I am in Sheffield, in the UK and this may well from country to country.

We have an online application system that is shared by the whole university and not controlled by the maths department. The form has a slot for a research plan, but in maths at least we ignore it - we have no expectation that beginning PhD students will have their own project in mind. If you just told us that you wanted to do Mathematical Biology, that would be sufficient for the application form.