On Sunday morning the farmer at Meikleholm Side, Annan Water, came across four pregnant sheep with serious injuries.
As a result, three of the animals had to be humanely destroyed by a vet.
And it is believed the animals may have been attacked by a dog or dogs in a case of sheep worrying.
Constable John Cowan said: “It appears from what we know that a dog or dogs have attacked the flock of sheep during the morning of Sunday January 28.
“We are asking the local community to help us trace the dogs and the owner.
“Anyone with any information which might help in this investigation is asked to call us at Lockerbie on the 101 number.”
Meanwhile in a second incident, a 30-year-old man was charged with sheep worrying after his dog allegedly chased two sheep around a field near Mouswald on Sunday.
For pictures – see video.
