A class of nonergodic Lotka-Volterra operators

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• 作者：M. Saburov
• 关键词：Lotka ; Volterra operator ; ergodicity ; quadratic stochastic operator
• 刊名：Mathematical Notes
• 出版年：2015
• 出版时间：May 2015
• 年：2015
• 卷：97
• 期：5-6
• 页码：759-763
• 全文大小：540 KB
• 参考文献：1.S. M. Ulam, A Collection ofMathematical Problems (New-York, London, 1960).
  2.M. I. Zakharevich “On the behaviour of trajectories and the ergodic hypothesis for quadratic mappings of a simplex,-Russian Math Survey 33(6), 207-08 (1978).MATH MathSciNet View Article
  3.R. N. Ganikhodzhaev and M. Kh. Saburov, “A generalized model of nonlinear operators of Volterra type and Lyapunov functions,-J. Sib. Fed. Univ. Math. Phys. 1(2), 188-96 (2008).
  4.F. Mukhamedov and M. Saburov, “On homotopy of Volterrian quadratic stochastic operators,-Appl. Math Info. Sci. 4(1), 47-2 (2010).MATH MathSciNet
  5.F. Mukhamedov and M. Saburov, “On dynamics of Lotka-Volterra type operators,-Bull. Malay. Math. Sci. Soc. 37(1), 59-4 (2014).MATH MathSciNet
  6.N. N. Ganikhodzhaev and D. V. Zanin, “On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex,-Russian Math Survey 59(357), 161-62 (2004).MathSciNet
  7.M. Saburov, “Some strange properties of quadratic stochastic Volterra operators,-World Appl. Sci. J. 21(1), 94-7 (2013).MathSciNet
  
• 作者单位：M. Saburov (1)
  
  1. International Islamic University of Malaysia, Kuantan, Malaysia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Russian Library of Science
  
• 出版者：MAIK Nauka/Interperiodica distributed exclusively by Springer Science+Business Media LLC.
• ISSN：1573-8876

文摘

On the basis of some numerical calculations, Ulam has conjectured that the ergodic theorem holds for any quadratic stochastic operator acting on a finite-dimensional simplex. However, Zakharevich showed that Ulam’s conjecture is false in general. Later, Ganikhodzhaev and Zanin generalized Zakharevich’s example to the class of quadratic stochastic Volterra operators acting on a 2D simplex. In this paper, we provide a class of nonergodic Lotka-Volterra operators which includes all previous operators used in this context.