Partial permanence and extinction in an N-species nonautonomous Lotka-Volterra competitive system
The authors study the \(N\)-dimensional (N-species) differential system \[ u'_k(t)=u_k(t) \left[a_k(t)-\lambda g_k(t)-\sum_{l=1}^N b_{kl}(t) u_l(t) \right], \qquad 1\leq k\leq N, \] where \(a_k(t)\), \(b_{kl}(t)\), \(g_k(t)\) are bounded continuous functions on \([0,\infty)\) and \(b_{kl}(t)\geq 0\) for \(t\in[0,\infty)\), \(\lambda\geq 0\) is a constant. They show that certain average conditions involving the averages of the growth rates and the interaction coefficients imply the total persistence of all species. They also give computable conditions on the coefficients which imply the ultimate extinction of certain species.
scientific article; zbMATH DE number 5623813