A paradox: Human population growth looks a lot like a simple logistic growth pattern. But the simplest interpretation of logistic growth doesn't seem to fit. Is this peculiar to humans, or does it have a biological explanation?
Ecologically, advances in agriculture over the last ~200 years have increased the Earth's carrying capacity for human population, and the human population appears to have grown and started flattening out as we approach the new carrying capacity -- this basically looks like logistic growth (following a change in the carrying-capacity parameter).
The simplest biological interpretation of logistic growth is that birth rates remain constant (driven by constant biological factors), but as the population approaches the carrying capacity, there are not enough resources to go around, so death rates rise to match the birth rate. In human populations, this rise in death rates would presumably manifest itself in the form of widespread famine and resource wars.
But that is not at all what we see in the human population! In fact, we have seen falling death rates due to advances in medicine. For some reason, birth rates have actually fallen to levels roughly commensurate with the death rates (and in most developed countries, the drop in birth rates has actually overcompensated for the drop in death rates, leading to an ageing population). So although the net population dynamics seem relatively simple, the simplest explanation for these dynamics fails spectacularly. This leads to several
Is there an underlying biological mechanism linking our approach toward the environment's carrying capacity to a resulting drop in birth rates?
There are compelling sociological explanations -- education and empowerment of women, urbanization, decreased child mortality, access to contraception, etc. all point individuals toward a reproductive strategy of investing more into a smaller number of children. But from a sociological perspective, it appears to be a mere accident that all of these forces are converging at once to save us from a scenario of high birth rates and high death rates. One suspects there might be something more fundamental going on here.
Are there examples of other species which do the same thing -- throttle down their birth rates as they approach the environment's carrying capacity?
Maybe there's an evolutionary advantage to this. A population which generates lots of individuals destined to fight one another or die of hunger before reproducing seems to be wasting a lot of energy, especially if individuals care for their young. An overcrowded, underfed population may be more vulnerable to catastrophic disease (especially in a social species), and reduced individual fitness may actually reduce the total amount of resources the population is able to extract from the environment. If a species relies on cooperation within social groups, then perhaps excess births lead to intragroup competition that undermines group fitness. The basic principle of group fitness -- that sometimes it is more advantageous to help others in one's social group than to act directly for one's own benefit -- may be magnified when populations are stable with little room for growth (just as in a mature marketplace, it may be easiest to cooperate with an established business, whereas in a new sector it may make more sense to grow a new business from the ground up).
So I would look for this phenomenon among species that form cooperative social groups and care for their young -- colony-forming insects and large social mammals spring to mind. From my amateur reading, it sounds like lions might be a non-example, regulating their population by pushing excess or unfit individuals out of the pride to their likely death (or at least, this suggests that the population is not regulated solely via birth rates). But it seems like this could be addressed experimentally: for example, if individuals from a stable population of species $X$ are introduced into a new environment that can easily support a larger population of $X$, then does their birth rate go up?
If the answer to (2) is "yes", then what biological mechanisms are behind this?
Such a mechanism must be interpreted in the context of individuals' reproductive behavior more generally. There will be (a) factors affecting the advantageousness of reproducing, with (b) environmental cues indicative of these factors, (c) factors influencing individuals' actual reproductive behavior, and (d) biological mechanisms by which (b) influences (c).
Among the (a)-type factors we're particularly interested in might be abundance of food, space, disease, social stability,... Candidates for (b) might include crowdedness, air quality, presence of greenery and wildlife, stress levels,... For (c), one might look at libido, fertility, desire for a mate or for children,... For (d), ideally one could get down to the level of things like hormone regulation.
Can we use the answer to (3) to inform an answer to (1)?
Here are a few crazy theories that might be supported.
In urban areas, the smell of other humans and lack of greenery are cues that the population is overcrowded and food is scarce, which via some dedicated hormone-regulation system common to all mammals suppresses the desire to mate or raise offspring (but in the United States, these factors are mitigated slightly by our constant showering and our sparser settlement patterns, leading to a birthrate higher than in much of the developed world.)
The gradual process of political integration and tolerance has led us to identify larger and larger subpopulations of humans as "us", and a sense of being part of a large group with a stable population both upregulates pro-social behavior and somewhat suppresses the desire to have children of one's own (and this tendency is especially pronounced among the well-off, who have a sense that society generally works in their interest, and that their non-reproductive behavior is particularly valuable to society, perhaps more valuable than effort they might spend rearing children).
The elevated stress levels of modern life are a cue that one can ill-afford the investment required to reproduce, suppressing the desire for children.