On the asymptotic behavior of some population models. II (Q1916710)

The author investigates the Lotka-Volterra system \(du_i/dt= u_i[b_i(t)- \sum_{j=1}^n a_{ij}(t)u_j]\), \(1\leq i\leq n\), where \(n\geq 2\) and \(a_{ij}\) and \(b_i\) are continuous bounded functions from \(\mathbb{R}\) to \(\mathbb{R}\). It is assumed that there exist constants \(c_1,\dots,c_n\), \(m\) such that \(c_ia_{ii}(t)\geq m+\sum_{i\in J_i}c_i|a_{ji}(t)|\) for all \(1\leq i\leq n\) and \(t\in\mathbb{R}\), where \(J_i=\{1,\dots,i-1, i+1,\dots,n\}\) for \(i=1,\dots,n\). It is proved that (1) if \(u\) and \(v\) are positive solutions then \(u(t)-v(t)\to 0\) as \(t\to\infty\), and (2) if \(a_{ij}\) and \(b_i\) are \(T\)-periodic then there is a unique nonnegative \(T\)-periodic solution \(u^0\) such that \(u(t)- u^0(t)\to 0\) as \(t\to\infty\) for any positive solution \(u\). The results are applied to some models of extinction of species.