The author studies the nonautonomous competition system \[ u'= u[a(t)- b(t)u- c(t)v],\quad v'=v[d(t)- e(t) u-f(t)v],\tag{1} \] where \(a(t),\dots,f(t)\) are continuous functions defined on all of \(\mathbb{R}\), bounded above and below by positive constants. He exhibits distinguished solutions defined by semitrivial (one component \(0\)) solutions to (1). The major results of the paper concern the structure of solutions which approach one or the other of the distinguished solutions asymptotically as \(t\to\infty\).
Increasingly specific results are obtained for uniformly continuous, almost-periodic, and periodic coefficients. As an example of the flavor of the results obtained, if the coefficients are \(T\)-periodic, the set of initial points of solutions which are asymptotic to one of the distinguished semitrivial solutions is either \((0,\infty)\times \{0\}\), \((0,\infty)\times (0,\infty)\), or a more complicated set and, in this third case, (1) has a positive, \(T\)-periodic solution.