Journal of Applied Mathematics and Decision Sciences
Volume 3 (1999), Issue 2, Pages 155-162

Global properties of the three-dimensional predator-prey Lotka-Volterra systems

A. Korobeinikov1 and G. C. Wake2

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.


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.