The Application of the Poincart-Transform to the Lotka-Volterra Model

It is a well known fact that the local stability of an equilibrium point in a system of ordinary differential equations doesnot necessarily imply its global stability. However, the usual methods used in the analysis of stability of equilibrium points in population models establishes only local stability. The restriction to sufficiently small perturbations of the initial conditions frequently rules out all perturbations of interest to the ecologist. The Lotka-Volterra model of n-species interaction is such an example. A Lotka-Volterra model can have a locally stable equilibrium without being globally stable; in fact, may not even be Lagrange-stable, i.e. there may exist an unbounded trajectory.