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 + × X 2 + X^+ = X_1^+ \times X_2^+ , the product of two cones in respective Banach spaces, if ( u ∗ , 0 ) (u^*,0) and ( 0 , v ∗ ) (0,v^*) are the global attractors in X 1 + × { 0 } X_1^+ \times \{0\} and { 0 } × X 2 + \{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 ∗ ) (u^*,0), (0,v^*) attracts all trajectories initiating in the order interval I = [ 0 , u ∗ ] × [ 0 , v ∗ ] I = [0,u^*] \times [0,v^*] . However, it was demonstrated by an example that in some cases neither ( u ∗ , 0 ) (u^*,0) nor ( 0 , v ∗ ) (0,v^*) is globally asymptotically stable if we broaden our scope to all of X + 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 ) (u^*,0) or ( 0 , v ∗ ) (0,v^*) among all trajectories in X + X^+ . Namely, one of ( u ∗ , 0 ) (u^*,0) or ( 0 , v ∗ ) (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.