Generalized Lotka–Volterra schemes and the construction of two-dimensional explodator cores and their liapunov functions via‘critical’ Hopf bifurcations

Abstract

A new class of generalized Lotka–Volterra schemes has been investigated. It is shown that such systems are conservative. Their first integrals can be used as Liapunov functions to prove the globally stable or explosive behaviour of a wide class of modifications. On modifying the value of an exponent in one rate law, a ‘critical’ Hopf bifurcation occurs and the stable and the explosive regions are separated by a critical value at which conservative oscillations take place. The explosive schemes can be regarded as two-dimensional explodator cores. Limit-cycle oscillators can be constructed using these cores and one or more limiting reactions.