Transitivity of a Lotka-Volterra map

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January 2008, 9(1): 75-82. doi: 10.3934/dcdsb.2008.9.75

Transitivity of a Lotka-Volterra map

Juan Luis García Guirao ^1, and Marek Lampart ^2,

1.	Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30203-Cartagena, Spain
2.	Mathematical Institute at Opava, Silesian University at Opava, Na Rybníčku 1, 746 01 Opava, Czech Republic

Received May 2007 Revised July 2007 Published October 2007

Abstract
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The dynamics of the transformation $F: (x,y)\rightarrow (x(4-x-y),xy)$ defined on the plane triangle $\Delta$ of vertices $(0,0)$, $(0,4)$ and $(4,0)$ plays an important role in the behaviour of the Lotka--Volterra map. In 1993, A. N. SharkovskiĬ (Proc. Oberwolfach 20/1993) stated some problems on it, in particular a question about the trasitivity of $F$ was posed. The main aim of this paper is to prove that for every non--empty open set $\mathcal{U} \subset \Delta$ there is an integer $n_{0}$ such that for each $n>n_{0}$ it is $F^{n}(\mathcal{U}) \supseteq \Delta \setminus P_{\varepsilon}$, where $P_{\varepsilon} = \{ (x,y) \in D : y<\beta, \mbox{ $where$ F(t,\varepsilon)=(\alpha,\beta) \mbox{ and } t \in[0,2] \}$ and $\varepsilon \rightarrow 0$ for $n \rightarrow \infty$. Consequently, we show that the map $F$ is transitive, it is not topologically exact and it is almost topologically exact. Additionally, we prove that the union of all preimages of the point $(1,2)$ is a dense subset of $\Delta$.

Keywords: Lotka-Volterra map, almost topologically exact system., transitive system, topologically exact system.

Mathematics Subject Classification: Primary: 37B99, 37B1.

Citation: Juan Luis García Guirao, Marek Lampart. Transitivity of a Lotka-Volterra map. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 75-82. doi: 10.3934/dcdsb.2008.9.75

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