Document Type
Article
Publication Date
3-1-2016
Publication Title
Proceedings of The American Mathematical Society
Abstract
A well-known result in [Hsu-Smith-Waltman, Trans. Amer. Math. Soc. (1996)] states that in a competitive semiflow defined on , the product of two cones in respective Banach spaces, if and are the global attractors in and 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 attracts all trajectories initiating in the order interval . However, it was demonstrated by an example that in some cases neither nor is globally asymptotically stable if we broaden our scope to all of . In this paper, we give two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of or among all trajectories in . Namely, one of or 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.
Repository Citation
Lam, King Yeung and Munther, Daniel, "A Remark on The Global Dynamics of Competitive Systems on Ordered Banach Spaces" (2016). Mathematics and Statistics Faculty Publications. 160.
https://engagedscholarship.csuohio.edu/scimath_facpub/160
DOI
10.1090/proc12768
Version
Postprint
Publisher's Statement
First published in Proceedings of The American Mathematical Society in 2016, published by the American Mathematical Society.
Volume
144
Issue
3
Comments
The first author was partially supported by NSF Grant DMS-1411476.