Balancing Survival and Extinction in Nonautonomous Competitive Lotka-Volterra Systems

We generalise and unify some recent results about extinction in nth-order nonautonomous competitive Lotka-Volterra systems. For each r ≤ n, we show that if the coefficients are continuous, bounded by strictly positive constants, and satisfy certain inequalities, then any solution with strictly positive initial values has the property that n − r of its components vanish, whilst the remaining r components asymptotically approach a canonical solution of an r-dimensional restricted system. In other words, r of the species being modeled survive whilst the remaining n − r are driven to extinction.