dynamics of a discrete-time plant-herbivore model

In this paper, we examine a discrete-time plantherbivore model.{xt+1 = xte^r[1−xt]−ayt , yt+1 = xte^r[1−xt][1 − e^−ayt ]phase portraits are drawn for different ranges of parameters. we use the liapunovschmidt reduction for attain a simpler and smaller system. transition route to chaos dynamics is established via perioddoubling bifurcations. conditions of occurrence the perioddoubling, neimarksacker and saddlenode bifurcations are analysed. we study stable and unstable manifolds for this system in equilibrium points. without the herbivore, the plant population follows the dynamics of the ricker model.