Contemporary Methods in Mathematical Physics and Gravitation

15

Authors: Qamar Din

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.