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Discrete Dynamics in Nature and Society
Volume 2017, Article ID 7893049, 6 pages
https://doi.org/10.1155/2017/7893049
Research Article

Convergence of a Logistic Type Ultradiscrete Model

1Tokyo Metropolitan Ogikubo High School, 5-7-20 Ogikubo, Suginami-ku, Tokyo 167-0051, Japan
2Department of Applied Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
3Department of Mathematics, Shimane University, 1600 Nishikawatsu-cho, Matsue-shi, Shimane 690-8504, Japan

Correspondence should be addressed to Masaki Sekiguchi; moc.liamg@smrafame

Received 5 May 2017; Accepted 16 July 2017; Published 27 August 2017

Academic Editor: Francisco R. Villatoro

Copyright © 2017 Masaki Sekiguchi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We derive a piecewise linear difference equation from logistic equations with time delay by ultradiscretization. The logistic equation that we consider in this paper has been shown to be globally stable in the continuous and discrete time formulations. Here, we study if ultradiscretization preserves the global stability property, analyzing the asymptotic behaviour of the obtained piecewise linear difference equation. It is shown that our piecewise linear difference equation has a threshold property concerning global attractivity of equilibria, similar to the stable logistic equations with time delay.