When an organism is adapted to its environment, and the environmental pressures are not changing, there will appear to be very little evolution going on. In actuality, the alleles of genes will continue to mix and recombine in new ways during sexual reproduction which means that with each new generation the organism is still putting out new versions of itself.

The DNA polymerase is also hardwired with a certain error rate which is very low, but just high enough to allow for a change in DNA here and there. Just rare enough to not really change much, but to allow for change to be possible. These changes too are put out in each new generation.

Typically if the species is excellently adapted to its environment, and the environment doesn't change all, these genetic changes in each new generation aren't likely to provide an improvement so you don't see much change. This is known as stabilizing pressure.

But as soon as a new selective pressure appears, it will become clear that that species never actually retired from the evolution game.

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Evolution is made up of two parts, broadly speaking: random variation and natural selection. (And sexual selection but I’m leaving that aside for now)

Random variation doesn’t slow down, it’s effectively always bubbling away making the next generation look and function slightly differently. These are effectively copying errors when the genetic code is replicated.

Natural selection can appear to slow down, or halt entirely. This is where organisms try to survive in the world long enough to reproduce, “may the best man win” so to speaks (all genders evolve, it’s just a phrase). As time goes by, the species either becomes better suited to its environment or goes extinct. And there is a limit to how well suited it can become (if nothing else, evolving can in principle get stuck at a dead end).

Such animals are called “living fossils”, and I believe alligators are an example of this. There are some plants also which have barely changed for millions of years, because they are very well suited to their environment and any variation tends to make them worse.

Yes, this principle should apply in nature; but at the same time this only applies to a fairly restrictive case

namely, if you take a population of organisms and put it in a new environment, at first its adaptation to this environment will improve very quickly, then it will keep making progress but slower

however, in nature you have to consider that (1) most organisms have been in their usual environments for a while so they're all in the slow adaptation phase; yet that (2) environments actually change all the time, because climate isn't perfectly stable and because the surrounding ecosystems aren't stable either (ecosystems are complex systems that are constantly disrupted by new pathogens, new species, species going extinct or becoming less/more abundant, etc., on top of natural phenomena)

So there's always adaptation, because it occurs towards a forever-moving target. In most cases, it's more accurate to consider that evolution is a perpetual thing

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Others have given great answers. I wanted to point out that the “Red Queen hypothesis” (Wikipedia link) is relevant here. In Lenski’s experiment, conditions were intentionally kept pretty much constant, but in the real world

> species must constantly adapt, evolve, and proliferate in order to survive while pitted against ever-evolving opposing species.… the effective environment of any homogeneous group of organisms deteriorates at a stochastically constant rate. …the evolutionary progress (= increase in fitness) of one species deteriorates the fitness of coexisting species, but because coexisting species evolve as well, no one species gains a long-term increase in fitness

Punctuated equilibrium is possibly more likely than some kind of slowing down. As you understand natural selection, what is the driving force? Some selective pressure from the environment is what causes all mutations and all culling of statistically slightly-less advantageous traits from propagating to the next generation. The list of those potential pressures is rather long, no?

Essentially, the rate of mutation in individuals may remain constant, but without some environnemental pressure exerting a selective effect on individuals with advantageous mutations, there is no reason for those mutations to become reinforced each generation.

If 1/1,000 people randomly have slightly longer fingers, but there is no pressure in the environnement that grants long-fingered people an advantage reproducing, you won’t see an increase in long-fingered people over time; there will remain a roughly 1/1,000 chance that anyone you meet is long-fingered.

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In the case of human evolution, where the species is adapting the environment to its survival needs, the species evolution will halt entirely.

If our ability to adapt the environment in friendly ways is broken then we will see more adaptation.

There is a popular fantasy of 'evolution' that it is a directed force in itself, whereas evolution is not a driver it is an outcome of conditions.

Witness rapid evolution of the COVID virus as it mutates around our hostile vaccines and systems of immunity.

Evolution never really stops. Alleles of genes mix in every generation in sexual reproduction. If the organism is well adapted to one or more environments, the resulting mutations in offspring create little change. Once the environment changes—which is pretty much a guarantee on Earth thanks to plate tectonics and ensuing climate change—then those mutations become critical to the organism’s survival. It’s a matter of what traits allow an organism to live long enough to successfully reproduce. That’s the central dogma of natural selection.

Some "living fossils" just don't change their appearance but change genetically

That’s super cool to learn, thank you!

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Given your explanation, would it be fair to say that species like that are "evolving in place?"

That is, they're still actively mutating, but the mutations are "undone" in successive generations.

I know I've simplified things a lot, but is this still a serviceable explanation of the concept?

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In the simplest model there are 3 kind of mutations: neutral, positive, negative.

The rate at which neutral mutations accumulate (which really means the number of neutral mutations you'll observe between you and your parent) is roughly constant.

But the rate at which positive and negative mutations accumulate depend a lot on: how positive and negative they are (obviously mutations killing the organism won't accumulate) and how many they are among the total number of possible mutations.

When taking a bacteria adapted to a temperature of 20°C and putting it at 30°C the positive mutations (positive for this new environment) will accumulate very fast, at the beginning, and once the bacteria will be adapted to the new temperature there will be much less possible positive mutations, so their accumulation rate will slow down a lot.

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check out the red queen hypothesis, don't know if it's still considered valid.

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