The exponential growth model and the logistic model typically introduced in the mathematics curriculum presume that a population grows exclusively. In reality, species can also die out and more sophisticated models that take the possibility of extinction into account are needed. In this article, two extensions of the logistic model are considered, one known as the "logistic model with Allee effect," which is a cubic model, and the other a further generalization, which is a quartic model. The approach used is designed to illustrate some techniques of model-building. (Contains 9 figures.)