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 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.