Ginoux, Jean-Marc; Rossetto, Bruno; Jamet, Jean-Louis Chaos in a three-dimensional Volterra-Gause model of predator-prey type. (English) Zbl 1092.37541 Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, No. 5, 1689-1708 (2005). Summary: 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 modelers.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 top-predator 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. Cited in 9 Documents MSC: 37N25 Dynamical systems in biology 34C28 Complex behavior and chaotic systems of ordinary differential equations 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 92D25 Population dynamics (general) PDFBibTeX XMLCite \textit{J.-M. Ginoux} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, No. 5, 1689--1708 (2005; Zbl 1092.37541) Full Text: DOI References: [1] Deng B., Am. Inst. Phys. 11 pp 514– [2] Evans G. T., Biol. Oceanogr. 3 pp 327– [3] DOI: 10.1016/0025-5564(77)90142-0 · Zbl 0363.92022 [4] Gause G. F., The Struggle for Existence (1935) [5] Glass L., From Clocks to Chaos (1988) · Zbl 0705.92004 [6] DOI: 10.2307/1940591 [7] DOI: 10.4039/Ent91385-7 [8] DOI: 10.1007/978-1-4757-2421-9 [9] Lotka A. J., Elements of Physical Biology (1925) · JFM 51.0416.06 [10] DOI: 10.1137/0152097 · Zbl 0774.92024 [11] Ramdani S., Int. J. Bifurcation and Chaos 10 pp 2729– [12] DOI: 10.1016/0304-3800(92)90023-8 [13] DOI: 10.1086/282272 [14] DOI: 10.1126/science.171.3969.385 [15] Verhulst P. F., Corresp. Math. Phys. pp 113– [16] Volterra V., Mem. Acad. Lincei III 6 pp 31– This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.