Dogs are pack animals so there is a leader of the dogs. Usually the resident dog is the boss when a new dog is added. However, when a new dog is added, if that dog is older, stronger or more dominant, it might try and take the lead position from the resident dog. This usually doesn't start until the new dog is comfortable in their new home. The alpha dog gets privileges that lower ranking dogs do not such as getting their choice of toys and coming in and out first. They also get to eat first and should be getting attention and treats first as well.
When the alpha no longer wishes to play, the other dog is supposed to stop playing. If they don't or bite too hard, the alpha dog reprimands them by growling or nipping at them. If they continue, a fight can occur. In addition, Gizmo might feel a bit threatened by Buster and feels he has to put Buster in his place often to stop him from trying to be the boss.
There are some things you can try to stop this behavior. One way is to treat Gizmo as the boss. Feed him first, give him treats first, show him attention before the other dog even if the other dog gets to you first. This will show Buster that Gizmo is the top dog. If Buster acknowledges Gizmos place, then Gizmo won't feel the same need to put Buster in his place.
The other thing is to start obedience training for both of them. It does not need to be a formal class but does need to be formal training.
The following site is helpful in helping owners train their dog. Be sure and click on the link to the page on obedience at the bottom. and links on subsequent pages leading to detailed instructions.
Training works best if you train at least 30 minutes a day (two 15 minute sessions). I would start making your dog work via the Nothing in life is free program (NILF). It is outlined below.
Additional training can also help. Both dogs should be leashed and if one dog even looks at the other dog, a correction should be done. Any sign of aggression including a prolonged look, hair raised on the shoulders, a growl or even a stiff legged walk, should be corrected. A correction is a quick tug of the leash and a firm low toned "NO". Once you have done this couple of times, you should notice the dogs ignoring each other. When that happens, you will want to reward them for the desired behavior. Again, use tasty treats like the hot dog slices. This teaches the dogs that you WILL not tolerate fighting in YOUR pack.
A better choice for a companion would have been a female dog as Gizmo would not have felt his place to be threatened. But you have Buster and the two will adjust to each other, it will Just take some time. As long as there is not a lot of bloodshed, you can try and see if they work things out. You can also do the training I mentioned which should also help the situation as well.
I hope this information is helpful to you. If you would like any additional information or have more questions please don’t hesitate to press the reply to expert or continue conversation button so I can address any issues you still have . If you do find this helpful, please take this opportunity to rate my answer positively so I am compensated for my time.