TL:DR; Mutation will ensure that no species is stable and will constantly evolve
Evolution is driven by differing survivability of individuals in response to environmental pressures (Natural Selection), rather than variability of the environment itself. All zebras (to pick a completely random example) evolved in, essentially, the same environment. The differences that drove their evolution was between the zebras. Those zebras that possessed better responses to that environment had more surviving descendants and, through the responses of those individuals, zebras as a species became better adapted to that environment over time.
The differences between individual zebras are encoded in their genomes, which is also how they pass those traits on to their descendants.
Now, one might suppose that in an unchanging environment eventually zebras would be perfectly adapted, and from then on never change.
But this is not the case because there is another factor in evolution. The genome is not unchanging. It is constantly subject to subtle naturally occurring mutations. Most are neutral, many are harmful, but a few are beneficial. A zebra with such a mutation will eventually have more descendants than zebras lacking it, and zebras will evolve, even in the absence of environmental change.
Finally consider that zebras do not exist in isolation and what is true for them is also true for the grass they eat and the lions that eat them, so their environment (even if climate and geology could somehow be stable for millions of years) constantly changes, and changes in environment mean that the great great grandchildren of todays perfect zebras will have to adapt to meet the new challenges and evolution will continue.