Perturbation of a globally stable steady state

Authors:
H. L. Smith and P. Waltman

Journal:
Proc. Amer. Math. Soc. **127** (1999), 447-453

MSC (1991):
Primary 34C35, 34E10, 58F30

MathSciNet review:
1487341

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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that within a parameterized family of semi-dynamical systems enjoying a mild uniform dissipative condition, the property that a locally asymptotically stable steady state is globally attracting is an open condition in the parameters.

**1.**Herbert Amann,*Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces*, SIAM Rev.**18**(1976), no. 4, 620–709. MR**0415432**

Herbert Amann,*Errata: “Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces” (SIAM Rev. 18 (1976), no. 4, 620–709)*, SIAM Rev.**19**(1977), no. 4, vii. MR**0467410****2.**Geoffrey Butler and Paul Waltman,*Persistence in dynamical systems*, J. Differential Equations**63**(1986), no. 2, 255–263. MR**848269**, 10.1016/0022-0396(86)90049-5**3.**Geoffrey Butler, H. I. Freedman, and Paul Waltman,*Uniformly persistent systems*, Proc. Amer. Math. Soc.**96**(1986), no. 3, 425–430. MR**822433**, 10.1090/S0002-9939-1986-0822433-4**4.**Jack K. Hale and Paul Waltman,*Persistence in infinite-dimensional systems*, SIAM J. Math. Anal.**20**(1989), no. 2, 388–395. MR**982666**, 10.1137/0520025**5.**M. A. Krasnosel′skiĭ,*Positive solutions of operator equations*, Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964. MR**0181881****6.**H. L. Smith,*On the basin of attraction of a perturbed attractor*, Nonlinear Anal.**6**(1982), no. 9, 911–917. MR**677616**, 10.1016/0362-546X(82)90010-4**7.**Hal L. Smith,*Monotone dynamical systems*, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR**1319817****8.**Hal L. Smith and Paul Waltman,*The theory of the chemostat*, Cambridge Studies in Mathematical Biology, vol. 13, Cambridge University Press, Cambridge, 1995. Dynamics of microbial competition. MR**1315301****9.**Horst R. Thieme,*Persistence under relaxed point-dissipativity (with application to an endemic model)*, SIAM J. Math. Anal.**24**(1993), no. 2, 407–435. MR**1205534**, 10.1137/0524026**10.**Eberhard Zeidler,*Nonlinear functional analysis and its applications. I*, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. MR**816732**

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Additional Information

**H. L. Smith**

Affiliation:
Department of Mathematics, Arizona State University, Tempe, Arizona 85287–1804

Email:
halsmith@asu.edu

**P. Waltman**

Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322

Email:
waltman@mathcs.emory.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-04768-1

Received by editor(s):
May 20, 1997

Additional Notes:
The first author was supported by NSF Grant DMS 9300974, and the second author was supported by NSF Grant DMS 9424592 and an award from the University Research Council of Emory University

Communicated by:
Linda Keen

Article copyright:
© Copyright 1999
American Mathematical Society