It has been said that Darwinian natural selection requires an increasing population. For example:
In fact, if a simple geometric growth rate is assumed (which was the assumption made by Charles Darwin in relation to his imagined "struggle for existence" in nature), ...How Populations Can Grow, Henry M. Morris, ICR
This is incorrect, because natural selection can apply when a population is shrinking. For example, suppose that some event kills off half of a species, but mostly kills just the tall ones (or short-beaked ones, or blondes, or whatever). If this characteristic is even partly heritable, then the next generation will have fewer members that have that characteristic. The species will, on average, be shorter (or whatever). Of course, they may not stay that way, but the gene pool was definitely changed.
Events like this have been observed in nature: some examples are described in The Beak of the Finch.
Natural selection can also apply when a population level stays the same. For example, some fraction of every human generation dies without having had children. If a heritable characteristic increased one's chances of being in that fraction, then the heritable characteristic would be under selection pressure, and might eventually vanish from the human gene pool. And it isn't necessary that the characteristic be linked to having no children. It will be under selection pressure if its possessors have slightly fewer children than average.