Oscillations and global attractivity in a discrete delay logistic model

Consider the discrete delay logistic model \[ {N_{t + 1}} = \frac {{\alpha {N_t}}}{{1 + \beta {N_{t - k}}}}, \qquad \left ( 1 \right )\] where $\alpha \in \left ( {1, \infty } \right ), \beta \in \left ( {0, \infty } \right )$, and $k \in \mathbb {N} = \left \{{0, 1, 2,...} \right \}$. We obtain conditions for the oscillation and asymptotic stability of all positive solutions of Eq. (1) about its positive equilibrium $\left ( {\alpha - 1} \right )/\beta$. We prove that all positive solutions of Eq. (1) are bounded and that for $k = 0$ and $k = 1$ the positive equilibrium $\left ( {\alpha - 1} \right )/\beta$ is a global attractor.