Carrying capacity
The equilibrium maximum of the population of an organism is known as the ecosystem's carrying capacity for that organism.  Generally it is the supportable population of an organism, given the food, habitat, water and other necessities available within an ecosystem.  For the human population more complex variables such as sanitation and medical care are sometimes considered as part of the necessary infrastructure.
As population density increases, birth rates often decrease and death rates typically increase. The difference between the birth rate and the death rate is the "natural increase."  The carrying capacity could support a positive natural increase, or could require a negative natural increase. Carrying capacity is thus the number of individuals an environment can support without significant negative impacts to the given organism and its environment. A factor that keeps population size at equilibrium is known as a regulating factor. The origins of the term lie in its use in the shipping industry to describe freight capacity, and a recent review finds the first use of the term in an 1845  report by the US Secretary of State to the Senate (Sayre, 2007). It was never used, as is widely assumed, by Thomas Malthus.What is best to remember about carrying capacity is that ultimately it is not a number but a relationship best described as a differential equation. With food availability, population size, and environmental factors, all varying inconsistently over time. While by this definition it is easy to see the need for computer modeling, early attempts at wildlife management often neglected this fact.
Below carrying capacity, populations typically increase, while above, they typically decrease.  Population size decreases above carrying capacity due to a range of factors depending on the species concerned, but can include insufficient space, food supply, or sunlight. The carrying capacity of an environment may vary for different species and may change over time due to a variety of factors including: food availability; water supply; environmental conditions; and living space.

# Temporary exceptions
It is possible for a species to exceed its carrying capacity temporarily.  Population variance occurs as part of the natural selection process but may occur more dramatically in some instances.  Due to a variety of factors a determinant of carrying capacity may lag behind another.  A waste product of a species, for example, may build up to toxic levels more slowly than the food supply is exhausted.  The result is a fluctuation in the population around the equilibrium point that is statistically significant.  These fluctuations are increases or decreases in the population until either the population returns to the original equilibrium point or a new one is established.  These fluctuations may be more devastating for an ecosystem compared to gradual population corrections since if it produces drastic decreases or increases the overall effect on the ecosystem may be such that other species within the ecosystem are in turn affected and begin to move with statistical significance around their equilibrium points.  The fear is a domino like effect where the final consequences are unknown and may lead to collapses of certain species or whole ecosystems.

# Examples
The moose and wolf population of Isle Royale National Park  in Lake Superior is one of the world's best studied predator-prey relationships. Without the wolves, the moose would overgraze the island's plants. Without the moose, the wolves would die. It seemed to the first scientists that studied the problem that the wolves would eventually overpopulate and kill all the moose calves and then die from famine.  This has not occurred, however, and in fact the wolves appear to be "limiting their own population size".
Easter Island has been cited as an example of a human population crash.  When fewer than 100 humans first arrived, the island was covered with trees with a large variety of food types, in 1722 the island was visited by Jacob Roggeveen, who estimated two to three thousand inhabitants with very few trees, "a rich soil, good climate" and "all the county was under cultivation". Half a century later it was described as "a poor land" and "largely uncultivated". The ecological collapse that followed has been variously attributed to overpopulation, slave raiders, European diseases including a Smallpox epidemic that killed so many the dead were left unburied and a Tuberculosis epidemic that killed a quarter of the population, civil war, cannibalism, and invasive species (such as the Polynesian rats that may have wiped out the ground nesting birds and eat the palm tree seeds). Whatever the combination of reasons, only 111 inhabitants were left on the island in 1877.  For whatever reason: Moai worship, survival, status, or pure ignorance, the question of how many humans the island could comfortably support never seems to have come up. Their known history includes a population crash that might have been avoided had they asked that simple question.
The Chincoteague Pony Swim  is a human assisted example.
Both herds are managed differently. The National Park Service owns and manages the Maryland herd while the Chincoteague Volunteer Fire Company owns and manages the Virginia herd. The Virginia herd, referred to as the "Chincoteague" ponies, is allowed to graze on Chincoteague National Wildlife Refuge, through a special use permit issued by the U.S. Fish and Wildlife Service. The size of both herds is restricted to approximately 150 adult animals each in order to protect the other natural resources of the wildlife refuge.
A further example is the Island of Tarawa,  where the finite amount of space is evident, especially since landfills cannot be dug to dispose of solid waste. With colonial influence and an abundance of food (relative to life before the year 1850), the population has expanded to the extent that overpopulation is transparently present.

# Fertility and carrying capacity interaction
If food supply of the environment is abundant, in humans for example, twinning may result .  As a result, parents then typically devote less care to each offspring in other ways as well, as the young may manage on their own with abundant food supply. Such parents have as many offspring as possible by starting early and quickly repeating breeding.  When environmental conditions deteriorate with an expanding population, they may K-shift (resort to small numbers of offspring) toward the more conservative strategy of betting on a few well-placed shots. When a species is already exploiting the environment near the limits of  carrying capacity (which includes food availability but also nesting sites etc.), a wise strategy is to play it safe by raising a limited number of offspring, devoting considerable care to each.
Since this also applies to humans, then two questions immediately arise: How is the "boom time" r-shift (resort to large numbers of offspring) implemented? (Is sexual maturity sped up, or is juvenile growth rate, or perhaps both?) And is the trigger, what aspects of the environment are "read" for the forecast? If one is ever to replace this corner-cutting "Quantity is Better than Quality" philosophy and effectively combat its fatalistic "Life is Cheap" corollary, we need to understand what drives it (the "hangover" that follows a reproductive "binge" is better known as a population crash).

# Mathematics
For a specific case example in the wild, see the Lotka-Volterra equation, which shows how limited resources will cause the predator population to decline due to famine. Note that depending on the situation, the impact of famine could be moderate (e.g. the prey is not the main source of food for the predator), or extreme (e.g. the prey becomes extinct due to over-predation, such as when humans stressed mammoth populations over the brink of extinction; if the prey is the only source of food, the predator will also suffer severe famine or become extinct).

# Humans
In the words of one researcher attempting to interpret the carrying capacity concept: "Over the past three decades, many scholars have offered detailed critiques of carrying capacity--particularly its formal application--by pointing out that the term does not successfully capture the multilayered processes of the human-environment link, and that it often has a blame-the-victim framework. These scholars most often cite the fluidity and nonequilibrium nature of this relationship, and the role of external forces in influencing environmental change, as key problems with the term." (Cliggett 2001)
In other words, the relationship of humans to their environment may be more complex than is the relationship of other species to theirs. Humans can consciously change the type and degree of their impact on their environment by, for example, increasing the productivity of land through more intensive farming techniques, leaving a defined local area, or scaling back their consumption: of course, humans may also irreversibly decrease the productivity of the environment or increase consumption (e.g. overconsumption).
Supporters of the concept argue that humans, like every species, have a finite carrying capacity.  Animal population size, living standards, and resource depletion vary, but the concept of carrying capacity still applies. The World3 model of Donella Meadows deals with carrying capacity at its core.
Carrying capacity on its most basic level is about organisms and food supply: X amount of humans need Y amount of food to survive. If the humans neither gain or lose weight in the long run the calculation is fairly accurate. If the quantity of food is constant at Y amount, carrying capacity has been reached.
Humans with the need to enhance their reproductive success (see Richard Dawkins 'the Selfish Gene') understand that food supply can vary and also that other factors in the environment can alter humans' need for food. A house for example might mean one does not need to eat as much to stay warm.
Over time monetary transactions have replaced barter and local production. However, purchases  impact regions thousands of miles away. Carbon dioxide from an automobile, for example, travels to the upper atomsphere. This lead Paul Ehrlich to develop the IPAT Equation where:
where:
I is the impact on the environment resulting from consumption
P is the population number
A is the consumption per capita (affluence)
T is the technology factor
(Ehrlich and Holdren 1971)
This is another way of stating the carrying capacity equation for humans that substitutes impact for resource depletion and adds the technology term to cover different living standards.  As can be seen from the equation money affects carrying capacity, but it is too general a term for accurate carrying capacity calcuation.
The concept of Ecological footprint was developed to examine differential consumption by humans. By calculating the average consumption of humans over a small area, projections  can be made for that type of population's impact on the environment.
Carrying capacity 'averages' the blame for these impacts. It blames the rich for using too many resources, as well as the poor for being too numerous. Carrying capacity calculates the 'average' use of food and resources, which is closer to the billions of poor in the world, than the hundreds of billionaires.
This type of discussion raises the question of whether it is possible to define a measure of sustainability that does not already contain implicit assumptions about the solution to the problem of resource over-use and environmental degradation. Only by showing these implicit assumptions can progress be made.

## Reduction of Earth's carrying capacity in the 21st century
After an expansion of agricultural capability on the Earth in the last quarter of the 20th century, there are many projections of a continuation of the decline in world agricultural capability (and hence carrying capacity) that began in the 1990s.  Most conspicuously China is forecast to decline in food production by 37 percent by the last half of the 21st century, placing a strain on the entire carrying capacity of the world as China's population will have expanded to about 1.5 billion people by the year 2050.  This reduction in agricultural capability in China (as in other world regions) is largely due to the world Water crisis, especially due to mining groundwater beyond sustainable yield, which process has been occurring in China since about 1950.

## Points of contention
The concept has been comprehensively critiqued by social scientists and demographers, who are concerned that it is simplistic and Malthusian in its assumptions. Some important criticisms appear in Sayre (2007). Ester Boserup, the Danish economist, argues that higher population densities increased agricultural productivity rather than leading to the reverse, because "necessity is the mother of invention".
A more complete synopsis of Ms Boserup's work is given by the AAAS Population & Environment Atlas:
A more sophisticated adaptation approach was put forward by Ester Boserup in her classic book The Conditions of Agricultural Growth. Boserup suggested that population growth was the principal force driving societies to find new agricultural technologies (Boserup, The Conditions of Agricultural Growth, Allen and Unwin, 1965, expanded and updated in Population and Technology, Blackwell, 1980.).
Unlike Julian Simon, Boserup did not claim that the process ran smoothly. She acknowledged that population pressure could cause serious resource shortages and environmental problems, and it was these problems that drove people to find solutions. Nor did she claim that things were always better after the adaptation.
They could often be worse. For example, when hunter-gatherers with growing populations depleted the stocks of game and wild foods across the Near East, they were forced to introduce agriculture. But agriculture brought much longer hours of work and a less rich diet than hunter-gatherers enjoyed. Further population growth among shifting slash-and-burn farmers led to shorter fallow periods, falling yields and soil erosion. Plowing and fertilizers were introduced to deal with these problems - but once again involved longer hours of work and degradation of soil resources(Boserup, The Conditions of Agricultural Growth, Allen and Unwin, 1965, expanded and updated in Population and Technology, Blackwell, 1980.).
2. If agricultural innovation could increase with population density, then carrying capacity may not be a decisive issue. Empirical justification for Boserups's view is found in the work of Mike Mortimore and Mary Tiffen (1994, ), working in high-density East Africa, and in several other studies they and others have conducted across the continent.
If this contention were true Africa would be self sustaining and Desertification would not be taking place at a rate higher than when the population was lower.
3.Centuries of evidence from Africa and Asia shows when local livelihood conditions deteriorate, migration and a multiplicity of livelihood strategies are developed as common human responses, well before any point of 'capacity' is reached.
This statement shows that the inhabitants are aware of carrying capacity, and their movements are in response to it. Also while these adaptations work with low density primitive populations, when densities increase as a result of abundant food supply over a long period and national boundaries stop natural migration, problems like Rwanda occur. 
4.Furthermore, the concept is simplistic. It requires belief in bounded spaces in which population-environment relationships may be measured and monitored.
As Occam's razor clearly states simplicity is an advantage, not a hindrance. While it is easier to quickly see the results of population biology on an island, with human populations worldwide subject to  national censuses, information on imports and exports of goods as well as people, much of our economic system depends on accurate information of food supply, what the average human eats, and how many humans there are. Humans even have 'clocks' of population and productive land.
5.The faith that these spatial units can be determined and then measured is unwarranted given recent knowledge of non-linear rangeland ecologies, and of population-environment relationships in general.
This is an article based on biological fact not faith. Nonlinear rangeland ecological thresholds refer to overshoot of carrying capacity, it doesn't negate carrying capacity it verifies it.
6.The world is not a petri dish, and people's use of it, is simply too chaotic and messy (Gausset et al, 2005).
A statement that in itself cannot be proved or negated, cannot logically be used to prove or disprove another statement. Carrying capacity has been proven to exist in the animal world beyond all doubt, it is the basis of evolution. To say carrying capacity doesn't apply to humans is tantamount to saying humans didn't evolve.