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A total 15 sheep have been killed in three suspected dog attacks.
The owner of a farm in Alkham Valley Road, Alkham, Dover, saw a grey, scraggy haired, dog, which looked similar to a lurcher, in amongst her sheep.
She shouted and the dog ran away in the direction of Green Lane, the incident happened last Saturday, at around 5.50pm.
Six of her sheep died and four still need medical care.
The next day, in Wolverton Hill, Alkham, seven sheep from another farm were found dead. It is believed they were attacked by an animal but it is not clear when.
Overnight on Monday the farm in Alkham Valley Road reported a second attack in which two more sheep were killed and one was seriously injured.
PC Dan Perry said: "We urge all dog owners who may be walking their pets in the countryside to keep them on leads around livestock. These were nasty incidents, which has led to distressing injuries and the deaths of livestock.
"When a dog has a taste for killing sheep it often returns..."
"As well as the suffering caused, attacks can lead to huge financial losses for farmers and when a dog has a taste for killing and maiming sheep it often returns.
"We know that the majority of dog owners do take care around livestock but we remind everyone to follow signs and stick to footpaths, as they are there to protect you and your dog, and to make your walk safe for all.
"It is also really important to check your properties and gardens are secure when you have dogs."
Under the Animals Act 1971, a person trying to protect livestock can kill or injure a dog that they reasonably believes is worrying the animals.
They can do this without any criminal or civil liability.
Anyone in charge of a dog could be committing an offence if the pet worries livestock on agricultural land.
This includes attacking or chasing livestock in a way that might reasonably be expected to cause injury, suffering or loss.
It can also be an offence to have a dog in a field or enclosed space where there are sheep when the dog is not on a lead or under close control.
Anyone who witnessed anything suspicious or who has information regarding the Alkham incidents should call Kent Police on 01843 222289 and should quote the reference number 04-1449 for the Alkham Valley Road cases and 05-0416 for the Wolverton Hill one.