Wolves, hunted to extinction in Wyoming and Montana in the twentieth century, occupy a vital place in the natural cycle of the area.
A straightforward test for subject-hood is to make a yes/no question and see what inverts with the auxiliary verb:
- Do [Wolves, hunted to extinction in Wyoming and Montana in the twentieth century] occupy a vital place in the natural cycle of the
area?
Here we see that bracketed part of the sentence appearing between the auxiliary Do and the main verb occupy. This shows the bracketed part of the sentence to be the Subject. It has inverted with Do. If we had an emphatic declarative version of the sentence, it would read:
- [Wolves, hunted to extinction in Wyoming and Montana in the twentieth century] do occupy a vital place in the natural cycle of the area.
As the bracketed section is the Subject, it cannot be part of the Predicate.