Three-dimensional competitive Lotka-Volterra systems with no periodic orbits. (English) Zbl 0910.34050

van den Driessche, P.; Zeeman, M. L.

SIAM J. Appl. Math. 58, No. 1, 227-234 (1998).

Summary: The following conjecture of M. L. Zeeman is proved. If three interacting species modeled by a competitive Lotka-Volterra system can resist each invasion at carrying capacity, then there can be no coexistence among the species. Indeed, two of the species are driven to extinction. It is proved that in the other extreme, if none of the species can resist an invasion from either of the others, then there is a stable coexistence of at least two of the species. In this case, if the system has a fixed point in the interior of a positive cone in \(\mathbb{R}^3\), then that fixed point is globally asymptotically stable, representing stable coexistence of all three species. Otherwise, there is a globally asymptotically stable fixed point in one of the coordinate planes of \(\mathbb{R}^3\), representing stable coexistence of two of the species.

MSC:

34C25	Periodic solutions to ordinary differential equations
92D25	Population dynamics (general)
34A26	Geometric methods in ordinary differential equations
92D40	Ecology

Keywords:

competition; Dulac criteria; global asymptotic stability; Lyapunov functions; Lotka-Volterra systems; periodic orbits