Hsu, Sze-Bi; Waltman, Paul; Ellermeyer, Sean F. A remark on the global asymptotic stability of a dynamical system modeling two species competition. (English) Zbl 0806.92016 Hiroshima Math. J. 24, No. 2, 435-445 (1994). We consider a strongly order preserving semiflow modeling the two-species competition in population biology. We motivate the principal result by three examples of competitive dynamical systems in an infinite dimensional space which is a product space. The first example is the model of the unstirred chemostat with equal diffusions, the second example is the classical Lotka-Volterra two-species competition model with diffusions and Dirichlet boundary conditions, and the third example is the two-species delayed chemostat model. Our principal result is the following:Let \(E_ 0= (0,0)\), \(\widehat{E}= (\widehat{u},0)\) and \(\widetilde {E}=(0, \widetilde{v})\) be the rest points in the Banach space \(C^ +_ 1\times C^ +_ 2\). If \(E_ 0\) and \(\widetilde{E}\) are repellers with respect to \({\overset {\circ} C}^ +_ 1\times {\overset {\circ} C}^ +_ 2\) and \(\widehat{E}\) is a local attractor, then \(E\) is a global attractor of \({\overset {\circ} C}^ +_ 1\times {\overset {\circ} C}^ +_ 2\) provided there are no rest points in \({\overset {\circ} C}^ +_ 1\times {\overset {\circ} C}^ +_ 2\). Reviewer: Sze-Bi Szu Cited in 15 Documents MSC: 92D25 Population dynamics (general) 46N60 Applications of functional analysis in biology and other sciences 37N99 Applications of dynamical systems 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 92D40 Ecology Keywords:strongly order preserving semiflow; two-species competition; product space; unstirred chemostat; Lotka-Volterra competition model; diffusions; Dirichlet boundary conditions; delayed chemostat model; rest points; Banach space; repellers; local attractor; global attractor PDFBibTeX XMLCite \textit{S.-B. Hsu} et al., Hiroshima Math. J. 24, No. 2, 435--445 (1994; Zbl 0806.92016)