If Pokemon are the animals of the Pokemon world, would that mean they'd eat eachother too? I mean, sure...You didn't see May's Eevee getting eaten by some Charizard, but really! Wild pokemon can't just eat leaves and dirt. Neither can real animals. Some can only eat meat. So, why is it that you don't see a Ninetails trying to catch a Rattata for supper?
Actually, I believe Pokémon do feed off each other. For instance, in Pidgeot's original Pokédex entries, it was said to 'skim the surface of water at high speed to pick off unwary prey such as Magikarp'. They just wouldn't show something like that graphically in the games or anime, so that's probably why you haven't seen it.
Also, just my two cents, but I've always thought that both animals and Pokémon coexisted.