Global Properties of the Three Dimensional Lotka-Volterra andLeslie-Gower Prey-Predator Models: A Comparative Analysis
by
Andrei Korobeinikov
University of Auckland
Coauthors: G. C. Wake

The generalisation of predator-prey models on three dimensional case leads to two possible cases: two prey--one predator system and one prey--two predator system. Applying the direct Lyapunov's method we analyse the global properties of the classical Lotka-Volterra and deterministic Leslie-Gower predator-prey models, both with assumption of unlimited grows of the prey population. The analysis of the three dimensional Lotka-Volterra system shows that, except for a pathological case, one species is always driven to extinction, and any solution of the system approaches asymptotically to that of the two dimensional Lotka-Volterra predator-prey system. In the pathological case coexistence of all three species is possible. In this case the three dimensional system is equivalent to the two dimensional system and can be completely solved. The global behaviour of the deterministic Leslie-Gower two prey-one predator system is the same as that of the equivalent Lotka-Volterra system in this respect, that is, except for a pathological case, one prey is always driven to extinction and any solution approaches asymptotically to that of the two-dimensional system.

But the global behaviour of the deterministic Leslie-Gower two predator-one prey system is completely different from that of the equivalent Lotka-Volterra system in that, for the Leslie-Gower model, all three species will always coexist.

The same approach can be used for systems with many prey and one predator and to systems with many predator species and the one prey and this leads to the same outcomes. This result leads to a better understanding of the mechanism of natural selection, and gives a new insight to pest control.

Date received: June 23, 1998