Oscillation and global attractivity in a discrete survival red blood cells model
Volume 30 / 2003
Abstract
We consider the discrete survival red blood cells model \begin{equation*} N_{n+1}-N_{n}=-\delta _{n}N_{n}+P_{n}e^{-aN_{n-k}}, \tag{$\ast$}\end{equation*} where $\delta _{n}$ and $P_{n}$ are positive sequences. In the autonomous case we show that $(\ast )$ has a unique positive steady state $N^{\ast }$, we establish some sufficient conditions for oscillation of all positive solutions about $N^{\ast }$, and when $k=1$ we give a sufficient condition for $N^{\ast }$ to be globally asymptotically stable. In the nonatonomous case, assuming that there exists a positive solution $\{ N_{n}^{\ast }\} ,$ we present necessary and sufficient conditions for oscillation of all positive solutions of $(\ast )$ about $\{ N_{n}^{\ast }\} $. Our results can be considered as discrete analogues of the recent results by Saker and Agarwal [12] and solve an open problem posed by Kocic and Ladas [8].
