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The carrying capacity of an environment is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. The carrying capacity of any natural system is not a constant amount, as it depends on a variety of dynamic factors such as technology, preferences, production and consumption, and certainly in natural populations, interactions of biotic factors with and within a system. In population ecology, carrying capacity is defined as the environment's maximal load, which is different from the concept of population equilibrium, which may be far below an environment's carrying capacity. The effect of carrying capacity on population dynamics may be modelled with a logistic function. Carrying capacity can be applied to many different population types, like natural biological populations in the field of ecology, and human populations. The term carrying capacity has been applied to a few different processes in the past before finally being applied to population limits in the 1950's.

Origins
The term carrying capacity was originally used to address matters related to international shipping in the 1840’s. It then became a term used in livestock management in the 1870’s before finally becoming a staple term used to define the biological limits of a natural system related to population size in the 1950’s.

Trends and Factors
The specific reason why a population stops growing is known as a limiting or regulating factor. Limiting factors for the human population include water availability, energy availability, renewable resources, non-renewable resources, heat removal, photosynthetic capacity, and land availability for food production. The difference between the birth rate and the death rate is the natural increase. If the population of a given organism is below the carrying capacity of a given environment, this environment could support a positive natural increase; should it find itself above that threshold the population typically decrease. Thus, the carrying capacity is the maximum number of individuals of a species that an environment can support.

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.

In the standard ecological algebra as illustrated in the simplified Verhulst model of population dynamics, carrying capacity is represented by the constant K:


 * $$ \frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right) $$

where

$N$ is the population size,

$r$ is the maximum growth rate,

$K$ is the carrying capacity of the local environment, and

$dN/dt$, the derivative of $N$ with respect to time $t$, is the rate of change in population with time.

Thus, the equation relates the growth rate of the population $N$ to the current population size, incorporating the effect of the two constant parameters $r$ and $K$. (Note that decrease is negative growth.) The choice of the letter $K$ came from the German Kapazitätsgrenze (capacity limit).

This equation is a modification of the original Verhulst model:


 * $$ \frac{dN}{dt} = rN - \alpha N^2$$

In this equation, the carrying capacity $K$, $$N^*$$, is


 * $$ N^* = \frac{r}{\alpha}. $$

Biological Populations
Carrying capacity is a commonly used method for biologists when trying to better understand biological populations and the factors which affect them. When addressing biological populations, carrying capacity can be used as a stable dynamic equilibrium, taking into account extinction and colonization rates. In population biology, logistic growth assumes that population size fluctuates above and below an equilibrium value. Although useful in theory and in laboratory experiments, the use of carrying capacity as a method of measuring population limits in the environment is less useful as it assumes no interactions between species.

Humans
Since the beginning of the late 1960's, talk of economic and population growth leading to the Earth's carrying capacity limits sparked a flame of environmental awareness. The use of carrying capacity as a method for measuring the Earth's limits in terms of the human population has been an extremely useful tool in promoting human awareness of the existing limits of economic activity. Carrying capacity becomes a very useful tool in neo-Malthusian arguments, where a numerical value can act as a great driver for collective political action. Along with creating awareness, carrying capacity can also be a useful tool in determining the population limits in cities when faced with rapid urbanization. Several estimates of the carrying capacity of the earth for humans have been made with a wide range of population numbers. A 2001 UN report said that two-thirds of the estimates fall in the range of 4 billion to 16 billion with unspecified standard errors, with a median of about 10 billion. Some of these issues have been studied by computer simulation models such as World3.

The application of the concept of carrying capacity for the human population, which exists in a non-equilibrium, has been criticized for not successfully being able to model the processes between humans and the environment.