Dynamics of the Non-autonomous Owen-Smith Model

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.