Following an open problem of S. Ahmad and A. C. Lazer [Nonlinear Anal., Theory Methods Appl. 34, 191–228 (1998; Zbl 0934.34037)], the author provides sufficient conditions for the extinction of the species \(y\) (and convergence of the remaining \(n\) species to a unique positive solution) in the \(n+1\)-dimensional Lotka-Volterra competitive system \[ \dot{x}_i = x_i(a_i(t)-\sum_{j=1}^n b_{ij} x_j - c_i y), \quad 1\leq i \leq n, \]
\[ \dot{y} = y (\alpha(t) - \sum_{j=1}^n q_j x_j - \beta y) \] with all coefficients nonnegative, \(b_{ii}, \beta >0\), functions \(a_i,\alpha\) bounded from above and below by positive constants (or, alteratively, \(T\)-periodic with a common period \(T\)) and such that they possess certain averages \([a_i]\) and \([\alpha]\). Using iterative schemes developed, e.g., by A. Tineo [Nonlinear World 3, 695–708 (1996; Zbl 0901.34050)], he also gives sufficient conditions for which the \(n+1\)-dimensional system has a positive \(T\)-periodic solution that attracts all other positive solutions.