Survival and extinction in competitive systems

In [S. Ahmad, I.M. Stamova, Almost necessary and sufficient conditions for survival of species, Nonlinear Anal. Real World Appl. 5 (2004)], implications of various inequalities involving interaction coefficients and averages of the growth rates were listed for a two-dimensional Lotka–Volterra system, and it was shown that the first condition can be extended to one that is almost necessary and sufficient for persistence of a three-dimensional Lotka–Volterra system. In this paper, we give similar extensions of all the cases. We also explore the relationship of the (I,J)(I,J) condition, a notion defined by Ahmad–Lazer recently (see [S. Ahmad, A.C. Lazer, Average growth and total permanence in a competitive Lotka–Volterra system, Ann. Mat. 185 (2006)]), to certain known conditions which imply persistence.