Nested tori in a 3-variable mass action model

Abstract

The dynamics of three competitive species are analysed through a mass action kinetic scheme derived from the Lotka-Volterra model. It is demonstrated that in a neighbourhood of bifurcation criticality, or crisis known as the explosive route to chaos, the system exhibits both nearly conservative and strongly dissipative solutions. The behaviour depends on the selection of initial conditions. All dissipative solutions are attracted to a fractal torus that produces a bead-like structure and supports chaotic solutions. In contrast, the enclosed bead interior constrains for a short time nearly conservative periodic orbits which promote formation of a nested tori family in the three-dimensional space. The long-time behaviour, however, exhibits either slowly contracting or slowly expanding scroll solutions. The model possesses qualitative and topological features found in medicine, neurobiology, chemistry, and fluid dynamics. Among these are examples such as oscillatory behaviour of the Belousov-Zhabotinskii chemical reaction and fluid dynamic properties of the Taylor-Couette flow.