Christophe LETELLIER
28/11/2023

In 1976, Michael Gilpin met Robert May and Otto E. Rössler - at the conference organized by the latter with Peter Ortoleva at the New York Academy of Science - where Otto suggested that a Volterra model with three species should provide chaos. Following this suggestion, Gilpin restarted from the work by Richard Vance [1] investigated the simplest three-species model [2]

This model is simple in the sense that the three species are treated in a similar way : they only differ by the values of the coefficient. Consequently, they are characterized by two different variable *per capita* predation rates. In his work, Vance observed aperiodic behaviors that he termed *quasi-cyclic*. From the exchanges with May and Rössler, Gilpin understood that chaos was possible to produce with the classical Volterra equations.

With the parameter values

Gilpin obtained a spiral chaos (Fig. 1), that is, a chaotic attractor which is topologically equivalent to the spiral Rössler attractor. Here the state portrait is the differential embedding induced by the *y* variable which allows to define the Poincaré section in a simple way as

The first-return map to this Poincaré section is a smooth unimodal map which is a sufficient condition for getting a period-doubling cascade, thanks to the Myrberg theorem. For these parameter values, the dissipation rate is -826 per revolution.

**Fig. 1. Spiral chaos in the three-species Volterra model.**

[1] **R. R. Vance**, Predation and resource partitioning in one predator-two prey model communities, *The American Naturalist*, **112**, 797-813, 1978.

[2] **M. E. Gilpin**, Spiral chaos in a predator-prey model, *The American Naturalist*, **113** (2), 306-308, 1979.