That is the major downside of shooting on auto modes - it is going to pick the best for what the camera sees in the situation. Camera do not know exactly what we want, nor what exactly what is going on. When you put the camera on a pet mode it likely is going for a shutter priority. Stopping motion is the first priority - everything else is a lesser priority still considering there should be decent exposure (not an underexposed pitch black image or overexposed white image).
This is where you're juggling three main pieces that control your overall and exposure (and where I hope I don't lose you or hope you don't feel I'm dumbing it down too much!
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1. Shutter Speed (as you know) - how slow or fast the shutter in the camera is opening and closing to expose for the image.
2. Depth of Field (DoF or F-stop) - The amount of light the lens allows into the camera. (possibly confusing part) The less available light, the smaller the f-stop number and the smaller the visible depth of field (or sharpness, what you are getting in your images). The more available light, the larger the possible f-stop number, the larger the visible depth of feild.
3. ISO - Lower ISO number allows for less light, while higher number allows and tells the camera there is more light. (when there may not be) Really high ISO numbers (1600+) start to create noise as the camera is having to 'guess' at what information was supposed to be there in low light situations.
So, what is happening is your camera is likely upping your shutter speed and since it does not see your depth of field as a priority it keeps it at a low number. (likely only compensating enough with ISO to get a decent exposure) If you would like the entire subject in focus I would say it is time to start playing on manual and playing around with your ISO, shutter speed and F-stops. Hopefully that helps a bit and I will try to add a bit more later if this is still a bit vague.