We had this issue with Buster when he came to us at 14 months old. We already had a cat who had decided several years before to move in and had enjoyed a good relationship with the old dogs. Enter a young hooligan who will chase anything that moves. It was the acquisition of the baby gate that saved the day in the short term. But in the longer term they came to a modus vivendi. Basically, the cat got fed up of being chased after, turned around, hissed, spat and took a haymaker of a left hook at Buster's nose. The point was made, because he was only ever looking for a game and thought that when Billie-Jo ran, she was joining in. With that misunderstanding sorted out and after we had been 'punished' by her spending a night camping in the garden in protest (she chose a very warm, dry night) peace and order returned. Buster will still chase, but since Billie-Jo simply ignores him, having learned that he is no threat, the chase is a short and slow one - more of a procession really!