A Remark on The Global Dynamics of Competitive Systems on Ordered Banach Spaces

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.