American Institute of Mathematical Sciences

Advanced
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

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
[1]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[2]

Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195
[3]

Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953
[4]

Linping Peng, Zhaosheng Feng, Changjian Liu. Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4807-4826. doi: 10.3934/dcds.2014.34.4807
[5]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020197
[6]

Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559
[7]

Fuke Wu, Yangzi Hu. Stochastic Lotka-Volterra system with unbounded distributed delay. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 275-288. doi: 10.3934/dcdsb.2010.14.275
[8]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083
[9]

Qi Wang, Chunyi Gai, Jingda Yan. Qualitative analysis of a Lotka-Volterra competition system with advection. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1239-1284. doi: 10.3934/dcds.2015.35.1239
[10]

Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171
[11]

Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743
[12]

Grant Cairns, Barry Jessup, Marcel Nicolau. Topologically transitive homeomorphisms of quotients of tori. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 291-300. doi: 10.3934/dcds.1999.5.291
[13]

Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665
[14]

Chiun-Chuan Chen, Li-Chang Hung. Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1451-1469. doi: 10.3934/cpaa.2016.15.1451
[15]

Yoshiaki Muroya. A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model). Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 999-1008. doi: 10.3934/dcdss.2015.8.999
[16]

Zhaohai Ma, Rong Yuan, Yang Wang, Xin Wu. Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2069-2092. doi: 10.3934/cpaa.2019093
[17]

Yuzo Hosono. Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 161-171. doi: 10.3934/dcdsb.2015.20.161
[18]

Hélène Leman, Sylvie Méléard, Sepideh Mirrahimi. Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 469-493. doi: 10.3934/dcdsb.2015.20.469
[19]

Yubin Liu, Peixuan Weng. Asymptotic spreading of a three dimensional Lotka-Volterra cooperative-competitive system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 505-518. doi: 10.3934/dcdsb.2015.20.505
[20]

Qi Wang. Some global dynamics of a Lotka-Volterra competition-diffusion-advection system. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3245-3255. doi: 10.3934/cpaa.2020142

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences