Zombies were originally conceived of as being dead bodies brought to life by black magic as I understand things.
As such, zombies were immortal. You could only really stop one by hacking it to bits, so the individual parts would no longer be able to move as a unit. And even so, the separated bits would still be animated by a mindless desire to eat human flesh...at least in one conception of "zombie."
But more modern stories have brought new ideas of what it means to be walking dead. The viral explanation is popular now. Humans get twisted by a-rabies-on-steroids type infection, leaving them mindless, with infinite pain tolerance, and hungry for homo sapiens. Any bite that does not kill will very quickly spread the disease--clearly zombie saliva is swarming with the pathogen. The only guaranteed way to kill one of the beasts is to put a bullet in its brain.
In these latter stories, it stands to reason that the zombies would still need to eat, right? After all, we're still taking about living organisms, just seriously twisted ones.
So what happens after the zombies eat all but the tiny handful of heroic humans still surviving? What do they eat then? Each other?
It would seem a likely choice for them to make. After all, while they would always find humans much yummier, since they themselves were once human, the taste should be close, right?
But what happens once they've been gnawing on each other a while? You see, in ecology terms, all life is supported by plants growing by the power of sunshine. You may eat a cow, but the cow eats wild grass or corn or some such, which gains energy to grow and store food via beaming rays of sunshine.
Meat eaters eating only meat eaters would make the pool of available energy (some always gets lost in the process of digestion) grow smaller and smaller. The number of zombies would grow less and less as a result, right? In maybe ten years, almost all of them would be dead.
UNLESS, the zombies, under the terrible pressure of starvation, learned to eat animals, who would supply energy that would either directly or indirectly come from plants. This is a whole different vision of a zombie, isn't it? Mindless man-things tromping tirelessly through the forest in pursuit of a white tailed deer...mostly they would not succeed in catching natural fauna, methinks.
But they would succeed every now and then, I think, and what would that mean? Would it not become possible that at least sometimes they would get in only a single bite before a tortured quadruped sped off in a sprint? Which would put that infected zombie saliva into the animal, wouldn't it?
Would the animals be subject to the zombie disease? Some infections are species specific, but others, including rabies itself, are not. There's a story here somewhere maybe, about some animals becoming zombies and others not...heroic human survivors being chased by zombie deer but never worrying much about snakes, which are immune to the disease...
Zombie deer may not be scary, but zombie dogs sure would be. So would zombie birds. Or bats. And is anyone ready to face a zombie grizzly bear?
I've really only played with a few aspects of zombie ecology. There are no doubt more story ideas ready to mindlessly charge forward from this notion...