Rational curves in Calabi-Yau threefolds

We study the set of rational curves of a certain topological type in general members of certain families of Calabi-Yau threefolds. For some families we investigate to what extent it is possible to conclude that this set is finite. For other families we investigate whether this set contains at least one point representing an isolated rational curve. Our study is inspired by Johnsen and Kleiman (Johnsen, T., Kleiman, S. (1996). Rational Curves of degree at most 9 on a general quintic threefold. Comm. Algebra 24(8)-2721-2753).