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

Abstract

In this paper, global dynamics and bifurcations of a two-dimensional discrete-time Lotka-Volterra model have been studied in the closed first quadrant R2. 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(α1-1)/α3,0, B0,(α4-1)/α6 and the unique positive equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5), 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(α1-1)/α3,0,B0,(α4-1)/α6 and a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5). Further it is shown that every positive solution of the discrete model is bounded and the set 0,α1/α3×0,α4/α6 is an invariant rectangle. It is proved that if α1<1 and α4<1, then equilibrium O(0,0) of the discrete model is a global attractor. Finally it is proved that the unique positive equilibrium C((α1-1)α6-α2(α4-1))/(α3α6-α2α5),(α3(α4-1)+α5(1-α1))/(α3α6-α2α5) is a global attractor. Some numerical simulations are presented to illustrate theoretical results.