Abstract
Discrete Lotka-Volterra systems in one dimension (the logistic equation) and two dimensions have been studied extensively, revealing a wealth of complex dynamical regimes. We show that three-dimensional discrete Lotka-Volterra dynamical systems exhibit all of the dynamics of the lower dimensional systems and a great deal more. In fact and in particular, there are dynamical features including analogs of flip bifurcations, Neimark-Sacker bifurcations and chaotic strange attracting sets that are essentially three-dimensional. Among these are new generalizations of Neimark-Sacker bifurcations and novel chaotic strange attractors with distinctive candy cane type shapes. Several of these dynamical are investigated in detail using both analytical and simulation techniques.
Highlights
We shall consider a natural discrete analog of the famous and ubiquitous LotkaVolterra model, developed independently by Alfred Lotka [1] and Vito Volterra [2] for the dynamics of a population comprising several interacting species, say x := ( x1, x2, xm ), as described in such references as [3] [4]
We show that three-dimensional discrete Lotka-Volterra dynamical systems exhibit all of the dynamics of the lower dimensional systems and a great deal more
We shall find and analyze flip bifurcations, certain higher dimensional Neimark-Sacker type bifurcations to be described in the sequel, several 3-dimensional chaotic regimes and a few unusual chaotic strange attractors corresponding to long-term population dynamic states
Summary
We shall consider a natural discrete analog of the famous and ubiquitous LotkaVolterra model, developed independently by Alfred Lotka [1] and Vito Volterra [2] for the dynamics of a population comprising several interacting species, say x := ( x1, x2 , , xm ) , as described in such references as [3] [4]. We shall find and analyze flip bifurcations, certain higher dimensional Neimark-Sacker type bifurcations to be described in the sequel, several 3-dimensional chaotic regimes and a few unusual chaotic strange attractors corresponding to long-term population dynamic states.
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