I don't agree that shooting has little impact. Hunting is used successfully in many places to keep pest animals at low densities. Poisoning can also reduce populations, but it has many setbacks. Very few dead animals an be found so it is hard to measure success, non target species are killed and animals that die are killed slowly and painfully. To have a lasting effect the poisoning must be regularly and not stop. There is no way of eliminating pests, or it would have been done earlier, all control methods are about keeping pests low densities. Here is a recent paper on the subject.
http://www.gamecouncil.nsw.gov.au/docs/ISC%20response.pdf
http://www.gamecouncil.nsw.gov.au/docs/ISC%20response.pdf