Almost necessary and sufficient conditions for survival of species

We consider a nonautonomous competitive Lotka-Volterra system of two species, satisfying two inequalities involving averages of the growth rates and the interaction coefficients, which imply persistence. We introduce a third species and give a third inequality, involving the average of the growth rate of the third species and solutions of a linear algebraic system, which guarantees persistence of the system. It is also shown that reversing this inequality implies non-persistence; more specifically, extinction of the third species with small positive initial values, in the autonomous case. Our conditions are simple and computable. (C) 2003 Elsevier Ltd. All rights reserved.