×

A remark on the global asymptotic stability of a dynamical system modeling two species competition. (English) Zbl 0806.92016

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

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
PDFBibTeX XMLCite