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A remark on the global dynamics of competitive systems on ordered Banach spaces


Authors: King-Yeung Lam and Daniel Munther
Journal: Proc. Amer. Math. Soc. 144 (2016), 1153-1159
MSC (2010): Primary 47H07, 47H20; Secondary 92D40
DOI: https://doi.org/10.1090/proc12768
Published electronically: July 10, 2015
MathSciNet review: 3447668
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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^+ \times X_2^+$, the product of two cones in respective Banach spaces, if $ (u^*,0)$ and $ (0,v^*)$ are the global attractors in $ X_1^+ \times \{0\}$ and $ \{0\}\times 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^*] \times [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.


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Additional Information

King-Yeung Lam
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email: lam.184@math.osu.edu

Daniel Munther
Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115
Email: d.munther@csuohio.edu

DOI: https://doi.org/10.1090/proc12768
Received by editor(s): January 28, 2015
Received by editor(s) in revised form: February 22, 2015
Published electronically: July 10, 2015
Additional Notes: The first author was partially supported by NSF Grant DMS-1411476
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society