Bifurcations and dynamics of a discrete predator-prey system

In this paper, we study the dynamics behaviour of a stratum of plant-herbivore which is modelled through the following F(x, y) = (f (x, y), g(x, y)) two-dimensional map with four parameters defined by f(x, y) = x exp (r (1 - x/k) - by), g(x, y) = x(1 - exp( -ay)), where x >= 0, y >= 0, and the real parameters a, b, r, k are all positive. We will focus on the case a not equal b. We study the stability of fixed points and do the analysis of the period-doubling and the Neimark-Sacker bifurcations in a standard way.