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Researchers are trying to find a way to stop grey squirrels reproducing, but can the menace ever be stopped?
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To some they are "tree rats", to others unwelcome foreign invaders. They have even been reported to launch unprovoked attacks on humans.
Hated for its stripping of tree bark, threats to wild birds, but most of all for its "displacement" of the red squirrel, there are many people who would be happy to see the grey squirrel eradicated.
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Now the Forestry Commission is carrying out research funded in part by the Department for the Environment, Food and Rural Affairs and Scottish Natural Heritage to see if contraceptives can be administered in an effort to tackle the population.
But can its spread across the UK ever be stopped and would contraception eradicate them completely?
The most realistic outcome, the commission says, is that contraception will, along with trapping and poisoning, control the population. Getting rid of the grey will never be a meaningful option.
"The concept of eradication has come up a number of times, there have been bounty schemes, but that has been found to be ineffective," says Brenda Mayle, programme manager at the Forestry Commission.
Contraception may be possible, eradication of the grey squirrel is not
The first efforts to tackle the grey squirrel menace were in the early 1930s. By the 1940s grey squirrel shooting clubs were common, with free shotgun cartridges issued by the authorities and a bounty of two shillings paid per tail. But the spread continued.
In recent years the desire to at least stop that spread has intensified with the discovery that the squirrel pox (SQPV) disease, which red squirrels suffer from, is carried by greys but does not cause symptoms in them.
Ms Mayle said research in the US into preventing reproduction in other animals, such as horses, had raised the hope that oral contraception could be used on grey squirrels.
A three-year-research project going on in the UK and US hopes to find a suitable drug. Ms Mayle said a permanent contraceptive effect was wanted and drugs used on animals mostly had a temporary effect.
The bait-and-delivery system would need to be carefully chosen so that other animals, particularly red squirrels, did not eat it.
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Currently, the grey squirrel population is controlled by trapping and poisoning by the anticoagulant warfarin. The poisoning is done using an L-shaped hopper that has a gate that only a squirrel could open. But poisoning is not allowed in areas where there are known to be reds.
If the search for a contraception solution is successful it will not spell rapid, dramatic results, Ms Mayle warns.
"The longevity of treated animals has increased. The rate of decline has not been as fast as expected," she suggests.
Ross Minett, campaigns director at Advocates for Animals, says welfare organisations would be interested in an approach that did not involve culling.
Tables have turned
"We share the concern of members of the public about the plight of the red squirrel, but the grey squirrel is here to stay. There is no way we are going to remove it.
"We are very interested in this idea [immunocontraception] if it could be an alternative to killing grey squirrels."
Mr Minett said a study done at Bristol University had shown that killing grey squirrels was not an effective way to conserve their red cousins. Instead cultivating more conifer forest and establishing island refuges that could be defended against greys was a more realistic option.
First introduced in 1876 to Henbury, Cheshire
It is illegal to release into the wild a captive grey
Spreads pox virus that kills reds
No concrete figure for population
"We don't believe it is ethically acceptable to kill on species to protect another. In live trapping they are bashed on the head. We believe that raises serious welfare issues."
Angus Macmillan, who runs the Grey Squirrels website, believes that control of the population is unnecessary and that the idea of exterminating one animal population to make another prosper is "bordering on ethnic cleansing".
"They shouldn't be controlled and I don't think they can be controlled. There are other methods of saving the red squirrel. Nature controls species."
Grey squirrels are persecuted, at least in part, for being seen as American invaders, Mr Macmillan suggests. But the red squirrel, which is being protected by greys being killed, might not be as native as people think. With the animals hunted as pests and affected by disease, Scotland's stocks were topped up in the 19th Century with squirrels from the continent.
Any contraceptive programme will not see an end to the killing of squirrels.
"It is not an alternative to lethal options," Ms Mayle insists.
In the 19th Century it was the red that was the pest, blamed for damaging trees. The tables are now turned.