I dont think it would ever be possible to be over populated with bevers in the UK as they are so shy and the UK is so full of people they would be frightened off most of the land .
We have bevers near us here in France . I have never seen one but have seen evidence of their noctural visits by the river bank . We do have coypu/ragondin as well and they are a pest although I am told they make an acceptable Paté.
David
Living in Anjou , France,
For the many not for the few
http://www.permies.com/t/80/31583/projects/Permie-Pennies-France#330873