Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 147926, 17 pages
http://dx.doi.org/10.1155/2011/147926
Research Article

On a Difference Equation with Exponentially Decreasing Nonlinearity

E. Braverman1 and S. H. Saker2,3

1Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W, Calgary, AB, Canada T2N 1N4
2Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia
3Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 7 April 2011; Accepted 1 June 2011

Academic Editor: Antonia Vecchio

Copyright © 2011 E. Braverman and S. H. Saker. 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 establish a necessary and sufficient condition for global stability of the nonlinear discrete red blood cells survival model and demonstrate that local asymptotic stability implies global stability. Oscillation and solution bounds are investigated. We also show that, for different values of the parameters, the solution exhibits some time-varying dynamics, that is, if the system is moved in a direction away from stability (by increasing the parameters), then it undergoes a series of bifurcations that leads to increasingly long periodic cycles and finally to deterministic chaos. We also study the chaotic behavior of the model with a constant positive perturbation and prove that, for large enough values of one of the parameters, the perturbed system is again stable.