Qualitative nature of a discrete predator-prey system

We study the qualitative behavior of a predator-prey model, where the carrying capacity of the predators environment is proportional to the number of prey. We investigate the boundedness character, steady-states, local asymptotic stability of equilibrium points, and global behavior of the unique positive equilibrium point of a discrete predator-prey model given by begin{equation*} x_{n+1}=frac{alpha x_n-beta x_n y_n}{1+ gamma x_n}, y_{n+1}=frac{delta x_ny_n}{x_n+eta y_n}, end{equation*} where parameters α, β, γ, δ, η and initial conditions are positive real numbers. Moreover, the rate of convergence of positive solutions is also discussed. Some numerical examples are given to verify our theoretical results.