When are population dynamics models useful? There seems to have been a lot of research about it, but how does it help? If I need data about how a population will evolve under what conditions, I need it because I need data for a decision (such as "can we kill 50% of population X without doing too much damage?"), right? But for that, the model needs to be aware of what causes what. And for that, I have to do experiments, right? Like "let's kill a significant amount of population X and see what happens in the next ten years". I really don't get it.
Population dynamics occupies a whole subset of mathematical biology. Perhaps the most pragmatic uses for modelling population dynamics come from the fields of epidemiology for modelling disease infection and transmission through a population (one such article), or ecology modelling things like forestation, fishing dynamics, predator-prey relationships (an example). Then there are more abstract uses, when you cannot measure a population to test a hypothesis because the labour is too intensive, or it's too costly, or no such surveillance mechanism exists. Theoretical models of population dynamics are built to gain an understanding of what the system as a whole may do under certain conditions. In this sense, population models are more of a theoretical exercise or a thought experiment.
Leonardo's already given you an excellent answer, but I thought I'd add my perspective. I'm a mathematical epidemiologist, so I'd at least like to believe these types of models are useful.
For me, there are a number of things population dynamics models are especially useful for:
- Highlighting data requirements. Yes, models need data, as you've mentioned. But they don't need all their data to come from one source, one study, or even one field. Models are also profoundly useful for showing where we don't have the data we need to fully understand a system. "To make a model where we understand A, we need the values for X, Y, and Z. X is well studied, but Y and Z aren't - though it turns out when we look over the entire parameter space for Z, nothing really changes in our answer. But guys? We could really use a study on Y."
- Eliminating guesswork. Models aren't perfect encapsulations of reality - there will always be some simplifying assumptions, etc. But it's better than "going with your gut" - especially for complex problems.
- Impossible studies. A ton of what mathematical epidemiology looks at it is areas where studies are either impossible, logistically difficult, or unethical. It would be very hard indeed to only be able to study pandemic response plans, or vaccination strategy only when we had an actual outbreak, or while a new vaccine is being rolled out.
- Highlighting potentially new directions. If you're considering an intervention, but no matter how effective you make it in your model it doesn't move the system much, it might not be worthwhile. Models can also highlight threshold effects - like the critical % of the population you'd need to vaccinate to achieve herd immunity.
Two previous answers listed many applications of population dynamics models. I want to add that they are also important for conservation of endangered species. For example classical stage-class model (Crouse et al 1987, free copy) indicate that the most effective way to protect sea turtles is reducing mortality of large juveniles.
Moreover, you don't have to perform such drastic experiments as killing 50% of a population to estimate your model parameters. Information about number of offspring, breeding success, natural mortality, etc can usually be gain without serious perturbation of wild populations. The number of individuals in every natural population fluctuates because of random reasons, so it is possible (but sometimes needs more field work) to calculate, (possibly nonlinear) regression between density of population and some demographic indicators and then extrapolate it to non-examined densities. For some small, short-living species it is sometimes possible to measure that correlations in laboratory or semi-wild conditions. For some long-living species, especially if they are sedentary, like trees, it may be better to compare specimens living in different distance from its neighborhoods. For poorly-known species it is possible to take missing information from related or similar species.
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.