Possums and stoats require a generous and sustained application of capitalism. Rewarding hunting/trapping eradication efforts.
I propose a controlled experiment - first wave Stewart Island, originally known as NZs South Island. Only ONE measure of success allowed: a growing native bird population. If after 10 years it is successful, move on up to the main island, NZ Middle Island, with a more expensive, larger scheme and lessons learnt.
How would such a scheme look like? What sort of reward per rat, possum or rabbit would make it worthwhile? What do farmers currently pay pest controllers per rabbit?
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