Global Dynamics and Bifurcations Analysis of a Two-Dimensional Discrete-Time Lotka-Volterra Model

In this paper, global dynamics and bifurcations of a two-dimensional discrete-time Lotka-Volterra model have been studied in the closed first quadrant R-2. It is proved that the discrete model has three boundary equilibria and one unique positive equilibrium under certain parametric conditions. We have investigated the local stability of boundary equilibria O(0,0), A((alpha(1) - 1)/alpha(3), 0), B(0, (alpha(4) - 1)/alpha(6)) and the unique positive equilibrium C(((alpha(1) - 1)alpha(6) - alpha(2) (alpha(4) - 1))/(alpha(3)alpha(6) - alpha(2)alpha(5)), (alpha(3)(alpha(4) - 1) + alpha(5)(1 - alpha(1)))/(alpha(3)alpha(6) - alpha(2)alpha(5))), by the method of linearization. It is proved that the discrete model undergoes a period-doubling bifurcation in a small neighborhood of boundary equilibria A((alpha(1) - 1)/alpha(3), 0), B(0, (alpha(4) - 1)/alpha(6)) and a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium C(((alpha(1) - 1)alpha(6) - alpha(2) (alpha(4) - 1))/(alpha(3)alpha(6) - alpha(2)alpha(5)), (alpha(3)(alpha(4) - 1) + alpha(5)(1 - alpha(1)))/(alpha(3)alpha(6) - alpha(2)alpha(5))). Further it is shown that every positive solution of the discrete model is bounded and the set [0, alpha(1)/alpha(3)] x [0, alpha(4)/alpha(6)] is an invariant rectangle. It is proved that if alpha(1) < 1 and alpha(4) < 1, then equilibrium O(0, 0) of the discrete model is a global attractor. Finally it is proved that the unique positive equilibrium C(((alpha(1) - 1)alpha(6) - alpha(2) (alpha(4) - 1))/(alpha(3)alpha(6) - alpha(2)alpha(5)), (alpha(3)(alpha(4) - 1)+ alpha(5)(1 - alpha(1)))/(alpha(3)alpha(6) - alpha(2)alpha(5))) is a global attractor. Some numerical simulations are presented to illustrate theoretical results.