There a plethora of ways to "cope" with a camera that has a lower FPS rating. The most over-looked technique is just "plain old experience" as an experienced pet photographer be sure to capitalize on your being able to anticipate the action. The "instant" shutter release, fast focusing and exposure determination of the DSLR will do much of the rest and be much more efficient at the task over a P&S.
Secondly a camera with the ability to continuous autofocus can also make your action-life a lot easier. This is the ability for the lens/camera to track a moving subject and continuously refocus on that subject - Nikon DSLRs do that quite well. If you combine continuous autofocus with panning (moving the camera with the subject) and maybe throw in a titch of strobe, you can do wonders.
The Nikon D3100 is a very respectable camera, my daughter uses a D5000 with great productivity. Either of these cameras bring a lot of features and capability to the challenge. The suggestion of the new 50mm f/1.8 AFS would be a good match for this kind of work on a DX camera and would nicely complement the kit lens that comes with the camera. You are likely to find that the kit zoom lens works very well outdoors for this task. The 50mm f/1.8 would also be a good choice for outdoors, but may not be quite long enough unless you can corral or confine your subjects to an exercise area. However, it's speed and bright field of view would be great assets for indoor photography.