Populations have growth limits

In nature we commonly observe that there are limits on how large a population can grow. Ecologists categorize these limits as being either density independent or density dependent. In this section we will review these two types of limits on population growth and discuss how we can incorporate them into population growth models.

Density-Independent Factors

As the name suggests, density-independent factors limit population size regardless of the population’s density. Common density-independent factors include natural disasters such as tornadoes, hurricanes, floods, and fires. However, other less dramatic changes in the environment, including extreme temperatures and droughts, can also limit populations. In all of these cases, the impact on the population is not related to the number of individuals in the population. Consider, for example, what happens when a hurricane hits a coastal forest. The number of trees that survive the hurricane is independent of the number of trees that were present before the hurricane arrived.

A classic study on density-independent effects was conducted by James Davidson and Herbert Andrewartha in Australia during the 1930s and 1940s. The apple thrip (Thrips imaginis) was a common insect pest in Australia that would occasionally increase to very large numbers and devastate apple trees and rose bushes throughout large regions of the country. From 1932 to 1946, the researchers surveyed thrip populations on rose bush flowers in an attempt to understand the causes of the large variations in population size. They suspected that these population changes were the result of density-independent factors such as seasonal variation in temperature and rainfall. Figure 12.5 shows the population sizes the researchers predicted and the population sizes they observed. As you can see, the predicted changes in thrip abundance closely matched the actual number of thrips they counted in their surveys.

Figure 12.5 Predicting density-independent growth of the apple thrip. Scientists surveyed thrip populations on the flowers of rose bushes from 1932 to 1946. The dashed line represents the predicted number of thrips each year based on a model that included temperature and rainfall variables that cause density-independent effects on thrips. The bars represent the number of thrips actually observed each year.

Data from J. Davidson and H. G. Andrewartha, The influence of rainfall, evaporation and atmospheric temperature on fluctuations in the size of a natural population of Thrips imaginis (Thysanoptera), The Journal of Animal Ecology 17 (1948): 200–222.

Understanding the role of density-independent factors remains an important goal of ecologists today. For example, in 2011 researchers from the U.S. Forest Service and the Canadian Forest Service reported on the impacts of bark beetles in western North America. Bark beetles are native insects that consume the phloem tissues of trees, causing them to die. During the past several decades, bark beetles have killed billions of coniferous trees covering millions of hectares from Mexico to Alaska (Figure 12.6). Under warmer temperatures, bark beetles develop faster and have more generations per year, but they can be killed by unusually cold temperatures. As a result, temperature is a major determinant of population size. Concerned about the possible effect of climate change on the population of bark beetles, researchers looked at expectations for increased climate warming and applied their knowledge of bark beetle ecology to project future growth. They concluded that bark beetle populations are likely to increase dramatically and cause more damage to conifer trees in the future.

Figure 12.6 Bark beetles. Many of these lodgepole pines in central British Columbia, Canada, have been severely damaged by bark beetles, which consume the phloem tissues of the pine trees. The mountain pine beetle (Dendroctonus ponderosae) is one of several species of bark beetles that is predicted to become more abundant with warmer global temperatures in the future.

Photos by Tom Nevesely/age fotostock and (inset) Spectrum Photofile.

Density-Dependent Factors

Density-dependent factors affect population size in relation to the population’s density. Ecologists find it useful to break these factors down into two categories. Negative density dependence occurs when the rate of population growth decreases as population density increases. Positive density dependence, also known as inverse density dependence, occurs when the rate of population growth increases as population density increases. Because positive density dependence was first proposed by the ecologist Warder Allee in 1931, it is also known as the Allee effect.

Some of the most common negative density-dependent factors include resources that are available in limited supply, such as food, nesting sites, and physical space. When a population is small, there is an abundance of resources for all individuals, but as the population increases, such resources are divided among more individuals. As a result, the per capita amount of resources declines and at some point it reaches a level at which individuals find it difficult to grow, reproduce, and survive. This was the concern Malthus expressed in the eighteenth century about the rapid growth of the human population.

Although competition for food is a common challenge for populations that grow to large sizes, it is not the only challenge. Crowded populations also can have higher levels of stress, transmit diseases at a greater rate, and attract the attention of predators. All of these factors can act to slow, and finally halt, population growth.

Negative Density Dependence in Animals

In a classic experiment that investigated density-dependent regulation of populations, Raymond Pearl introduced pairs of fruit flies (Drosophila melanogaster) at different densities into laboratory bottles containing identical amounts of food. As the number of individuals in the bottles increased, competition for food became more intense and the number of progeny produced per pair of adult flies decreased, as shown in Figure 12.7. With even larger populations, the life span of the adults also sharply declined, demonstrating that both the young and old fruit flies were affected by the competition for food.

Figure 12.7 Limits on population growth. Larger populations cause increased competition. This graph shows that as competition among fruit flies increases, they experience a decline in the number of progeny produced per adult breeding pair per day. Increased competition also leads to a decline in the life span of adult flies.

Data from R. Pearl, The growth of populations, Quarterly Review of Biology 2 (1927): 532–548.

Although negative density dependence is easily demonstrated in the laboratory, a number of natural experiments have confirmed the existence of the phenomenon in nature. For example, the common tern (Sterna hirundo) is a bird that nests on beaches. In the 1970s, the tern population on the eastern coast of North America began to expand into an area known as Buzzards Bay in Massachusetts. It first colonized Bird Island where its numbers quickly grew from about 200 to 1,800 individuals by 1990, as shown in Figure 12.8. For these terns the factor limiting population growth appears to be the availability of suitable nesting sites. By 1991, most suitable breeding sites on Bird Island were occupied and the population leveled off. The following year, birds began to colonize Ram Island, where the population increased to just over 2,000 birds and then leveled off by the year 2000. Around this time, the terns began to colonize Penikese Island. These data demonstrate that although the terns have a high intrinsic growth rate, a limited number of nesting sites ultimately limits the population size.

Figure 12.8 Negative density dependence in terns. As the tern population expanded in Buzzards Bay, Massachusetts, it colonized Bird Island. Rapid growth of the population filled most of the available nesting sites and the birds then colonized Ram Island. Once again, the population grew and occupied most of the available nesting sites. The terns then colonized Penikese Island.

Data courtesy of Ian C. T. Nisbet.

Negative Density Dependence in Plants

Plants also experience reduced fecundity and increased mortality at high densities. When plants are grown at high densities, each plant has access to fewer resources such as sunlight, water, and soil nutrients. As a result, it is harder for each plant to grow. We can see this outcome in a study of flax plants (Linum usitatissimum) that were grown at a wide range of densities and then dried to determine their mass. The data from this experiment are shown in Figure 12.9. When seeds were sown at a density of 60 seeds per m², the average dry weight of individuals was between 0.5 and 1 g. When seeds were sown at densities of 1,440 and 3,600 per m², most of the individuals weighed less than 0.5 g. Plants with reduced growth typically experience reduced fecundity. As a result, the high competition caused by high plant densities will cause a population of flax plants to grow more slowly.

Figure 12.9 Negative density dependence in flax. When flax seeds are sown at higher densities, the average plant is smaller. Smaller plants are less fecund, so high densities cause plant populations to increase at a slower rate. The dashed red line in each panel represents the mean dry mass.

Data from J. Harper, A Darwinian approach to plant ecology, Journal of Ecology 55 (1967): 247–270.

Under very high densities, competition among conspecifics can cause plants to die. In an experiment using horseweed (Erigeron canadensis), seeds were sown at a density of 100,000 per m². Competition among the tiny seedlings was intense; as you can see in Figure 12.10a, over an 8-month period most of them died. A close examination of the two y axes in the figure illustrates that there was a hundredfold decrease in population density but a thousandfold increase in the average weight of the surviving individuals. As a result, the total mass of the plants increased tenfold over time.

Figure 12.10 Self-thinning curve. Horseweed seeds were sown at a density of 100,000 per m². (a) Over time, the number of survivors declined by 100 while the average mass of the surviving plants increased by 1,000. (b) When we plot plant density against average mass per plant on log scales, we see it follows a line with a slope of approximately - ³/₂.

Data from J. Harper, A Darwinian approach to plant ecology, Journal of Ecology 55 (1967): 247–270.

In Figure 12.10b, you can see the same data, but now we have graphed the density of surviving plants over time against the average dry mass per plant, using a log scale. When we graph these data, we see that the data fall on a line with a slope of approximately . Plant ecologists call this relationship a self-thinning curve. This relationship has been observed frequently across a wide variety of species. The insights that emerge from the self-thinning curve have a number of practical uses. For example, the curve can be used to predict survival and growth of crop plants that might be sown at different densities in agricultural fields or the growth and survival of tree seedlings that might be planted at different densities in forest plantations.

In plants, animals, and other taxonomic groups, negative density dependence tends to bring populations under control and to maintain their abundances at a level close to the maximum number that can be supported by the environment. In many cases, populations are affected by density-independent and density-dependent factors. Next we consider the fascinating process of positive density dependence.

Positive Density Dependence

Whereas negative density dependence causes population growth to decrease as population density increases, positive density dependence causes population growth to increase as population density increases. Positive density dependence typically occurs when population densities are very low and can be caused by several different mechanisms. In some cases, very low densities make it hard for individuals to find mates or, in the case of flowering plants, to obtain pollen, which can reduce reproductive success and slow population growth. Very low densities can also lead to the harmful effects of inbreeding, as discussed in Chapter 4. There can also be problems associated with small population sizes that, by chance, have uneven sex ratios. Small populations with a low proportion of females can suffer low population growth rates. Finally, individuals living in smaller populations can face a higher predation risk than those living in large populations, as we discussed in Chapter 10. In short, while negative density dependence causes slow population growth due to overcrowding, positive density dependence causes slow population growth due to undercrowding.

Positive density dependence has been demonstrated in a wide variety of species. Many plant species, for example, avoid the costs of inbreeding through a number of mechanisms that prevent self-fertilization. They depend on receiving pollen from other plants, but this can be difficult when individuals live at low densities and are widely spread over an area. In a study of cowslip (Primula veris), a plant of European grasslands, researchers were interested in determining why many of the smaller populations were declining. When they looked at reproduction in populations of different sizes, they found that populations of fewer than 100 individuals produced fewer seeds per plant, as can be seen in Figure 12.11. This probably occurred because the small populations were not as good at attracting pollinators and plants received less pollen.

Figure 12.11 Positive density dependence in cowslip plants. At low densities, plants appear to be pollen-limited, which causes them to produce fewer seeds. At higher densities, each plant produces many more seeds.

Data from M. Kéry et al., Reduced fecundity and offspring performance in small populations of the declining grassland plants Primula veris and Gentiana lutea, Journal of Ecology 88 (2000): 17–30.

A recent study found that positive density dependence also occurs in parasites. In 2012, researchers reported on reproduction in salmon lice (Lepeophtheirus salmonis), which feed on the skin of salmon. When breeding, male and female lice form a mating pair. Researchers in British Columbia counted the number of mating pairs on each fish and found that the probability of an individual forming a mating pair was positively correlated with the number of lice of the opposite sex. Individuals living at low densities have a hard time finding a mate and therefore the lice experienced positive density dependence.

Although we often think of positive and negative density dependence as isolated phenomena, populations can be regulated by both processes, as shown in Figure 12.12. As we move from low densities to intermediate densities, the effects of positive dependence play an important role. Increased densities provide more individuals for breeding and the growth rate of the population can improve. Above some intermediate density, resources start to become limiting and negative density dependence begins to play a role. As densities continue to increase, negative density dependence continues to slow population growth and, eventually, the population growth rate falls to zero.

Figure 12.12 Positive and negative density dependence. When populations are very small, an increase in density can cause an increase in the population’s per capita growth rate. However, when populations are large and resources start to become limiting, further increases in density can cause a decrease in the population’s per capita growth rate.

An example of the dynamic between positive and negative effects of density on population growth can be found in a population of herring (Clupea harengus) that spawns near Iceland. As you can see in Figure 12.13, the herring experience low population growth at low densities, high population growth at intermediate densities, and then a return to low population growth at high densities. The existence of positive density dependence in the herring population concerns fisheries managers because it indicates that if the herring were driven to low densities by fishing, it would be difficult for the population to bounce back. In fact, the populations could experience negative growth rates and become extinct. Fortunately, most fish populations do not appear to experience positive density dependence.

Figure 12.13 Positive and negative density dependence in herring. The highest population growth is measured as a ratio between the number of young fish and the number of adult fish. The highest growth rate of the population occurs at intermediate densities. At higher densities, there is negative density dependence. At lower densities, there is positive density dependence.

Data from R. A. Myers et al., Population dynamics of exploited fish stocks at low population levels, Science 269 (1995): 1106–1108.

The herring study highlights why those who manage populations in the wild need to understand positive density dependence. For example, when a species has declined to very low numbers and we wish to improve its numbers, management strategies need to consider how to avoid inbreeding and how to ensure that each female can encounter a sufficient number of mates to fertilize all of her eggs. The concept of positive density dependence also offers an opportunity for ecologists to help control undesirable pest species. For example, several pest insects have been controlled by releasing sterile males into the population. This skews the sex ratio and causes the females to breed with sterile males, leading to a lower growth rate in the pest population. In other cases, researchers have simply reduced the size of an undesirable population to the point that individuals have a hard time finding mates.

The Logistic Growth Model

Although positive and negative density dependence both occur in nature, the effects of negative density dependence have received much more attention from population modelers. Researchers have developed population growth models that mimic the behavior of many natural populations: rapid initial growth followed by slower growth as populations grow toward their maximum size. The maximum population size that can be supported by the environment is known as the carrying capacity of the population, denoted as K.

To model the slowing growth of populations at high densities, we use the logistic growth model. The logistic growth model builds on the exponential growth model, but we add a term in parentheses that accounts for a decline in growth rate as the population approaches its carrying capacity:

When the number of individuals in the population is small relative to the carrying capacity, the fraction is close to 0 and the term inside the parentheses approaches 1. When this happens, the equation becomes nearly identical to the exponential growth model. However, when the number of individuals in the population approaches the carrying capacity, the term inside the parentheses approaches 0. As a result, the population’s rate of growth approaches 0. When we plot the growth of a population over time using the logistic growth curve, we obtain a sigmoidal, or S-shaped curve, which is shown in Figure 12.14. The middle point on the curve, where the population experiences its highest growth rate, is known as the inflection point. Above the inflection point, the population growth begins to slow.

Figure 12.14 The logistic growth model. When the population is small, growth is exponential. When the population is large, growth slows. The inflection point indicates the point at which the population experiences its fastest rate of growth. Above the inflection point, growth begins to slow. When the population reaches its carrying capacity, K, growth is zero. As a result of this growth pattern, the logistic growth model is represented by an S-shaped growth curve.

We can further understand how the logistic growth model behaves by examining the effect of population size on the rate of increase of the population . As shown in Figure 12.15a, as the population increases from a very small size, the rate of increase grows because the number of reproductive individuals increases. After reaching one-half of the carrying capacity, which corresponds to the inflection point of the S-shaped curve, the rate of increase begins to slow because the reproductive individuals are each obtaining fewer resources.

Figure 12.15 The effect of population size on the rate of increase and the per capita rate of increase. (a) As populations grow from a small number to their carrying capacity, the rate of population growth increases until it reaches the inflection point, after which it decreases. (b) On a per capita basis, the rate of increase continually declines.

We can also examine how population size affects the rate of increase on a per capita basis, which is:

As you can see in Figure 12.15b, individuals in the population continually decline in their ability to contribute to the growth of the population. These insights help us understand that the logistic growth curve shows an initial rapid increase in growth due to the increasing number of individuals in the population, followed by a slowing rate of growth as a large number of individuals experience limited per capita resources.

The logistic equation describes the growth of many populations. One of the classic studies was conducted by Russian biologist Georgyi Gause in 1934. Gause raised two species of protists, Paramecium aurelia and Paramecium caudatum, in test tubes and added a fixed amount of food each day. Both species initially experienced exponential growth, which then slowed until they reached carrying capacity. Because the two species differ in many ways, their populations stabilized at different carrying capacities, which are illustrated as blue lines in Figure 12.16. Gause suspected that the cause of this maximum population size was the amount of available food. To test this hypothesis, he did the experiment again, but this time doubled the amount of food. As you can see from the orange line in each graph, the two species once again experienced rapid initial growth followed by slower growth that stabilized. With twice as much food available, however, the two species stabilized at population sizes that were twice as large as in the first experiment. This early experiment confirmed that logistic growth can happen in real organisms and that the increased availability of a limiting resource can increase the carrying capacity of a population.

Figure 12.16 Logistic growth in two species of Paramecium. (a) When Paramecium aurelia was raised under low or high food supply, the population size—measured in terms of the total volume of protist cells—grew rapidly at first and then slowed as they reached their carrying capacity. When twice as much food was provided, the species achieved a population size that was twice as large. (b) A similar pattern was found for another species of protist, Paramecium caudatum.

Data from G. Gause, The Struggle for Existence (Zoological Institute of the University of Moscow, 1934).

Predicting Human Population Growth With the Logistic Equation

The logistic growth model was first developed in 1838 by the Belgian mathematician Pierre François Verhulst. Verhulst had read Thomas Malthus’s 1798 essay and sought to formulate a natural law governing the growth of populations. Nearly a century later, in 1920, Raymond Pearl and Lowell Reed independently developed the logistic growth model when they examined human population growth in the United States.

Using census data that had been collected every 10 years, Pearl and Reed confirmed the observation by Malthus that the population in the United States initially grew exponentially from 1790 to 1910. They noticed, however, that by 1910 this growth began to slow, as illustrated in Figure 12.17, and that the rate of growth appeared to be declining over time. Based on these data, Pearl and Reed applied their logistic growth equation to the census data and predicted that the population of the United States, which was 91 million in 1910, had a carrying capacity of 197 million. However, they were careful to note that this prediction depended on the population supporting itself using only the land of the United States.

Figure 12.17 The growing human population in the United States. A census conducted every 10 years from 1790 to 1910 indicated that the U.S. population was continuing to increase. However, the rate of increase started to decline by 1910. Based on these data, Pearl and Reed predicted that the population would reach a maximum of 197 million people. In reality, the U.S. population has continued to grow.

Data from R. Pearl and L. J. Reed, On the rate of growth of the population of the United States since 1790 and its mathematical representation, Proceedings of the National Academy of Sciences 6 (1920): 275–288; and U.S. Census Bureau (http://2010.census.gov/2010census/data/apportionment-pop-text.php).

As you can see in the figure, the U.S. population has greatly exceeded this prediction; by 2012, the population was more than 310 million people. There are several reasons for this. One important reason is that technological advances have allowed farmers in the United States to produce food much more efficiently. Moreover, a substantial amount of the food consumed by the U.S. population comes from other regions of the world. In addition to the increases in available food, we have had major improvements in public health and medical treatment that have increased survival rates substantially, particularly for infants and children. Finally, the logistic equation did not incorporate the millions of immigrants that came to the United States after 1910.