The joining and mathematical representation of Malthus’ assumptions of exponential growth and
existing limits to growth was an important first step in the development of the paradigm today
known as carrying capacity. In conclusion, the considerable impact of Malthus’ assumptions can
be attributed to their broadly favorable reception in his lifetime, to his influence on Darwin, and
consequently on biology, and also to the fact that the new science of demography initially adopted
Malthusian ideas.
3. Carrying capacity in biology and ecology
Logistic growth of populations is a basic as- sumption in population biology as ‘[o]ver long
periods of time, in all populations of organisms… N, the population size, fluctuates up and down
around some average value.’ Wilson and Bossert, 1971, p. 102. This logistic growth dynamic con-
sists of an exponential population growth slowed down by an upper limit carrying capacity.
In laboratory experiments populations of ani- mal species often exhibit the logistic growth pat-
tern. However, when this model with its simplified assumptions is applied in population biology, it is
particularly restrictive since the basic model al- lows no time-lags and no interactions between
species. Furthermore, it implies a carrying capac- ity independent of past population sizes, and
finally, the prediction of the model rests on the assumption of independent life and reproduction
of the individuals. However, population dynamics cannot be predicted deterministically. Rather,
‘chaos’, interactions between species, and environ- mental factors, have been attracting significant
attention in the research of modern population dynamics.
In spite of its simplified assumptions, experi- mental evidence indicates that the logistic growth
model can depict dynamics of simple populations. Nicholson showed in his now famous experiment
with blowflies in population cage experiments that food availability is the sole limiting factor for
their population growth and reproduction. This experiment shows that ‘…egglaying of population
is limited by the food availability, such that a high adult population density results in a low produc-
tion of eggs and a low density of adults results in a high egg production’ Christiansen and Fenchel,
1977, p. 4. Also, the doubling of K doubled the population of the blowflies as the model pre-
dicted. May 1973, p. 101 was able to model the experimental results of Nicholson by successfully
fitting a logistic growth model incorporating a time lag t – T into the equation:
dNt dt
= rNt
K − Nt − T K
For further experimental studies of logistic growth see Hutchinson, 1979, pp. 21 – 27.
However, dynamics of many natural popula- tions are much more complex than this, and
consequently can only crudely be described by the logistic equation Pulliam and Haddad, 1994, p.
142. Characteristics of natural surroundings which are not reflected in Nicholson’s experiment
are, for instance, the interdependence between the population and its surrounding abiotic and biotic
system e.g. weather, predators, diseases, intra- specific
interactions like
saturation density,
changes of the environment, stochastic events, and finally differing and possibly unpredictable
time lags. Also, a transgression of the initial level of carrying capacity can induce a dynamic which
results in a new lower carrying capacity.
The logistic curve in biological models of popu- lation growth also builds upon the ecological
concept of carrying capacity as first formulated by Hawden and Palmer in 1922 as they observed the
effects of introduction of reindeer populations in Alaska. After a huge growth in the number of
these animals, their population fell rapidly and eventually reached an equilibrium. Hawden and
Palmer defined carrying capacity as the number of stock which a range can support without injury to
the range Pulliam and Haddad, 1994, p. 141. Their definition was an advance in that it intro-
duced the idea of interaction between the popula- tion of animal species and the state of the
environment. It shifted the attention from equi- librium population at dNdt = 0, r = 0 to an
equilibrium quality of the environment range. Yet, this definition of carrying capacity indicates
the difficulty of evaluating objectively the interac-
tion between a population and the environment. What qualifies as an injury, what does it mean to
say ‘without injury’? Normative judgments may well be inescapable in such evaluations.
A similar definition was introduced a decade later by Leopold Hawden and Palmer, 1994, p.
141. He defined carrying capacity as the maxi- mum density which a particular range is capable
of supporting Dhondt, 1988, p. 339. Since the definitions of HawdenPalmer and Leopold many
others have been formulated which restate, vary, or develop these basic ones see Dhondt, 1988;
Pulliam and Haddad, 1994. All these definitions, however, are, as Pulliam and Haddad 1994
point out, a poor descriptor of the dynamics of many natural populations. An important reason
for this is the complexity of actual ecosystems. Their biotic interactions and multiple steady
states are characterized by nonlinear dynamics and population thresholds which are all influ-
enced and modified by environmental variations in space and time e.g. exogenous disturbance.
Hardin states: ‘‘There is no hope of ever making carrying capacity figures as precise as, say the
figures for chemical valence or the value of the gravitational constant. On St. Matthew Island the
growth of reindeer moss is no doubt greater in some summers than others…’’ Hardin, 1986, p.
600.
In principle, there are at least three major rea- sons why the simple logistic model of population
growth may be a poor predictor in practice: 1. Exogenous environmental forces may cause
variation in the carrying capacity, K, or in the Malthusian parameter, r, or in the lag involved
in the response of a population. While these variations are frequently temporary, they can
also be permanent.
2. Variations in population sizes may cause K, r, or the lag-factor to alter permanently. For
example, a sudden and large increase in the population of a species may permanently de-
stroy environmental resources which the spe- cies utilizes to some extent, or populations in
particular ranges may slowly and irreversibly degrade their environment. These environmen-
tal alterations are internal to the system, often exhibit hysteresis and can occur either in the
absence of exogenous environmental change or due to human intrusion.
3. Exogenous environmental variation combined with certain sizes of population may bring
about permanent alterations in the coefficients of the logistic model. In other words, interac-
ti6e permanent effects can easily occur between the state of the environment and particular
population sizes.
McLeod 1997 analyzed different models and methods for determining and calculating carrying
capacity, and showed that complex characteris- tics, uncertainties and stochastic environments
cannot be overcome or captured by these models. Rather, the concept of carrying capacity can only
be calculated for deterministic and slightly vari- able systems, and only for cases where behaviour
and ecological relationships of the species change slowly on the human time scale Cohen, 1995a, p.
247. In variable environments, carrying capacity might be useful as a measurement of short-term
potential densities as a function of resource availability but not of long-term equilibrium den-
sities McLeod, 1997, p. 540.
4. Application of carrying capacity to environmental impacts of human activity
