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A REMARK ON THE GLOBAL DYNAMICS OF COMPETITIVE SYSTEMS ON ORDERED BANACH SPACES.
- Source :
-
Proceedings of the American Mathematical Society . Mar2016, Vol. 144 Issue 3, p1153-1159. 7p. - Publication Year :
- 2016
-
Abstract
- A well-known result in [Hsu-Smith-Waltman, Trans. Amer. Math. Soc. (1996)] states that in a competitive semiflow defined on X+ = X+1 × X+2, the product of two cones in respective Banach spaces, if (u*, 0) and (0,v*) are the global attractors in X+1 × {0} and {0} × X+2 respectively, then one of the following three outcomes is possible for the two competitors: either there is at least one coexistence steady state, or one of (u*, 0), (0,v*) attracts all trajectories initiating in the order interval I = [0, u*] × [0, v*]. However, it was demonstrated by an example that in some cases neither (u*,0) nor (0,v*) is globally asymptotically stable if we broaden our scope to all of X+. In this paper, we give two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of (u*, 0) or (0, v*) among all trajectories in X+. Namely, one of (u*,0) or (0,v*) is (i) linearly unstable, or (ii) linearly neutrally stable but zero is a simple eigenvalue. Our results complement the counterexample mentioned in the above paper as well as applications that frequently arise in practice. [ABSTRACT FROM AUTHOR]
- Subjects :
- *BANACH spaces
*OPERATOR spaces
*TOPOLOGY
*VECTOR spaces
*EIGENVALUE equations
Subjects
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 144
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 112315660
- Full Text :
- https://doi.org/10.1090/proc12768