Exercises that begin, ``A model of such and such is the equation ...'' quickly become unsatisfactory because the model has come magically onto the scene, deus ex machina. Nonetheless, an instructor seeking such examples can hunt for equations among undergraduate texts in science and engineering or talk with colleagues in those disciplines.
Many population models require minimal background, and the mathematics and the physical behavior are rich. Useful references include those in the review [5] (at a higher level) as well as [6] and [9] (simpler but more limited).
An excellent introduction to the ideas of mathematical modeling in a variety of physical and biological settings is Lin and Segel's classic text [8]. Other texts are discussed in the review [2]. The COMAP library is a source of models and exercises of many sorts, e.g. [6]. Several introductory differential equations texts treat modeling, including [1, 4, 7].