Global stability in a discrete Lotka-Volterra competition model

We consider the Euler difference scheme for two-dimensional Lotka-Volterra competition equations and show that the difference scheme has positive and bounded solutions. In addition, we present sufficient conditions that the solutions of the scheme converge to the equilibrium points of the scheme. The convergence is shown based on the two approaches: first, partition of the domain used for the boundedness of the solutions and second, calculation of the movement of the species started in each partitioned region. Numerical examples are presented to verify the results.