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Many thanks for replies.
Ratter, we don't want to use poison if we can avoid it, as we think the mice probably die a slow and painful death, and we are both animal lovers, although we accept that the mice have got to go! My husband favours snap traps (if that is the correct word). He says the mouse dies instantly and he thinks this is the kindest method.
Also, a friend's dog died of poisoning after eating a mouse or small rodent-like animal that had been poisoned. My friend took the dog to the vet but nothing could be done to save it, it was tragic. After this experience, we are worried about taking the risk of something like this happening again.
Honey, I mentioned a collar for the cat to my husband, but he is not keen on approaching the neighbour with this suggestion, he says the neighbour might then say to us, "so you are the ones who keep feeding the birds that are messing up our paths etc ........" I am sure you know what I mean. The street we have moved to is very cat-orientated, the previous owners of our house told us he was always getting upset about the neighbourhood cats killing the birds, it has apparently been a problem for a long time, and I am sure he would have already approached the neighbours with a suggestion about the cats wearing collars, to no avail.
Most pet owners would be sympathetic to the birds' problem with cats, but I guess in truth, not all cat owners are as passionate about wildlife as us! (No offence meant to cat owners who are passionate about wildlife, I am sure there are lots of them around).
I think Ratter's suggestion about terrorising the cat without actually harming it is a good idea, and hubby has suggested we buy a water pistol to shoot at it every time we see it in the garden. Don't want to do it any permanent harm, just give it a short, sharp shock. Just hope the neighbours don't spot us doing it or we could be in for a rough ride!