Generalized monotonicity, integral conditions and partial survival

An interesting paper of Vance and Coddington concerning the scalar equation x = xf(t, x) is improved by a drastic weakening of the assumptions. Surprising as it may seem, this weakening simplifies the analysis and leads to stronger conclusions. An extension to scalar inequalities of similar structure is applied to a large class of vector equations describing the evolution of n interacting species. Thus we obtain new criteria for extinction and for partial survival, in the sense that each of the separate species may come arbitrarily close to extinction but the population as a whole does not.