Necessary and sufficient conditions for permanence and extinction in a three dimensional competitive Lotka-Volterra system

A three dimensional nonautonomous competitive Lotka-Volterra system is considered in this paper. It is shown that if the growth rates are positive, bounded and continuous functions, and the averages of the growth rates satisfy certain inequalities, then any positive solution has the property that one of its components vanishes. Moreover, if one of the above inequalities is changed, then all components of any positive solution have positive infimum.