Global asymptotic behavior of the difference equation \(y_{n+1}=\frac{\alpha \cdot \text{e}^{-(ny_n+(n-k)y_{n-k})}}{\beta +ny_n+(n-k)y_{n-k}}\) (Q1021893)

The authors are investigating the boundedness and the global asymptotic behavior of the solutions of the difference equation \[ y_{n+1}=\frac{\alpha \cdot \exp\left[-ny_n-(n-k) y_{n-k}\right]}{\beta+ ny_n+(n-k) y_{n-k}},\quad n=0,1,2,\dots \] where the parameters \(\alpha\) and \(\beta\) are positive real numbers, the delay \(k\) is a positive integer, and the initial conditions \(y_{-k},\dots,y_{-1},y_0\) are real numbers. It appears to me that the above difference equation is non-autonomous (\(n\) is explicitly present in the equation). As such, utilizing ideas that have only been developed for autonomous equations led to incorrect results. It is worth mentioning that an ``equilibrium'' of a non-autonomous difference (or differential) equation, if it exists, is called \textit{frozen time} or \textit{instantaneous} equilibrium, and in general it is not a solution of the given equation. Interested readers may wish to review \textit{S. Wiggins}' [Introduction to applied nonlinear dynamical systems and chaos. 2nd ed. New York, NY: Springer (2003; Zbl 1027.37002), pp. 5--7]. In particular, local stability cannot be inferred from linearizing about an instantaneous equilibrium.