Journal of Applied Mathematics and Decision Sciences
Volume 3 (1999), Issue 2, Pages 155-162
doi:10.1155/S1173912699000085
Global properties of the three-dimensional
predator-prey Lotka-Volterra systems
1Department of Mathematics, University of Auckland, New Zealand
2Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand
Copyright © 1999 A. Korobeinikov and G. C. Wake. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The global properties of the classical three-dimensional Lotka-Volterra two prey-one
predator and one prey-two predator systems, under the assumption that competition can
be neglected, are analysed with the direct Lyapunov method. It is shown that, except for a
pathological case, one species is always driven to extinction, and the system behaves asymptotically
as a two-dimensional predator-prey Lotka-Volterra system. The same approach can be easily
extended to systems with many prey species and one predator, or many predator species and one
prey, and the same conclusion holds. The situation considered is common for New Zealand wild life,
where indigenous and introduced species interact with devastating consequences for the indigenous
species. According to our results the New Zealand indigenous species are definitely driven to
extinction, not only in consequence of unsuccessful competition, but even when competition is
absent. This result leads to a better understanding of the mechanism of natural selection, and
gives a new insight into pest control practice.