Chaos in a Three-Dimensional Volterra-gause Model of Predator-prey Type

The aim of this paper is to present results concerning a three-dimensional model including a prey, a predator and top-predator, which we have named the Volterra–Gause model because it combines the original model of V. Volterra incorporating a logisitic limitation of the P. F. Verhulst type on growth of the prey and a limitation of the G. F. Gause type on the intensity of predation of the predator on the prey and of the top-predator on the predator. This study highlights that this model has several Hopf bifurcations and a period-doubling cascade generating a snail shell-shaped chaotic attractor. With the aim of facilitating the choice of the simplest and most consistent model a comparison is established between this model and the so-called Rosenzweig–MacArthur and Hastings– Powell models. Many resemblances and differences are highlighted and could be used by the modellers. The exact values of the parameters of the Hopf bifurcation are provided for each model as well as the values of the parameters making it possible to carry out the transition from a typical phase portrait characterizing one model to another (Rosenzweig–MacArthur to Hastings–Powell and vice versa). The equations of the Volterra–Gause model cannot be derived from those of the other models, but this study shows similarities between the three models. In cases in which the toppredator has no effect on the predator and consequently on the prey, the models can be reduced to two dimensions. Under certain conditions, these models present slow–fast dynamics and their attractors are lying on a slow manifold surface, the equation of which is given.