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You probably mean the stage where creatures in an ecological succession do not change much because the established plants and animals are the ones that finally survived the struggle and suppressed all the unsuccessful rivals. Successful rivals have achieved some sort of balance. We call that the climax community. A typical example would be a well-established forest, especially a tropical rain forest.
Remember that climax communities are nothing cut-and-dried. Sooner or later time and chance happeneth to them all and they are replaced by disturbed or "ruderal" communities where the whole struggle starts over. Even in a climax rainforest when an old tree falls, forming a clearing, ruderal plants take over, growing quickly. It then takes years or decades for the climax vegetation to dominate and smother the ruderal plants.
Note that many climax trees will not take on the typical climax shape, with long, straight, bare trunks and the foliage mainly at the top, unless they have to grow through a shading tangle of ruderal plants. For instance, some forest yellow-woods (Podocarpus) if grown in the open will form shrubby or domed shapes just a few metres tall, but if they have to grow through a tangle of short-lived ruderal plants such as Virgilia, they grow straight and tall.
As you can see, the idea of a mature ecological community is neither clear-cut nor rigid. But it is generally useful.
