1979 Gilpin’s predator-prey model

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.