Consider the equation u'(t) = l0(u)(t) - l1(u)(t) + f(u)(t) for almost every t ϵ R, where li : Cloc(R;R) → Lloc(R;R) (i = 0, 1) are linear positive continuous operators and f : Cloc(R;R) → Lloc(R;R) is a continuous operator satisfying the local Carathéodory conditions. Efficient conditions guaranteeing the existence of a global solution, which is bounded and non-negative in the neighbourhood of -8, to the equation considered are established provided that l0, l1 and f are Volterra-type operators. The existence of a solution that is positive on the whole real line is discussed as well. Furthermore, the asymptotic properties of such solutions are studied in the neighbourhood of -∞. The results are applied to certain models appearing in the natural sciences.