- Email: [email protected]
Carrying Capacity, Concept of
CARRYING CAPACITY, CONCEPT OF Gregg Hartvigsen State University of New York, College at Geneseo
I. II. III. IV. V. VI. VII. VIII. IX.
Introduction Origin of the Concept of Carrying Capacity Definitions of Carrying Capacity Do Populations Have Carrying Capacities? Determining the Carrying Capacity of a Population Current Research on Carrying Capacity The Importance of the Concept of Carrying Capacity to Biodiversity The Human Carrying Capacity Conclusions
GLOSSARY density dependence The condition that environmental factors influence population growth rate in relation to population size. Density dependence usually is seen as an linear, inverse relationship between population growth rate and population density (i.e., population growth decreases as density increases) and may occur if individuals compete or predators are more effective as a prey population increases. density independence The absence of environmental factors that influence population growth as a function of density. This may occur if mortality removes a fixed percentage of a population, independent of population size. logistic growth Regulated population growth that follows the logistic equation dN/dt ⫽ rN(1 ⫺ N/K). Populations growing according to this equation in-
crease rapidly at low densities and the growth rate decreases as they approach carrying capacity (K). population A group of individuals of a particular species that live in a region. A population is usually a subset of the entire species. population regulation The constraint of positive population growth. The study of population regulation deals with the factors that cause this constraint, such as competition for food or predation. population stability The tendency for populations to return to a previous size after a disturbance, such as reductions due to hunting or disease or increases due to immigration. Stable populations may be locally stable (return after small disturbances) or globally stable (return after severe or catastrophic disturbances).
CARRYING CAPACITY is the maximum number, density, or biomass of a population that a specific area can support. This number is likely to change over time and depends on changes in environmental factors (e.g., rainfall and temperature), resources (e.g., food, hiding places, and nesting sites), and the presence of predators, disease agents, and competitors over time. The concept of carrying capacity has been explicitly recognized for approximately 150 years and its use has waxed and waned during this time. Currently, the use of carrying capacity to describe any particular population is made only with great caution, although the concept remains
Encyclopedia of Biodiversity, Volume 1 Copyright 2001 by Academic Press. All rights of reproduction in any form reserved.
641
642
CARRYING CAPACITY, CONCEPT OF
intuitive and fosters questions that address our fundamental understanding of what factors regulated populations over time and space.
I. INTRODUCTION Populations, or groups of individuals within a species, change over time. There is general agreement among ecologists that population growth is bounded by biotic and abiotic environmental factors that result in approximate, maximum numbers of organisms that can be supported in different habitats. A population’s carrying capacity is difficult to measure and likely varies over time and through space. The concept of carrying capacity has played an important role in the fields of basic ecological research, wildlife management, and conservation biology. The concept of carrying capacity also involves determining how many people Earth can support. Dialogues about human carrying capacity are often quite contentious and illustrate the difficulty surrounding the concept of carrying capacity. The concept of carrying capacity is alive and well, although some have argued that it should be abandoned altogether. On the surface, the concept is easy to understand and intuitive and, therefore, is likely to stay with us for some time. However, recent developments in our understanding of the dynamics of population change over time have greatly modified what is considered an area’s carrying capacity. Therefore, we need to recognize both the strengths and the weaknesses of the concept of carrying capacity.
II. ORIGIN OF THE CONCEPT OF CARRYING CAPACITY Humans have long been aware of the limitations of their own population growth. As early as the Old Testament one can argue that the concept of population limitation was recognized. In the book of Genesis (28:3) are the following words: ‘‘And God Almighty bless thee, and make thee fruitful, and multiply thee, that thou mayest be a multitude of people.’’ Although rather vague, as quoted from the King James version, a ‘‘multitude’’ in English is defined as ‘‘a great number of people’’ (Merriam-Webster Dictionary) that eventually would be spread over the earth and would not be an ever-increasing population. The number at which a population reaches and remains sustainable is referred to as the ‘‘carrying capacity.’’
Recognition of carrying capacity probably occurred long before written history began. It is likely that the earliest agriculturists, perhaps 10,000 years ago, were keenly aware of the number of mouths that an area could sustainably feed and that increasing numbers of people required increases in food production (i.e., area of land in cultivation). Long before agriculture, huntergatherer groups likely were aware of the sustainable number of members that regions could support, although it may be argued that high mortality rates inhibited these early populations from pushing the limits of sustainability. During difficult times populations reached or exceeded what we might think of as a carrying capacity, possibly imposing nomadic lifestyles in which groups had to intermittently move after local resources were depleted. Speculating on past population dynamics hints at the trouble with the concept of carrying capacity: It must represent a dynamic value that changes over time and is highly dependent on many interacting factors, such as environmental variability and, for early humans, what hungry animals awaited their forays. This makes the concept difficult to use. Charles Darwin, on page 116 in his 1859 book Origin of Species, quoted in his third chapter on the struggle of existence from Malthus’ 1798 ‘‘An Essay on the Principle of Population as It Affects the Future Improvement of Society’’:
‘‘A struggle for existence inevitably follows from the high rate at which all organic beings tend to increase. Every being, which during its natural lifetime produces several eggs or seeds, must suffer destruction during some period of its life, and during some season or occasional year, otherwise, on the principle of geometrical increase, its numbers would quickly become so inordinately great that no country could support the product. Hence, as more individuals are produced than can possibly survive, there must in every case be a struggle for existence, either one individual with another of the same species, or with the individuals of distinct species, or with the physical conditions of life. It is the doctrine of Malthus applied with manifold force to the whole animal and vegetable kingdoms; for in this case there can be no artificial increase of food, and no prudential restraint from marriage. Although some species may be now increasing, more or less rapidly, in numbers, all cannot do so, for the world would not hold them.’’
CARRYING CAPACITY, CONCEPT OF
Malthus’ clear recognition of the importance of limitations of growth in populations helped Darwin to lay the foundation for his theory of natural selection, which is built on the premise that populations are regulated primarily by competition which leads to differential reproduction. The earliest concise description of carrying capacity derives from Pierre Francois Verhulst, a Belgian who lived in the mid-nineteenth century. Verhulst, perplexed by accounts that the human population appeared to be increasing exponentially, derived a mathematical formula which he called the ‘‘logistic’’ equation that would account for a slowing in the population growth rate as a function of population size. The same relationship was rediscovered in 1920 by Raymond Pearl and Lowell Reed, who used the logistic equation to predict the population of the United States based on census data collected from 1790 to 1910 (apparently neglecting the fact that the area of the United States increased more than three-fold during this time). The resulting application of the logistic equation to U.S. census data led Pearl and Reed to greatly underestimate the U.S. population, predicting it would level off at about 197 million in the Year 2050 (Fig. 1). The failure of Pearl and Reed to accurately predict the population of the United States, currently at about 275 million, reveals at least one important aspect of mathematical models. It indicates that, for the U.S. hu-
FIGURE 1 Data on the growth of the U.S. population from 1790 to 1910 (䊉) and from 1920 to 1990 (䊊). The dotted line represents Pearl and Reeds’ fit of the logistic equation, yielding a carrying capacity (K) of about 197 million people, estimated to be reached in about the Year 2050. The population in the Year 2000 is about 275 million (not shown).
643
man population, some aspect of the logistic equation must be wrong. Two candidate problems include violations of the assumptions that the area of the United States remained constant and that the U.S. population lacked immigration. These differences between data and model predictions can help us to better understand the problem at hand. In this example, the violations of model assumptions leads us to re-evaluate the factors that influence population growth and ultimately regulate it. The prediction from the model also clearly did not hold for a variety of socioeconomic and human health reasons and may have been wrong for statistical reasons. Using the logistic equation (based on the filled data points in Fig. 1) to predict population size well into the future, without providing confidence limits, is clearly tenuous at best. These problems have led us to be more careful in our predictions of how populations change over time and whether a carrying capacity can be predicted from such data or even exists.
III. DEFINITIONS OF CARRYING CAPACITY Carrying capacity is the maximum population that a given area can sustain. The measures commonly used include the number of individuals or the total biomass of a population, which are each highly dependent on differences in physiology and age structure among species and across large taxonomic groups. The use of the term carrying capacity has changed over time, but most models suggest that population growth is rapid when density is low and decreases as populations increase toward some maximum. In addition, any definition of this concept improves as we narrow the time and area for the population that we are studying. Population descriptions, therefore, are often depicted as densities, accounting for the number of individuals per unit area. Population density usually varies over time and from place to place. In practice, we generally use population size or density to describe carrying capacity, which is determined either by resource availability or by the influence of enemies (predators and/or pathogens). Various definitions of carrying capacity arose in the twentieth century, ranging from the suggestion that carrying capacity is that level below which predators have no effect on a population to the population size which can be maximally supported in a given region (previously referred to as the ‘‘saturation level’’). There also has been a distinction made between ‘‘ecological
644
CARRYING CAPACITY, CONCEPT OF
carrying capacity,’’ which refers to the limitation of a population due to resources, and a management-oriented, maximum sustainable yield for a population, referred to as an ‘‘economic carrying capacity,’’ which is usually lower than ecological carrying capacity. These definitions clearly lead to difficulty for wildlife managers who have been preoccupied with attempting to determine whether populations are either too high or too low. These debates continue, as exemplified by range management decisions in Yellowstone National Park and issues regarding the increasing frequency of reintroduction programs of top predators. Carrying capacity may best be expressed mathematically. One of the simplest forms of population change over time can be represented as the differential equation dN/dt ⫽ rN, where dN/dt represents the instantaneous change in a population over a short time period, r is the intrinsic growth rate of the population, and N is the size of the population. This yields what is often referred to as a ‘‘J’’ curve, or exponential growth (Fig. 2). In discrete time this relationship is referred to as geometric growth. In 1838, Verhulst modified the exponential growth equation and derived the logistic equation that depicted population growth rate as being inversely related to population size. To slow population growth he added an additional term yielding dN/dt ⫽ rN(1 ⫺ N/K), where K is the population carrying capacity. The term ‘‘1 ⫺ N/K’’ slows growth rate linearly toward zero as the population (N) approaches the carrying capacity (K). This results in a sigmoidal S-shaped curve for an increasing population over time (Fig. 2). If the popula-
tion exceeds K (N ⬎ K), then 1 ⫺ N/K is negative, causing growth rate dN/dt to be negative and the population to decline monotonically toward K. An important attribute to bear in mind is that the logistic equation is deterministic, meaning that if we use the equation to predict population size at the end of a fixed amount of time we will derive the same population each time we start the population over. This assumption is usually violated in field conditions in which random effects, such as accidental deaths, failure to find mates, or fluctuations in environmental conditions, are common. Therefore, it has been argued that we should not expect real populations to behave according to the logistic equation. This simple equation has been challenged repeatedly by critics without apparent damage. This resilience of a theory is rather rare in science, which is a discipline that prides itself on being able to quickly dispel hypotheses (or equations) given even a small amount of contradictory data. However, the intuitive nature of the idea that populations are regulated by factors such as food supply helps the logistic equation to remain a staple in ecological texts and classrooms. The reason this equation and carrying capacity (K) endure is that the equation’s shortcomings help us better understand the dynamics of real populations, ensuring its utility for many years to come. The discrete, or difference, form of the logistic equation yields a different prediction of population behavior compared to the previously described continuous version. In particular, the discrete form was the equation used by Sir Robert May to first describe how a simple, deterministic equation could produce chaotic population dynamics, a pattern that emerges when intrinsic growth is relatively high. This chaotic behavior appears to mimic realistic changes in populations over time. Several long-term data records conform to chaotic dynamics, including the change in the number of lynx captured over time in Canada (Fig. 3).
IV. DO POPULATIONS HAVE CARRYING CAPACITIES?
FIGURE 2 Comparison of unregulated exponential growth (solid line) with regulated logistic growth (dotted line).
This question has been addressed using a variety of techniques, including observational types of studies that rely on long-term time series data sets such as the number of lynx captured over time (Fig. 3), highly controlled laboratory experiments (Fig. 4), and mathematical models to determine potential mechanisms through time series reconstruction. The short answer
CARRYING CAPACITY, CONCEPT OF
FIGURE 3 The number of lynx trapped in the Mackenzie River district from 1821 to 1934 (after Elton and Nicholson, 1942).
is that there likely are carrying capacities for most species but that determining these at any one point in time and space is quite difficult. Many controlled laboratory experiments have been published that show populations behaving in a fashion consistent with the logistic model (i.e., populations reach a carrying capacity). One of the earliest studies was completed by Gause in 1934 (Fig. 4). Most laboratory populations tend to increase and then reach some level at which they fluctuate around what might represent a carrying capacity. It is interesting to note that populations rarely exhibit a smooth transition between a growth phase and gradual reductions in growth rate
645
ending in stable populations, despite the controlled environmental conditions. An additional caveat to consider is that prior studies that concluded population growth patterns differed significantly from the expected logistic growth likely suffered disproportionately during the review process and failed to make it into print. Determining how natural populations change over time is surprisingly difficult. The first requisite information necessary to determine whether a population is at or near a constant size, considered here to be a proxy for the habitat’s carrying capacity, is simply the population’s size over time. This often has to be determined over long periods of time in ways that are accurate, reliable, and repeatable. In field studies, it is rare to have the luxury to repeatedly estimate population size, a technique allowing us to assess the accuracy of our estimates. Determining that a population fluctuates may represent real changes in populations or represent either natural variability (statistical ‘‘errors’’) or actual errors in our estimates. Assuming we overlook these shortcomings in our data, what do populations do? In general, populations usually fluctuate over time. We may be able to correlate these changes with biotic or abiotic factors or some function of the two with time. Sometimes the fluctuations cannot be distinguished from random noise. Some populations, including the classic examples of lynx, hares, and lemmings, cycle periodically (Fig. 3). The persistence of population cycles over long periods of time has led to great speculation regarding the factors that might lead to periodicity. Recent work suggests that simple causal mechanisms of cycling are unlikely and that a combination of random environmental factors and nonlinear, density-dependent factors influence populations.
V. DETERMINING THE CARRYING CAPACITY OF A POPULATION
FIGURE 4 The change in the density of Paramecium caudatum over time in the laboratory. The dotted line represents the best fit logistic equation (after Gause, 1934).
Determining the carrying capacity of any particular population at a particular time is not trivial. Many different techniques have been suggested and tested, including three primary techniques that can be used to attempt to detect a change in population growth rate as a function of population size: mathematical modeling of specific mechanisms, tactical experimental tests in the laboratory and field, and statistical analysis of time series data. Ultimately, a combination of these techniques will enable us to understand the importance
646
CARRYING CAPACITY, CONCEPT OF
of regulation in populations and the degree to which populations appear to be governed by a carrying capacity. I have already discussed the importance of the logistic equation and will briefly introduce the empirical approaches. There has been much interest in and work completed to determine what factors regulate the change in growth of populations over time. The factors that slow population growth rate change over time and from location to location and differ for different species. Regulating factors also are likely to interact with each other, thus complicating the determination of a population’s carrying capacity. In a classic study, Davidson and Andrewartha in 1948 used a partial regression technique to analyze an experiment designed to test the relative influence of biotic and abiotic factors on regulating a small herbivorous insect population. They concluded that 78% of the population variance was due to abiotic or weather-related factors. In particular, the number of individual thrips in the spring was related mostly to the preceding autumn climate. This study was influential because it provided strong evidence that this population of thrips was regulated not by biotic factors such as competition or predation but rather by abiotic factors. A second method used to detect the presence of density dependence on population regulation is the analysis of time series data. The best data are those that have been collected over consecutive years and that exceed the periodicity of both observed environmental and population fluctuations (generally ⬎10 years). These data can be subjected to tests that investigate the relationship of change from year to year as a function of the population during the previous year or years in order to detect whether the population appears to be regulated. Such analyses, however, are unable to provide any information on the underlying mechanisms that might lead to population regulation. Therefore, time series analysis is an excellent exploratory tool that can be used to investigate the possibility that a population is regulated. This information can then suggest experiments designed to partition variance among potential candidate mechanisms. Determination of a population’s carrying capacity is best done through a combination of modeling, experimentation, and time series analysis. Research efforts, however, need to be directed toward investigating the underlying mechanisms that govern population regulation. Without an understanding of the relative importance of these regulating factors, it will be diffi-
cult to determine whether populations are regulated and whether we can detect a population’s carrying capacity.
VI. CURRENT RESEARCH ON CARRYING CAPACITY Two main areas of research continue to drive our quest to understand population regulation and the strength and importance of carrying capacity. The persistence of these questions indicates the need to clarify the mechanisms that influence population change over time.
A. Determining the Relative Strengths of Factors That Regulate Populations Although some researchers have argued that populations are unregulated, most agree that negative feedback mechanisms operate on populations, resulting in decreased growth at high densities. This may occur through changes in the abundance of food, through increased predation or disease, or through a combination of these biotic factors and abiotic factors such as local climate. These factors may reduce birth rates or increase death rates, or both. Although there are circumstances in which these rates change at low population densities (e.g., the Allee effect, which states that very small populations are likely to decrease due to such factors as difficulty in finding mates or pollen limitation), their regulation at high densities is likely to be common. This change in birth and death rates as a function of density is referred to as ‘‘density dependence.’’ A population that is regulated has intrinsic, extrinsic, or a combination of these factors that slows population growth. Under such conditions a population’s per capita growth rate decreases with increasing population size through reduced birth rates and/or increased death rates. This relationship, in logistic growth, is assumed to be linear. The existence of a carrying capacity, however, is not dependent on the shape of this function, so the violation of this linearity assumption does not weaken the concept of carrying capacity. A better understanding of this relationship, generally determined through carefully designed experiments, will help us understand the importance of regulation on population dynamics.
CARRYING CAPACITY, CONCEPT OF
Krebs et al. (1995) suggested that hare and lynx cyclic population dynamics are likely influenced by different sets of factors, including food availability and predation driving the dynamics of hare populations and the lynx population is driven primarily by changes in the number of hares. In a more highly controlled experiment using three trophic levels, Hartvigsen et al. (1995) determined that plant performance was controlled by the interaction of top-down and bottom-up factors, including the level of plant resource availability and the presence or absence of herbivores and herbivore predators. These studies suggest that complex, interacting biotic and abiotic factors likely influence population dynamics.
B. Determining Population Carrying Capacity The logistic growth equation attempts to model regulated population change over time and relies on several important assumptions, including the absence of time lags (population dynamics is independent of prior events), migration or immigration, genetic variability or selection, population age structure, and the fact that density dependence is linear (each individual added to the population has a similar effect on the population’s per capita growth rate). Violations of these assumptions have been found in various populations and have led to more refined, realistic, and complicated forms of the logistic equation. In addition, the model assumes that carrying capacity (K) is constant over time and space. This assumption occasionally may be valid in situations in which a population is regulated by habitat availability. This might occur, for example, where the number or area of nesting sites is fixed. It is easy to conjure up situations, however, when this assumption would be violated over very small spatial or temporal scales. It is not likely that K would be constant since populations are usually limited by resources, competitors, enemies, and often combinations of these factors that vary with the environment over time. Under these conditions changes in resource availability can influence population size directly or indirectly through its often nonlinear effect on the population of competitors and/or predators. In addition, there is great concern about the stability and persistence of threatened and endangered species (see Section VII). Work in this area has begun to recognize the importance of species interactions, immigration and emigration among subpopulations, the intro-
647
duction of exotic species, and other factors that violate the assumptions of simple logistic growth. The movement of individuals among subpopulations enables the possibility of increased long-term persistence of populations by reducing large-scale fluctuations and the spreading of risk that a species will become extinct in the event that a single local population disappears (becomes extirpated). This area of research, referred to as ‘‘metapopulation biology,’’ involves determining long-term viability of these subdivided species, and there is currently much research activity in this area.
VII. THE IMPORTANCE OF THE CONCEPT OF CARRYING CAPACITY TO BIODIVERSITY The concept of carrying capacity suggests that species are likely to have some upper limit to their population. If the upper limit is ‘‘hard,’’ then we expect populations to achieve this state and remain relatively constant. Populations, however, as demonstrated in Figs. 3 and 4, do not behave in such a simple fashion and have rather ‘‘soft’’ limits. As such, populations usually exhibit random, cyclic, or chaotic dynamics. These dynamics generally lead to increased chances that populations will reach the absorbing state of zero (become either locally extirpated or globally extinct). We must be concerned about the dynamics of relatively small populations over time. The probability that a population will go extinct is generally related to the degree to which it fluctuates (population amplitude and frequency). Therefore, processes that cause populations to increase fluctuations are likely to lead to species loss and associated reductions in biodiversity. Thus, conservation efforts may be needed that will buffer populations and associated habitats from extreme fluctuations. Conservation efforts are often directed toward increasing a population’s carrying capacity. It should be kept in mind, however, that constant environments also may lead to species losses. The intermediate disturbance hypothesis has gained much empirical support and suggests that the maximum number of species that an area can support occurs when disturbances are intermediate in either frequency or impact. We should be concerned that our management efforts do not reduce the carrying capacity of target species.
648
CARRYING CAPACITY, CONCEPT OF
VIII. THE HUMAN CARRYING CAPACITY The best estimates of human population indicate that it has continued to grow exponentially over recorded history, although the actual growth rate has changed over time. Attempts to fit data on the human population to the logistic equation have failed (Fig. 1), and current indications are that no human carrying capacity can be predicted from simple population statistics. However, we might ask whether our population growth rate is likely to slow down in the foreseeable future and, ultimately, reach a stable carrying capacity or whether it will overshoot its carrying capacity and eventually collapse. Joel Cohen (1995) found that estimates of the human carrying capacity have ranged between 1 billion and 1 trillion people, with the majority of estimates falling between 4 and 16 billion (the current population is about 6 billion). These estimates suggest that we are approaching an apparent limit for our species. Regardless of which estimate seems most appropriate as an upper limit for humans on Earth, the growth rate of our population will eventually slow to ⱕ0. This can occur as a result of increasing death rates and/or decreasing birth rates. I predict that as our population grows in the coming decades there will be an increase in mortality due to diseases. The effect of disease agents on controlling population growth will likely increase due to increases in human contact rates and rapid transit, increasing evolution of drug resistance, and increasing virulence rates. These factors also may reduce birth rate, which of course presents a more pleasant alternative to slow population growth. Can we avoid a population crash? I venture the guess that we cannot. Any long-term stabilization of the human population will require a decrease in the current global birth rate. We certainly cannot hope to achieve a relatively stable population without invoking a substantially higher death rate than the current rate, which is not a comforting thought. It is difficult to imagine, however, that the influence of disease will operate in a simple density-dependent fashion. Instead, it seems more plausible that diseases will ‘‘break out’’ more often with increasing population size and with larger scale consequences, bringing about a strong reduction in our population—a response seen in many other populations that have increased beyond their carrying capacities. One last area of hope is that individuals will lower their consumption rates, thereby adjusting the human
carrying capacity. It is unlikely that Earth can support tens of billions of people with lifestyles matching those of people in the developed nations such as the United States. Therefore, there remains a chance that changes in human behavior will allow our population to gently transition toward a sustainable, zero growth rate population.
IX. CONCLUSIONS The concept of carrying capacity has a history that spans at least thousands of years. The formal definition is about 150 years old and is generally coupled to the asymptotic population in the logistic growth equation (K). Critics argue that because of ongoing confusion and the multitude of definitions attached to the concept we would be better off to simply abandon the term. We also must be concerned that the term not be used to advance any particular political agenda associated with determining how large populations of any particular species ‘‘should’’ be in particular areas. This entry cautiously suggests that the concept remains useful. Since most populations are likely to be at least occasionally limited by factors that depend on the population’s density, we need to continue advancing our knowledge of how populations behave and use this information to guide the design of laboratory and field experiments aimed at determining the mechanisms that regulate populations. Only by using the combination of field and laboratory techniques, grounded in a theoretical framework that has roots going back to the simple logistic equation, will we hope to understand and conserve populations, including our own, over long periods of time.
See Also the Following Articles POPULATION DENSITY • POPULATION DYNAMICS • POPULATION STABILIZATION (HUMAN) • POPULATION VIABILITY ANALYSIS (PVA) • SUSTAINABILITY, CONCEPT AND PRACTICE OF
Bibliography Caughley, G. (1979). What is this thing called carrying capacity? In North American Elk: Ecology, Behavior and Management (M. S. Boyce and L. D. Hayden-Wing, Eds.). The University of Wyoming, Laramie.
CARRYING CAPACITY, CONCEPT OF Cohen, J. E. (1995). How Many People Can the Earth Support? Norton, New York. Darwin, C. (1859). On the Origin of Species by Means of Natural Selection. Murray, London. Davidson, J., and Andrewartha, H. G.(1948). The influence of rainfall, evaporation, and atmospheric temperature on fluctuations in the size of a natural population of Thrips imaginis (Thysanoptera). J. Anim. Ecol. 17, 200–222. Dhondt, A. A. (1988). Carrying capacity: A confusing concept. Acta Oecol. 9, 337–346. Elton, C., and Nicholson, M. (1942). The ten-year cycle in numbers of lynx in Canada. J. Anim. Ecol. 11, 215–244. Gause, G. F. (1934). The Struggle for Existence. Macmillan, New York.
649
Hartvigsen, G., Wait, D. A., and Coleman, J. S. (1995). Tri-trophic interactions influenced by resource availability: Predator effects on plant performance depend on plant resources. Oikos 74, 463– 468. Krebs, C. J., Boutin, S., Boonstra, R., Sinclair, A. R. E., Smith, J. N. M., Dale, M. R. T., Martin, K., and Turkington, R. (1995). Impact of food and predation on the snowshoe hare cycle. Science 269, 1112–1115. Pearl, R., and Reed, L. J. (1920). On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proc. Natl. Acad. Sci. USA 6, 275–288. Turchin, P. (1999). Population regulation: A synthetic view. Oikos 84, 153–159.
I. II. III. IV. V. VI. VII. VIII. IX.
Introduction Origin of the Concept of Carrying Capacity Definitions of Carrying Capacity Do Populations Have Carrying Capacities? Determining the Carrying Capacity of a Population Current Research on Carrying Capacity The Importance of the Concept of Carrying Capacity to Biodiversity The Human Carrying Capacity Conclusions
GLOSSARY density dependence The condition that environmental factors influence population growth rate in relation to population size. Density dependence usually is seen as an linear, inverse relationship between population growth rate and population density (i.e., population growth decreases as density increases) and may occur if individuals compete or predators are more effective as a prey population increases. density independence The absence of environmental factors that influence population growth as a function of density. This may occur if mortality removes a fixed percentage of a population, independent of population size. logistic growth Regulated population growth that follows the logistic equation dN/dt ⫽ rN(1 ⫺ N/K). Populations growing according to this equation in-
crease rapidly at low densities and the growth rate decreases as they approach carrying capacity (K). population A group of individuals of a particular species that live in a region. A population is usually a subset of the entire species. population regulation The constraint of positive population growth. The study of population regulation deals with the factors that cause this constraint, such as competition for food or predation. population stability The tendency for populations to return to a previous size after a disturbance, such as reductions due to hunting or disease or increases due to immigration. Stable populations may be locally stable (return after small disturbances) or globally stable (return after severe or catastrophic disturbances).
CARRYING CAPACITY is the maximum number, density, or biomass of a population that a specific area can support. This number is likely to change over time and depends on changes in environmental factors (e.g., rainfall and temperature), resources (e.g., food, hiding places, and nesting sites), and the presence of predators, disease agents, and competitors over time. The concept of carrying capacity has been explicitly recognized for approximately 150 years and its use has waxed and waned during this time. Currently, the use of carrying capacity to describe any particular population is made only with great caution, although the concept remains
Encyclopedia of Biodiversity, Volume 1 Copyright 2001 by Academic Press. All rights of reproduction in any form reserved.
641
642
CARRYING CAPACITY, CONCEPT OF
intuitive and fosters questions that address our fundamental understanding of what factors regulated populations over time and space.
I. INTRODUCTION Populations, or groups of individuals within a species, change over time. There is general agreement among ecologists that population growth is bounded by biotic and abiotic environmental factors that result in approximate, maximum numbers of organisms that can be supported in different habitats. A population’s carrying capacity is difficult to measure and likely varies over time and through space. The concept of carrying capacity has played an important role in the fields of basic ecological research, wildlife management, and conservation biology. The concept of carrying capacity also involves determining how many people Earth can support. Dialogues about human carrying capacity are often quite contentious and illustrate the difficulty surrounding the concept of carrying capacity. The concept of carrying capacity is alive and well, although some have argued that it should be abandoned altogether. On the surface, the concept is easy to understand and intuitive and, therefore, is likely to stay with us for some time. However, recent developments in our understanding of the dynamics of population change over time have greatly modified what is considered an area’s carrying capacity. Therefore, we need to recognize both the strengths and the weaknesses of the concept of carrying capacity.
II. ORIGIN OF THE CONCEPT OF CARRYING CAPACITY Humans have long been aware of the limitations of their own population growth. As early as the Old Testament one can argue that the concept of population limitation was recognized. In the book of Genesis (28:3) are the following words: ‘‘And God Almighty bless thee, and make thee fruitful, and multiply thee, that thou mayest be a multitude of people.’’ Although rather vague, as quoted from the King James version, a ‘‘multitude’’ in English is defined as ‘‘a great number of people’’ (Merriam-Webster Dictionary) that eventually would be spread over the earth and would not be an ever-increasing population. The number at which a population reaches and remains sustainable is referred to as the ‘‘carrying capacity.’’
Recognition of carrying capacity probably occurred long before written history began. It is likely that the earliest agriculturists, perhaps 10,000 years ago, were keenly aware of the number of mouths that an area could sustainably feed and that increasing numbers of people required increases in food production (i.e., area of land in cultivation). Long before agriculture, huntergatherer groups likely were aware of the sustainable number of members that regions could support, although it may be argued that high mortality rates inhibited these early populations from pushing the limits of sustainability. During difficult times populations reached or exceeded what we might think of as a carrying capacity, possibly imposing nomadic lifestyles in which groups had to intermittently move after local resources were depleted. Speculating on past population dynamics hints at the trouble with the concept of carrying capacity: It must represent a dynamic value that changes over time and is highly dependent on many interacting factors, such as environmental variability and, for early humans, what hungry animals awaited their forays. This makes the concept difficult to use. Charles Darwin, on page 116 in his 1859 book Origin of Species, quoted in his third chapter on the struggle of existence from Malthus’ 1798 ‘‘An Essay on the Principle of Population as It Affects the Future Improvement of Society’’:
‘‘A struggle for existence inevitably follows from the high rate at which all organic beings tend to increase. Every being, which during its natural lifetime produces several eggs or seeds, must suffer destruction during some period of its life, and during some season or occasional year, otherwise, on the principle of geometrical increase, its numbers would quickly become so inordinately great that no country could support the product. Hence, as more individuals are produced than can possibly survive, there must in every case be a struggle for existence, either one individual with another of the same species, or with the individuals of distinct species, or with the physical conditions of life. It is the doctrine of Malthus applied with manifold force to the whole animal and vegetable kingdoms; for in this case there can be no artificial increase of food, and no prudential restraint from marriage. Although some species may be now increasing, more or less rapidly, in numbers, all cannot do so, for the world would not hold them.’’
CARRYING CAPACITY, CONCEPT OF
Malthus’ clear recognition of the importance of limitations of growth in populations helped Darwin to lay the foundation for his theory of natural selection, which is built on the premise that populations are regulated primarily by competition which leads to differential reproduction. The earliest concise description of carrying capacity derives from Pierre Francois Verhulst, a Belgian who lived in the mid-nineteenth century. Verhulst, perplexed by accounts that the human population appeared to be increasing exponentially, derived a mathematical formula which he called the ‘‘logistic’’ equation that would account for a slowing in the population growth rate as a function of population size. The same relationship was rediscovered in 1920 by Raymond Pearl and Lowell Reed, who used the logistic equation to predict the population of the United States based on census data collected from 1790 to 1910 (apparently neglecting the fact that the area of the United States increased more than three-fold during this time). The resulting application of the logistic equation to U.S. census data led Pearl and Reed to greatly underestimate the U.S. population, predicting it would level off at about 197 million in the Year 2050 (Fig. 1). The failure of Pearl and Reed to accurately predict the population of the United States, currently at about 275 million, reveals at least one important aspect of mathematical models. It indicates that, for the U.S. hu-
FIGURE 1 Data on the growth of the U.S. population from 1790 to 1910 (䊉) and from 1920 to 1990 (䊊). The dotted line represents Pearl and Reeds’ fit of the logistic equation, yielding a carrying capacity (K) of about 197 million people, estimated to be reached in about the Year 2050. The population in the Year 2000 is about 275 million (not shown).
643
man population, some aspect of the logistic equation must be wrong. Two candidate problems include violations of the assumptions that the area of the United States remained constant and that the U.S. population lacked immigration. These differences between data and model predictions can help us to better understand the problem at hand. In this example, the violations of model assumptions leads us to re-evaluate the factors that influence population growth and ultimately regulate it. The prediction from the model also clearly did not hold for a variety of socioeconomic and human health reasons and may have been wrong for statistical reasons. Using the logistic equation (based on the filled data points in Fig. 1) to predict population size well into the future, without providing confidence limits, is clearly tenuous at best. These problems have led us to be more careful in our predictions of how populations change over time and whether a carrying capacity can be predicted from such data or even exists.
III. DEFINITIONS OF CARRYING CAPACITY Carrying capacity is the maximum population that a given area can sustain. The measures commonly used include the number of individuals or the total biomass of a population, which are each highly dependent on differences in physiology and age structure among species and across large taxonomic groups. The use of the term carrying capacity has changed over time, but most models suggest that population growth is rapid when density is low and decreases as populations increase toward some maximum. In addition, any definition of this concept improves as we narrow the time and area for the population that we are studying. Population descriptions, therefore, are often depicted as densities, accounting for the number of individuals per unit area. Population density usually varies over time and from place to place. In practice, we generally use population size or density to describe carrying capacity, which is determined either by resource availability or by the influence of enemies (predators and/or pathogens). Various definitions of carrying capacity arose in the twentieth century, ranging from the suggestion that carrying capacity is that level below which predators have no effect on a population to the population size which can be maximally supported in a given region (previously referred to as the ‘‘saturation level’’). There also has been a distinction made between ‘‘ecological
644
CARRYING CAPACITY, CONCEPT OF
carrying capacity,’’ which refers to the limitation of a population due to resources, and a management-oriented, maximum sustainable yield for a population, referred to as an ‘‘economic carrying capacity,’’ which is usually lower than ecological carrying capacity. These definitions clearly lead to difficulty for wildlife managers who have been preoccupied with attempting to determine whether populations are either too high or too low. These debates continue, as exemplified by range management decisions in Yellowstone National Park and issues regarding the increasing frequency of reintroduction programs of top predators. Carrying capacity may best be expressed mathematically. One of the simplest forms of population change over time can be represented as the differential equation dN/dt ⫽ rN, where dN/dt represents the instantaneous change in a population over a short time period, r is the intrinsic growth rate of the population, and N is the size of the population. This yields what is often referred to as a ‘‘J’’ curve, or exponential growth (Fig. 2). In discrete time this relationship is referred to as geometric growth. In 1838, Verhulst modified the exponential growth equation and derived the logistic equation that depicted population growth rate as being inversely related to population size. To slow population growth he added an additional term yielding dN/dt ⫽ rN(1 ⫺ N/K), where K is the population carrying capacity. The term ‘‘1 ⫺ N/K’’ slows growth rate linearly toward zero as the population (N) approaches the carrying capacity (K). This results in a sigmoidal S-shaped curve for an increasing population over time (Fig. 2). If the popula-
tion exceeds K (N ⬎ K), then 1 ⫺ N/K is negative, causing growth rate dN/dt to be negative and the population to decline monotonically toward K. An important attribute to bear in mind is that the logistic equation is deterministic, meaning that if we use the equation to predict population size at the end of a fixed amount of time we will derive the same population each time we start the population over. This assumption is usually violated in field conditions in which random effects, such as accidental deaths, failure to find mates, or fluctuations in environmental conditions, are common. Therefore, it has been argued that we should not expect real populations to behave according to the logistic equation. This simple equation has been challenged repeatedly by critics without apparent damage. This resilience of a theory is rather rare in science, which is a discipline that prides itself on being able to quickly dispel hypotheses (or equations) given even a small amount of contradictory data. However, the intuitive nature of the idea that populations are regulated by factors such as food supply helps the logistic equation to remain a staple in ecological texts and classrooms. The reason this equation and carrying capacity (K) endure is that the equation’s shortcomings help us better understand the dynamics of real populations, ensuring its utility for many years to come. The discrete, or difference, form of the logistic equation yields a different prediction of population behavior compared to the previously described continuous version. In particular, the discrete form was the equation used by Sir Robert May to first describe how a simple, deterministic equation could produce chaotic population dynamics, a pattern that emerges when intrinsic growth is relatively high. This chaotic behavior appears to mimic realistic changes in populations over time. Several long-term data records conform to chaotic dynamics, including the change in the number of lynx captured over time in Canada (Fig. 3).
IV. DO POPULATIONS HAVE CARRYING CAPACITIES?
FIGURE 2 Comparison of unregulated exponential growth (solid line) with regulated logistic growth (dotted line).
This question has been addressed using a variety of techniques, including observational types of studies that rely on long-term time series data sets such as the number of lynx captured over time (Fig. 3), highly controlled laboratory experiments (Fig. 4), and mathematical models to determine potential mechanisms through time series reconstruction. The short answer
CARRYING CAPACITY, CONCEPT OF
FIGURE 3 The number of lynx trapped in the Mackenzie River district from 1821 to 1934 (after Elton and Nicholson, 1942).
is that there likely are carrying capacities for most species but that determining these at any one point in time and space is quite difficult. Many controlled laboratory experiments have been published that show populations behaving in a fashion consistent with the logistic model (i.e., populations reach a carrying capacity). One of the earliest studies was completed by Gause in 1934 (Fig. 4). Most laboratory populations tend to increase and then reach some level at which they fluctuate around what might represent a carrying capacity. It is interesting to note that populations rarely exhibit a smooth transition between a growth phase and gradual reductions in growth rate
645
ending in stable populations, despite the controlled environmental conditions. An additional caveat to consider is that prior studies that concluded population growth patterns differed significantly from the expected logistic growth likely suffered disproportionately during the review process and failed to make it into print. Determining how natural populations change over time is surprisingly difficult. The first requisite information necessary to determine whether a population is at or near a constant size, considered here to be a proxy for the habitat’s carrying capacity, is simply the population’s size over time. This often has to be determined over long periods of time in ways that are accurate, reliable, and repeatable. In field studies, it is rare to have the luxury to repeatedly estimate population size, a technique allowing us to assess the accuracy of our estimates. Determining that a population fluctuates may represent real changes in populations or represent either natural variability (statistical ‘‘errors’’) or actual errors in our estimates. Assuming we overlook these shortcomings in our data, what do populations do? In general, populations usually fluctuate over time. We may be able to correlate these changes with biotic or abiotic factors or some function of the two with time. Sometimes the fluctuations cannot be distinguished from random noise. Some populations, including the classic examples of lynx, hares, and lemmings, cycle periodically (Fig. 3). The persistence of population cycles over long periods of time has led to great speculation regarding the factors that might lead to periodicity. Recent work suggests that simple causal mechanisms of cycling are unlikely and that a combination of random environmental factors and nonlinear, density-dependent factors influence populations.
V. DETERMINING THE CARRYING CAPACITY OF A POPULATION
FIGURE 4 The change in the density of Paramecium caudatum over time in the laboratory. The dotted line represents the best fit logistic equation (after Gause, 1934).
Determining the carrying capacity of any particular population at a particular time is not trivial. Many different techniques have been suggested and tested, including three primary techniques that can be used to attempt to detect a change in population growth rate as a function of population size: mathematical modeling of specific mechanisms, tactical experimental tests in the laboratory and field, and statistical analysis of time series data. Ultimately, a combination of these techniques will enable us to understand the importance
646
CARRYING CAPACITY, CONCEPT OF
of regulation in populations and the degree to which populations appear to be governed by a carrying capacity. I have already discussed the importance of the logistic equation and will briefly introduce the empirical approaches. There has been much interest in and work completed to determine what factors regulate the change in growth of populations over time. The factors that slow population growth rate change over time and from location to location and differ for different species. Regulating factors also are likely to interact with each other, thus complicating the determination of a population’s carrying capacity. In a classic study, Davidson and Andrewartha in 1948 used a partial regression technique to analyze an experiment designed to test the relative influence of biotic and abiotic factors on regulating a small herbivorous insect population. They concluded that 78% of the population variance was due to abiotic or weather-related factors. In particular, the number of individual thrips in the spring was related mostly to the preceding autumn climate. This study was influential because it provided strong evidence that this population of thrips was regulated not by biotic factors such as competition or predation but rather by abiotic factors. A second method used to detect the presence of density dependence on population regulation is the analysis of time series data. The best data are those that have been collected over consecutive years and that exceed the periodicity of both observed environmental and population fluctuations (generally ⬎10 years). These data can be subjected to tests that investigate the relationship of change from year to year as a function of the population during the previous year or years in order to detect whether the population appears to be regulated. Such analyses, however, are unable to provide any information on the underlying mechanisms that might lead to population regulation. Therefore, time series analysis is an excellent exploratory tool that can be used to investigate the possibility that a population is regulated. This information can then suggest experiments designed to partition variance among potential candidate mechanisms. Determination of a population’s carrying capacity is best done through a combination of modeling, experimentation, and time series analysis. Research efforts, however, need to be directed toward investigating the underlying mechanisms that govern population regulation. Without an understanding of the relative importance of these regulating factors, it will be diffi-
cult to determine whether populations are regulated and whether we can detect a population’s carrying capacity.
VI. CURRENT RESEARCH ON CARRYING CAPACITY Two main areas of research continue to drive our quest to understand population regulation and the strength and importance of carrying capacity. The persistence of these questions indicates the need to clarify the mechanisms that influence population change over time.
A. Determining the Relative Strengths of Factors That Regulate Populations Although some researchers have argued that populations are unregulated, most agree that negative feedback mechanisms operate on populations, resulting in decreased growth at high densities. This may occur through changes in the abundance of food, through increased predation or disease, or through a combination of these biotic factors and abiotic factors such as local climate. These factors may reduce birth rates or increase death rates, or both. Although there are circumstances in which these rates change at low population densities (e.g., the Allee effect, which states that very small populations are likely to decrease due to such factors as difficulty in finding mates or pollen limitation), their regulation at high densities is likely to be common. This change in birth and death rates as a function of density is referred to as ‘‘density dependence.’’ A population that is regulated has intrinsic, extrinsic, or a combination of these factors that slows population growth. Under such conditions a population’s per capita growth rate decreases with increasing population size through reduced birth rates and/or increased death rates. This relationship, in logistic growth, is assumed to be linear. The existence of a carrying capacity, however, is not dependent on the shape of this function, so the violation of this linearity assumption does not weaken the concept of carrying capacity. A better understanding of this relationship, generally determined through carefully designed experiments, will help us understand the importance of regulation on population dynamics.
CARRYING CAPACITY, CONCEPT OF
Krebs et al. (1995) suggested that hare and lynx cyclic population dynamics are likely influenced by different sets of factors, including food availability and predation driving the dynamics of hare populations and the lynx population is driven primarily by changes in the number of hares. In a more highly controlled experiment using three trophic levels, Hartvigsen et al. (1995) determined that plant performance was controlled by the interaction of top-down and bottom-up factors, including the level of plant resource availability and the presence or absence of herbivores and herbivore predators. These studies suggest that complex, interacting biotic and abiotic factors likely influence population dynamics.
B. Determining Population Carrying Capacity The logistic growth equation attempts to model regulated population change over time and relies on several important assumptions, including the absence of time lags (population dynamics is independent of prior events), migration or immigration, genetic variability or selection, population age structure, and the fact that density dependence is linear (each individual added to the population has a similar effect on the population’s per capita growth rate). Violations of these assumptions have been found in various populations and have led to more refined, realistic, and complicated forms of the logistic equation. In addition, the model assumes that carrying capacity (K) is constant over time and space. This assumption occasionally may be valid in situations in which a population is regulated by habitat availability. This might occur, for example, where the number or area of nesting sites is fixed. It is easy to conjure up situations, however, when this assumption would be violated over very small spatial or temporal scales. It is not likely that K would be constant since populations are usually limited by resources, competitors, enemies, and often combinations of these factors that vary with the environment over time. Under these conditions changes in resource availability can influence population size directly or indirectly through its often nonlinear effect on the population of competitors and/or predators. In addition, there is great concern about the stability and persistence of threatened and endangered species (see Section VII). Work in this area has begun to recognize the importance of species interactions, immigration and emigration among subpopulations, the intro-
647
duction of exotic species, and other factors that violate the assumptions of simple logistic growth. The movement of individuals among subpopulations enables the possibility of increased long-term persistence of populations by reducing large-scale fluctuations and the spreading of risk that a species will become extinct in the event that a single local population disappears (becomes extirpated). This area of research, referred to as ‘‘metapopulation biology,’’ involves determining long-term viability of these subdivided species, and there is currently much research activity in this area.
VII. THE IMPORTANCE OF THE CONCEPT OF CARRYING CAPACITY TO BIODIVERSITY The concept of carrying capacity suggests that species are likely to have some upper limit to their population. If the upper limit is ‘‘hard,’’ then we expect populations to achieve this state and remain relatively constant. Populations, however, as demonstrated in Figs. 3 and 4, do not behave in such a simple fashion and have rather ‘‘soft’’ limits. As such, populations usually exhibit random, cyclic, or chaotic dynamics. These dynamics generally lead to increased chances that populations will reach the absorbing state of zero (become either locally extirpated or globally extinct). We must be concerned about the dynamics of relatively small populations over time. The probability that a population will go extinct is generally related to the degree to which it fluctuates (population amplitude and frequency). Therefore, processes that cause populations to increase fluctuations are likely to lead to species loss and associated reductions in biodiversity. Thus, conservation efforts may be needed that will buffer populations and associated habitats from extreme fluctuations. Conservation efforts are often directed toward increasing a population’s carrying capacity. It should be kept in mind, however, that constant environments also may lead to species losses. The intermediate disturbance hypothesis has gained much empirical support and suggests that the maximum number of species that an area can support occurs when disturbances are intermediate in either frequency or impact. We should be concerned that our management efforts do not reduce the carrying capacity of target species.
648
CARRYING CAPACITY, CONCEPT OF
VIII. THE HUMAN CARRYING CAPACITY The best estimates of human population indicate that it has continued to grow exponentially over recorded history, although the actual growth rate has changed over time. Attempts to fit data on the human population to the logistic equation have failed (Fig. 1), and current indications are that no human carrying capacity can be predicted from simple population statistics. However, we might ask whether our population growth rate is likely to slow down in the foreseeable future and, ultimately, reach a stable carrying capacity or whether it will overshoot its carrying capacity and eventually collapse. Joel Cohen (1995) found that estimates of the human carrying capacity have ranged between 1 billion and 1 trillion people, with the majority of estimates falling between 4 and 16 billion (the current population is about 6 billion). These estimates suggest that we are approaching an apparent limit for our species. Regardless of which estimate seems most appropriate as an upper limit for humans on Earth, the growth rate of our population will eventually slow to ⱕ0. This can occur as a result of increasing death rates and/or decreasing birth rates. I predict that as our population grows in the coming decades there will be an increase in mortality due to diseases. The effect of disease agents on controlling population growth will likely increase due to increases in human contact rates and rapid transit, increasing evolution of drug resistance, and increasing virulence rates. These factors also may reduce birth rate, which of course presents a more pleasant alternative to slow population growth. Can we avoid a population crash? I venture the guess that we cannot. Any long-term stabilization of the human population will require a decrease in the current global birth rate. We certainly cannot hope to achieve a relatively stable population without invoking a substantially higher death rate than the current rate, which is not a comforting thought. It is difficult to imagine, however, that the influence of disease will operate in a simple density-dependent fashion. Instead, it seems more plausible that diseases will ‘‘break out’’ more often with increasing population size and with larger scale consequences, bringing about a strong reduction in our population—a response seen in many other populations that have increased beyond their carrying capacities. One last area of hope is that individuals will lower their consumption rates, thereby adjusting the human
carrying capacity. It is unlikely that Earth can support tens of billions of people with lifestyles matching those of people in the developed nations such as the United States. Therefore, there remains a chance that changes in human behavior will allow our population to gently transition toward a sustainable, zero growth rate population.
IX. CONCLUSIONS The concept of carrying capacity has a history that spans at least thousands of years. The formal definition is about 150 years old and is generally coupled to the asymptotic population in the logistic growth equation (K). Critics argue that because of ongoing confusion and the multitude of definitions attached to the concept we would be better off to simply abandon the term. We also must be concerned that the term not be used to advance any particular political agenda associated with determining how large populations of any particular species ‘‘should’’ be in particular areas. This entry cautiously suggests that the concept remains useful. Since most populations are likely to be at least occasionally limited by factors that depend on the population’s density, we need to continue advancing our knowledge of how populations behave and use this information to guide the design of laboratory and field experiments aimed at determining the mechanisms that regulate populations. Only by using the combination of field and laboratory techniques, grounded in a theoretical framework that has roots going back to the simple logistic equation, will we hope to understand and conserve populations, including our own, over long periods of time.
See Also the Following Articles POPULATION DENSITY • POPULATION DYNAMICS • POPULATION STABILIZATION (HUMAN) • POPULATION VIABILITY ANALYSIS (PVA) • SUSTAINABILITY, CONCEPT AND PRACTICE OF
Bibliography Caughley, G. (1979). What is this thing called carrying capacity? In North American Elk: Ecology, Behavior and Management (M. S. Boyce and L. D. Hayden-Wing, Eds.). The University of Wyoming, Laramie.
CARRYING CAPACITY, CONCEPT OF Cohen, J. E. (1995). How Many People Can the Earth Support? Norton, New York. Darwin, C. (1859). On the Origin of Species by Means of Natural Selection. Murray, London. Davidson, J., and Andrewartha, H. G.(1948). The influence of rainfall, evaporation, and atmospheric temperature on fluctuations in the size of a natural population of Thrips imaginis (Thysanoptera). J. Anim. Ecol. 17, 200–222. Dhondt, A. A. (1988). Carrying capacity: A confusing concept. Acta Oecol. 9, 337–346. Elton, C., and Nicholson, M. (1942). The ten-year cycle in numbers of lynx in Canada. J. Anim. Ecol. 11, 215–244. Gause, G. F. (1934). The Struggle for Existence. Macmillan, New York.
649
Hartvigsen, G., Wait, D. A., and Coleman, J. S. (1995). Tri-trophic interactions influenced by resource availability: Predator effects on plant performance depend on plant resources. Oikos 74, 463– 468. Krebs, C. J., Boutin, S., Boonstra, R., Sinclair, A. R. E., Smith, J. N. M., Dale, M. R. T., Martin, K., and Turkington, R. (1995). Impact of food and predation on the snowshoe hare cycle. Science 269, 1112–1115. Pearl, R., and Reed, L. J. (1920). On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proc. Natl. Acad. Sci. USA 6, 275–288. Turchin, P. (1999). Population regulation: A synthetic view. Oikos 84, 153–159.