They have had little to no help from the Otago Regional Council, and are at a loss as to where to turn next.
Hay said the onus was "absolutely" on the landowner to deal with the situation, and the couple were keen to work with other Moeraki property owners to tackle the problem.
But the nature of the town presented its own issues.
"In a place like Moeraki, there are a lot of crib owners that don't need a lawnmower."
Hay said the couple regularly had shooters on their property trying to keep the numbers under control.
"We had two guys in there last weekend, they did two nights and one day … and they shot just under 700 rabbits. Someone went in the next day and got 90. And they're still everywhere."
In the past five weeks, 1540 rabbits had been shot.
"Every time you drive down the driveway, you could still probably count like, 20."
Because some of the farm is near yellow-eyed penguin habitat, poison was not an option, even if the Hays were keen to go down that route.
"We've also got a township of a whole lot of people with pet cats and dogs, so they're not going to want you poisoning either. And then some of our shooters shoot for pet-food companies. They come in every week or every couple of weeks and shoot for us. And they're actually really good, but if we poison, they can't shoot."
The rabbit presence has had a noticeable effect on grass growth.
"Last February, we sowed a paddock and it didn't grow until spring, because it just gets eaten off the whole time, and in winter they hunker down a bit.
"We've definitely had failed crops, absolutely. Just because they eat it."
ORC biosecurity and rural liaison manager Andrea Howard said while the council did not undertake rabbit pest control, which, under the Biosecurity Act, was designated the responsibility of landowners, it had a leadership role to play in co-ordinating and facilitating better rabbit management and undertaking property inspections to check rabbit densities against its Otago Regional Pest Plan rules.
"In addition to our inspection and compliance role, we have been trialling a few collaborative approaches to rabbit management, which involved facilitating a poisoning operation across areas of private and public land," she said.
"We are looking to reproduce a similar approach in North Otago this year, as well as in other rabbit-prone areas around Otago. We plan on starting community engagement over the winter period."
In other areas, the approach had involved working with public organisations and private landowners to develop a co-ordinated control effort.
This initial management work would be supported by providing better education for landowners on the best ways to control rabbits on their properties, and exploring community appetite for longer-term solutions such as the establishment of community-owned entities to undertake (or contract out) consistent rabbit control on an ongoing basis, Howard said.
