Bifurcations of a two-dimensional discrete time plant-herbivore system

In this paper, bifurcations of a two dimensional discrete time plant-herbivore system formulated by Allen et al. (1993) have been studied. It is proved that the system undergoes a transcritical bifurcation in a small neighborhood of a boundary equilibrium and a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium. An invariant closed curve bifurcates from the unique positive equilibrium by Neimark-Sacker bifurcation, which corresponds to the periodic or quasi-periodic oscillations between plant and herbivore populations. For a special form of the system, which appears in Kulenović and Ladas (2002), it is shown that the system can undergo a supercritical Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium and a stable invariant closed curve appears. This bifurcation analysis provides a theoretical support on the earlier numerical observations in Allen et al. (1993) and gives a supportive evidence of the conjecture in Kulenović and Ladas (2002). Some numerical simulations are also presented to illustrate our theocratical results.