Mr. R.I.N.G., you are making one very large assumption about the nature of different species of animals - i.e., that killing those not of their own species is instinctual and natural.
This is not the case. By and large, animals require reasons to take the risk and expend the energy involved in killing other animals. To begin with, only predator species have any hunting instinct, and the most common trigger for it is hunger, not proximity. Most well-fed lions, tigers, wolves, etc., will ignore vulnerable prey species most of the time, to the point that an experienced prey animal will not hesitate to walk straight past a predator that gives evidence of already being full.
To go on with, not all animals are prey. The most common domestic animals in modern western society - cats and dogs - are both carnivorous. Carnivores make poor eating, as their flesh tastes sharp and their vital organs build up toxic levels of nutrients like vitamin A in the process of digesting the flesh of others.
The stereotypical enmity between cats and dogs exists primarily as a result of a nasty tendency over the years for people who didn't like cats to buy dogs and sic their dogs on cats as an amusement. The breeds that are the biggest threat to their neighborhood cats, in the absence of a vicious owner, are those specifically bred to hunt smaller animals (e.g., coon hounds) and those specifically bred to fight (e.g. "pit bulls"). These animals have had their agressive tendencies artificially enhanced, and are carefully monitored by responsible owners.
Even animals which know how to hunt can be taught to regard certain animals or classes of animals as non-prey, especially if the process is begun at a young age. If you don't make a big positive fuss over your cat the first time it brings you a dead mouse, you can kiss that potential mouser good-by. If you beat your coon hound for killing a kitten, it is far less likely to kill the next kitten it runs across. When I was researching my parrot book, I found a story of a cat who was a terror to cagebirds, but who left the family parrot alone because it talked to him in a human-like way and the cat classified it as human.
Lastly, hunting for food is complex learned behavior, as any cat owner who has fruitlessly tried to get her cat to actually kill the mouse it's playing with can tell you. I've had cats my entire life, and only had two good mousers, though all of them instinctively chased moving objects. Chasing is one thing; catching and killing quite another. If you examine the domestic animals around you, most of those that kill are those that have been taught to do it, either by the human being it regards as master/mommy/source of food and social reinforcement or by a mentor animal. I really wish Eric the Mighty Hunter had lived long enough to train the present generation of cats at Casa Griffin, because they're hopeless - rodents are toys as far as they're concerned.
The primary other reason for one domestic animal to kill one another is competition, which the wise owner can greatly reduce and which their own instinctive social codes are also usually good at controlling. These deaths are however most likely to be intra-specific, since pigs and dogs seldom compete for the same food source and do not usually belong to the same social hierarchy. Most herd and flock animals, which would normally spend considerable energy sorting out who has access to mates and whose offspring die, have their breeding strictly controlled by castration and separation. It's not uncommon for tom cats to kill kittens it doesn't recognize as its own, and in stressful social situations the maintenance of the intraspecies pecking order (a term derived from the literal behavior of chickens) can be perilous to the life of the lowest-status individual.
This is all normal behavior, observable both in the wild and under domestication. Interspecies relations in a domestic setting may well be more amicable than intraspecies ones, as the animals involved are already used to regarding another species - humans - as family/pack, and this makes it easy to extend family/pack privileges to alien creatures, and the different talents and requirements of the different species reduce competition and increase the benefits of cooperation. Bigger and older animals feel parental towards smaller and younger ones; dogs who can push open doors rely on cats to figure out the latch; etc.
This is all observation, not a study.