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. [ABSTRACT FROM AUTHOR]