Carrying capacity is defined as the "maximum population size that an environment can sustain indefinitely." For most species, there are four variables that factor into calculating carrying capacity: food availability, water supply, living space, and environmental conditions. Even with this relatively basic quartet, carrying capacity can be rather tricky to determine. It's simplest to compute when a clear limiting factor is realized. In the case of a population on a small, isolated island, the limiting factor might be space. For a species dwelling in a desert, it might be water.
In the case of humans, the arithmetic is much, much more complicated. Our meta-population is diverse, widespread, and affected by a large multitude of unique variables including technological advancement, disease, and energy consumption. Plus, our species has the ability to deny our reproductive instincts and procreate without producing offspring. Brilliant thinkers can estimate human carrying capacity all they want, but any prediction is subject to the whims of time and the inherent difficulty to grasp the sheer complexities of the variables.
Sometime over the next millennium -- a relatively short time when you consider the age of the Earth -- the human population's carrying capacity will be realized. Assuming space as the limiting factor, and a population growth rate of 1% (It's currently 1.14%), the population density of Earth will equal one human being per square meter of land in a little over 1,000 years. That's quite a packed Earth.
of overpopulation: It's hard to find the guy in the red and white striped shirt!
But we know it won't get to this point, as other factors such as food availability and environmental degradation will likely stem our growth first. More importantly, as people become wealthier and more educated, they tend to produce fewer children. This phenomenon is already occurring in Europe, where the fertility rate has dropped below 2.1 births per woman -- the number considered to be the replacement rate of the human population. Because of this, some demographers think the world population will simply stop growing after we hit a population of around 9 or 10 billion within the next 50 to 100 years.
However, if the human population does continue to grow, it may be that we humans will not know our carrying capacity until it comes upon us. I wonder, will we reach it subtly, without even realizing it, or will it smack us in the face? Humans are considered a "K-selected" species, in that our population size is relatively stable and prone to fluctuate around a carrying capacity, so adapting to it may be easy. But as we've seen in the past, humans are not invulnerable to population collapse. We may fall hard before rising again.
So, in reality, the ultimate question isn't, "What is the human race's carrying capacity?" It's "what will we do if and when we get there?"
That question will likely be even harder to answer.