manifold

If M is a compact Riemannian manifold then we show that for typical continuous function defined on M, the upper box dimension of graph(f) is as big as possible and the lower box dimension of graph(f) is as small as possible.

In this paper, we examine a discrete-time plant-herbivore model. We investigate stability of 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 Liapunov-Schmidt reduction for attain a simpler and smaller system. Transition route to chaos dynamics is established via period-doubling bifurcations. Conditions of occurrence the perioddoubling, Neimark-Sacker and saddle-node 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.