Difference equations arise in many contexts in biological, economic and social sciences., can exhibit a complicated dynamical behavior, from stable equilibria to a bifurcating hierarchy of cycles. There are a lot of fascinating problems, which are often concerned with both mathematical aspects of the fine structure of the trajectories and practical applications. In this paper, we investigate the generalized rational difference equation, a kind of fractional linear maps with two delays. Sufficient conditions for the global asymptotic stability of the zero fixed point are given. For the positive equilibrium, we find the region of parameters in which the positive equilibrium is local asymptotic stable and attracts all positive solutions. As for general solutions, two specific and easy checked conditions on the initial values are obtained to guarantee corresponding solutions to be eventually positive. The upper or lower bound are also provided according to different parameters. Of particular interest for this generalized equation would be the existence of periodic solutions and their stabilities. We get the necessary and sufficient conditions for the existence of period two solutions depending on the combination of delay terms. In addition, the sufficient conditions for the existence of 2r− and 2d−periodic solutions are obtained too.In the end of the paper, we give examples to illustrate our results.
| Published in | American Journal of Applied Mathematics (Volume 10, Issue 1) |
| DOI | 10.11648/j.ajam.20221001.12 |
| Page(s) | 9-14 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Rational Difference Equation, Delay, Eventually Bounded, Eventually Positive, Periodic Solution
| [1] | R. Sacker, Semigroups of maps and periodic difference equations, Journal of Difference Equations & Applications, 16 (1) (2010), 1–13. |
| [2] | S. Elaydi, R. Sacker, Population Models with Allee Effect: A New Model, Journal of Biological Dynamics, 4 (4) (2010), 397–408. |
| [3] | C. Haskell, Y. Yang, R. Sacker, Resonance and attenuation in the n−periodic Beverton-Holt equation, Journal of Difference Equations & Applications 19 (7) (2013), 1174–1191. |
| [4] | Garren R. J. Gaut, Katja Goldring, Francesca Grogan, Cymra Haskell and Sacker Robert J., Difference equations with the Allee effect and the periodic Sigmoid Beverton-Holt equation revisited, Journal of Biological Dynamics 6 (2) (2012), 1019–1033. |
| [5] | Y. Hao, C. Li, Researches on the Properties of Periodic Solutions of Beverton-Holt Equation, Journal of Harbin Institute of Technology(New series) https://kns.cnki.net/kcms/detail/23.1378.t.20211008. 2125.002.html (2021). |
| [6] |
E. A. Grove, G. Ladas, M. Predescu and M. Radin, On the global charecter of  , Math. Sci. Res. Hot-line, 5 (2001), 25–39.
|
| [7] |
E. Chatterjee, R. DeVault and G. Ladas, On the global charecter of  , International Journal of Applied Mathematical Sciences, 2 (2005), 39–46.
|
| [8] |
E. Camouzis, E. Chatterjee and G. Ladas, On the Dynamics of  , Journal of Mathematical Analysis and Applications, 331 (2007), 230–239.
|
| [9] | M.R.S. Kulenovi´ c, G. Ladas and N.R. Prokup, On a rational difference equation, Computers and Mathematics with Applications, 41 (2001), 671–678. |
| [10] | E. Camouzis and G. Ladas, Dynamics of Third Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC Press, Boca Raton, London, and New York, 2008. |
| [11] | Esha Chatterjee and Sk. Sarif Hassan, On the Asymptotic Character of a Generalized Rational Difference Equation, Journal of Discrete and Continuous Dynamical Systems, 38 (Number 4, April 2018), 1707–1718. |
| [12] | R. C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, Ann Arbor, London, and Tokyo, 1995. |
| [13] | C.W. Clark, A delayed recruitment model of population dynamics with an application to baleen whale populations, Journal of Math. Biol., 3 (1976), 381–391. |
| [14] |
E. A. Grove, G. Ladas, M. Predescu and M. Radin, On the global character of the difference equation  , Journal of Difference Equations and Applications, 9 (2003), 171–199.
|
| [15] |
R. DeVault, G. Ladas and S. W. Schultz, On the recursive sequence  , Proc. Amer. Math. Soc., 126 (1998), 3257–3261.
|
APA Style
Yingchao Hao, Cuiping Li. (2022). Some Characters of a Generalized Rational Difference Equation. American Journal of Applied Mathematics, 10(1), 9-14. https://doi.org/10.11648/j.ajam.20221001.12
ACS Style
Yingchao Hao; Cuiping Li. Some Characters of a Generalized Rational Difference Equation. Am. J. Appl. Math. 2022, 10(1), 9-14. doi: 10.11648/j.ajam.20221001.12
AMA Style
Yingchao Hao, Cuiping Li. Some Characters of a Generalized Rational Difference Equation. Am J Appl Math. 2022;10(1):9-14. doi: 10.11648/j.ajam.20221001.12
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ajam.20221001.12,
author = {Yingchao Hao and Cuiping Li},
title = {Some Characters of a Generalized Rational Difference Equation},
journal = {American Journal of Applied Mathematics},
volume = {10},
number = {1},
pages = {9-14},
doi = {10.11648/j.ajam.20221001.12},
url = {https://doi.org/10.11648/j.ajam.20221001.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221001.12},
abstract = {Difference equations arise in many contexts in biological, economic and social sciences., can exhibit a complicated dynamical behavior, from stable equilibria to a bifurcating hierarchy of cycles. There are a lot of fascinating problems, which are often concerned with both mathematical aspects of the fine structure of the trajectories and practical applications. In this paper, we investigate the generalized rational difference equation, a kind of fractional linear maps with two delays. Sufficient conditions for the global asymptotic stability of the zero fixed point are given. For the positive equilibrium, we find the region of parameters in which the positive equilibrium is local asymptotic stable and attracts all positive solutions. As for general solutions, two specific and easy checked conditions on the initial values are obtained to guarantee corresponding solutions to be eventually positive. The upper or lower bound are also provided according to different parameters. Of particular interest for this generalized equation would be the existence of periodic solutions and their stabilities. We get the necessary and sufficient conditions for the existence of period two solutions depending on the combination of delay terms. In addition, the sufficient conditions for the existence of 2r− and 2d−periodic solutions are obtained too.In the end of the paper, we give examples to illustrate our results.},
year = {2022}
}
TY - JOUR T1 - Some Characters of a Generalized Rational Difference Equation AU - Yingchao Hao AU - Cuiping Li Y1 - 2022/02/18 PY - 2022 N1 - https://doi.org/10.11648/j.ajam.20221001.12 DO - 10.11648/j.ajam.20221001.12 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 9 EP - 14 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20221001.12 AB - Difference equations arise in many contexts in biological, economic and social sciences., can exhibit a complicated dynamical behavior, from stable equilibria to a bifurcating hierarchy of cycles. There are a lot of fascinating problems, which are often concerned with both mathematical aspects of the fine structure of the trajectories and practical applications. In this paper, we investigate the generalized rational difference equation, a kind of fractional linear maps with two delays. Sufficient conditions for the global asymptotic stability of the zero fixed point are given. For the positive equilibrium, we find the region of parameters in which the positive equilibrium is local asymptotic stable and attracts all positive solutions. As for general solutions, two specific and easy checked conditions on the initial values are obtained to guarantee corresponding solutions to be eventually positive. The upper or lower bound are also provided according to different parameters. Of particular interest for this generalized equation would be the existence of periodic solutions and their stabilities. We get the necessary and sufficient conditions for the existence of period two solutions depending on the combination of delay terms. In addition, the sufficient conditions for the existence of 2r− and 2d−periodic solutions are obtained too.In the end of the paper, we give examples to illustrate our results. VL - 10 IS - 1 ER -
Information
Email: