ecology - When are population dynamics models useful?

I'll throw one more application into the pot. Population dynamics also forms the foundations of population genetics, population ecology, and more recently plays an important role in frameworks such as evolutionary game theory and eco-evolutionary dynamics.

Here the models are also used as a type of theoretical exercise or thought experiment (as a previous answer suggests). In the development of evolutionary theory we simply cannot observe the process over the timescales we require to test the hypotheses we make. Thus, the development of population models allows us to explore 'possible worlds', as Robert May once put it, to see what kinds of adaptations or population structures we would expect to see, given the assumptions we put in.

We are also seeing an increasing number of population models and dynamical models used in conjunction with experiments on microbes in the field of experimental evolution. Here we can observe evolution in real time, and many of the assumptions about well-mixedness and large population sizes that are often made in modelling populations are actually fairly accurate.