Consider the case where a species is in an intolerant environment that has very specific demands on survival. Then there will be very little variation in characteristics of the species because any departure from maximal fitness will ensure nonsurvival. In a few generations the species will be finely honed to the environment.
Then let's say that the environment changes. There may be a volcanic eruption that changes the carbon dioxide level, or the predator species gets more efficient, or a food species goes through a population minimum. The finely tuned species, having very little variation, will not be fit for the new environment and will become extinct. This is shown in Figure EG0509a where the range of characteristics tolerated by the environment makes a change greater than the original survival range.
The sequence shows the environment changing enough that there is no survival overlap with the original population.
The moral of this process is that too close adaptation to an environment is not conducive to species survival. If the environment changes, a population that is too "fit" for the old environment is likely to become extinct. In fact the population, because they were all close to being "the fittest" for the old environment, were not able to survive a change in environment. Being "the fittest" in that circumstance was not at all good for the species whether or not it was good for the individual in the old niche.
This is the classic picture of an environment changing and a species going extinct. This is the picture of what happened to the dinosaurs, with some doubt because we don't know if and how the environment changed. We have the sense that if their environment changes fast enough almost any species can go extinct.
It doesn't have to happen like that.
Lets look at a different environment. We'll make this one less harsh so that an individual member of the species can squeak by even if it isn't very well matched to the environmental conditions. That's probably more typical than the severe environment.
The less than perfectly adapted individual has a chance to pass on its characteristics, perhaps by mating with an individual that is maladapted in some different way. Their less than perfectly adapted children may land in some other spot in parameter space. In a few generations the species will show examples of all the variations that that environment will tolerate. Now, when the environment changes, there will be some members of the species who will be still more maladapted and they won't survive.
But there will also be individuals who are better adapted to the changed environment than they were to the old environment. They could survive in the old environment because it was tolerant, but they survive even better in the new one. The old fashioned name for these individuals was "hopeful monsters". We now understand that they were just lucky; lucky to start out in an environment that tolerated their variation and lucky that their maladaptation turned out to be more adapted to the changed environment. This is illustrated in Figure EG0509b.
They will survive and they will continue to contribute their variation until the species again shows all the possible variations that the new environment will tolerate. The result of this gedankenexperiment can be stated as a principle: The species that has within its gene pool the maximum amount of variation that the environment will tolerate is the most likely to survive changed environmental conditions as a species.
If the environment keeps changing slowly (relative to the gestation period of the species) the population can change to keep up with the environmental changes. It is even possible that the process can proceed to the point that there is no overlap between the original population and the new one. This is precisely the case of the evolution of a new species from an old one.
This process can happen, not just to an entire species, but to a subpopulation of the original species that is more-or-less isolated. Thus it is possible to have a new species in one venue and the old species surviving in some other venue. This is what happened to Darwin's Galapagos finches.
"Survival of the Fittest" does not explain the evolution of species because those individuals that are fittest for the original environment are not the fittest in the changed environment. If only the fittest survive, as in the case of the intolerant environment, the species does not survive the change in environment. The more variation the species has, and the more variation the environment will tolerate, the more likely this scenario will become. This is exactly described by "The Survival of The Just-Barely-Fit".
The real basis for the evolution of species, is that those species survive (albeit in a different form which may be considered a 'new species' ) which have the largest tolerance for variations that are barely able to survive as individuals.