Please note that recently introduced cats do need to work out where each fits into the social hierarchy and this will inevitably involve staring, posturing, and hissing until the dust has settled. It may even involve one major tooth and claw brawl for the pair to learn who really is the dominate cat. If you intervene too often during this transition, you will likely only extend this limbo period which can cause even greater stress to you and your cats than if you just let them work it out. Some cats will never be friends – such as Sushi and Mittens, but others will bond beautifully like Riki and Sushi, and Riki and Mittens (Riki is the affable, agreeable go-between who loves everyone!), but they should be able to create a truce.
If your newly introduced cats are really have a hard time adjusting to each other, try playing with both of them at the same time. Use a string toy that they can both follow and pounce on together (preferably not at exactly the same moment, as that would really freak them out!). Over time, the serotonin rush that they get with chasing “prey” sometimes will help overcome the bad feelings they have about each other.
Do you have other ideas about how to deal with a bully cat? If so, please share with us in the comment section below.