If one species in a community dies out or moves, the community will

The community will slowly decline, or be just fine, or be better off.

Explanation:

Wolves being killed in Yellowstone showed that Yellowstone suffered, because there was nothing to kill the elk, so their population exploded and they ate everything.

If certain rats were gone in certain areas, then the community/environment would be fine, because rats eat birds and other small animals (they eat too many of them).

And if stray cats were gone then certain forests would be better off and see an increase in birds population, stray cats eat birds.