowicz, Successful invasion of a food web in a chemostat, Math. Biosci. 93 (1989), no. 2, 249–268. MR 984280, DOI 10.1016/0025-5564(89)90025-4