  In this paper we study the dynamics of the general case of Owen-Smith
metaphysiological model, to explore the effects of seasonality on population
fluctuations. The study will include the permanence, herbivore extinction,
global asymptotic stability and existence of positive periodic solutions. Under
certain assumptions, we obtained sufficient and necessary conditions which
guarantee the permanence of herbivore and vegetation species, and existence of
periodic solutions. The techniques we used here are, comparison method of
differential equations, Lyapunov method and Brouwer's fixed-point Theorem.
