Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2011, Article ID 147926, 17 pages doi:10.1155/2011/147926 Research Article On a Difference Equation with Exponentially Decreasing Nonlinearity E. Braverman 1 and S. H. Saker 2, 3 1 Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W, Calgary, AB, Canada T2N 1N4 2 Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Correspondence should be addressed to E. Braverman, maelena@math.ucalgary.ca Received 7 April 2011; Accepted 1 June 2011 Academic Editor: Antonia Vecchio Copyright q 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. 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. 1. Introduction In 1950–1980 various biological and medical phenomena were for the first time described using differential equations, among them the blood cell production 1–3. The general approach was to describe the cell production dynamics as N ′  pt, Nt - dt, Nt, 1.1 where N is the number of cells, pt, N is the cell production rate at time t under the condition that the population size is N, and dt, N is the mortality rate. It was a common belief that the mortality is proportional to the amount of blood cells in the circulation dt, Nt  μNt, where μ can be time dependent or not, and that higher per capita production rates correspond