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

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.