Global Asymptotic Stability For Damped Oscillators And Its Application To Lotka-Volterra Model

We consider the half-linear di?erential equation with an unbounded damped term,p(+ h(t)φp(x′) + ωpφp(x) = 0,where ω>0 and Φp(z)=|z|p-zz with p> 1. The divergence speed of the damping coefficient h(t) is assumed to be determined by some parameters. By using the relations between the index number p and the parameters, we describe some criteria judging whether the equilibrium of this equation is globally asymptotically stable or not. and we present parameter diagrams to clarify the relations between them. Then we consider the Lotka-Volterra predator-prey system whose prey population receives time-variation of the environment. For the interior equilibrium of this system, we give a necessary and sufficient condition of being globally asymptotically stable and a sufficient condition of being uniformly globally asymptotically stable in turn.