Carrying capacity is most often presented in ecology textbooks as the constant K in the logistic population growth equation, derived and named by Pierre Verhulst in 1838, and rediscovered and published independently by Raymond Pearl and Lowell Reed in 1920:

(integral form)

where N is the population size or density, r is the intrinsic rate of natural increase (i.e., the maximum per capita growth rate in the absence of competition), t is time, and a is a constant of integration defining the position of the curve relative to the origin. The expression in brackets in the differential form is the density-dependent unused growth potential, which approaches 1 at low values of N, where logistic growth approaches exponential growth, and equals 0 when N= K, where population growth

Figure 1 The definition of carrying capacity most frequently used in basic ecology textbooks. (a) Logistic population growth model, showing how population size (N) eventually levels off at a fixed carrying capacity (K) through time (t). (b) Logistic population growth rate (dN/dt) as a function of population size. Note that the growth rate peaks at 0.5K and equals zero at K.

Figure 1 The definition of carrying capacity most frequently used in basic ecology textbooks. (a) Logistic population growth model, showing how population size (N) eventually levels off at a fixed carrying capacity (K) through time (t). (b) Logistic population growth rate (dN/dt) as a function of population size. Note that the growth rate peaks at 0.5K and equals zero at K.

simplistic and much more of heuristic than practical value; very few populations undergo logistic growth. Nonetheless, ecological models often include Kto impose an upper limit on the size of hypothetical populations, thereby enhancing mathematical stability.

Of historical interest is that neither Verhulst nor Pearl and Reed used 'carrying capacity' to describe what they called the maximum population, upper limit, or asymptote of the logistic curve. In reality, the term 'carrying capacity' first appeared in range management literature of the late 1890s, quite independent of the development of theoretical ecology (see below). Carrying capacity was not explicitly associated with K of the logistic model until Eugene Odum published his classic textbook Fundamentals of Ecology in 1953.

The second use in basic ecology is broader than the logistic model and simply defines carrying capacity as the equilibrial population size or density where the birth rate equals the death rate due to directly density-dependent processes.

The third and even more general definition is that of a long-term average population size that is stable through time. In this case, the birth and death rates are not always equal, and there may be both immigration and emigration (unlike the logistic equation), yet despite population fluctuations, the long-term population trajectory through time has a slope of zero.

The fourth use is to define carrying capacity in terms of Justus Liebig's 1855 law of the minimum that population size is constrained by whatever resource is in the shortest supply. This concept is particularly difficult to apply to natural populations due to its simplifying assumptions of independent limiting factors and population size being directly proportional to whatever factor is most limiting. Moreover, unlike the other three definitions, the law ofthe minimum does not necessarily imply population regulation.

Note that none of these definitions from basic ecology explicitly acknowledges the fact that the population size of any species is affected by interactions with other species, including predators, parasites, diseases, competitors, mutu-alists, etc. Given that the biotic environment afforded by all other species in the ecosystem typically varies, as does the abiotic environment, the notion of carrying capacity as a fixed population size or density is highly unrealistic. Additionally, these definitions of carrying capacity ignore evolutionary change in species that may also affect population size within any particular environment.

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