Population ecologists make use of a variety of methods to model population dynamics. An accurate model should be able to describe the changes occurring in a population and predict future changes.

Population Growth

The two simplest models of population growth use deterministic equations (equations that do not account for random events) to describe the rate of change in the size of a population over time. The first of these models, exponential growth, describe populations that increase in numbers without any limits to their growth. The second model, logistic growth, introduces limits to reproductive growth that become more intense as the population size increases. Neither model adequately describes natural populations, but they provide points of comparison.

Exponential Growth

In developing his theory of natural selection, Charles Darwin was influenced by the English clergyman Thomas Malthus. Malthus published his book in 1798, stating that populations with abundant natural resources grow very rapidly. However, they limit further growth by depleting their resources. The early pattern of accelerating population size is called exponential growth (Figure 1).

The best example of exponential growth in organisms is seen in bacteria. Bacteria are prokaryotes that reproduce quickly, about an hour for many species. If 1000 bacteria are placed in a large flask with an abundant supply of nutrients (so the nutrients will not become quickly depleted), the number of bacteria will have doubled from 1000 to 2000 after just an hour. Each of the 2000 bacteria will divide in another hour, producing 4000 bacteria. After the third hour, there should be 8000 bacteria in the flask. The important concept of exponential growth is that the growth rate—the number of organisms added in each reproductive generation—is itself increasing; that is, the population size is increasing at a greater and greater rate. After 24 cycles, the population would have increased from 1000 to more than 16 billion bacteria. A J-shaped growth curve is produced when the population size, N, is plotted over time (Figure 1).

The bacteria-in-a-flask example does not truly represent the real world, where resources are usually limited. However, when a species is introduced into a new habitat that it finds suitable, it may show exponential growth for a while. In the case of the bacteria in the flask, some bacteria will die during the experiment and thus not reproduce; therefore, the growth rate is lowered from a maximal rate in which there is no mortality.

Logistic Growth

Extended exponential growth is possible only when infinite natural resources are available; this is not true in the real world. Charles Darwin recognized this fact in his description of the “struggle for existence,” which states that individuals will compete, with members of their own or other species, for limited resources. The successful ones are more likely to survive and pass on the traits that made them successful to the next generation at a greater rate (natural selection). Population ecologists developed the logistic growth model to model the reality of limited resources.

Carrying Capacity and the Logistic Model

Both (a) and (b) graphs plot population size versus time. In graph (a), exponential growth results in a curve that gets increasingly steep, resulting in a J-shape. In graph (b), logistic growth results in a curve that gets increasingly steep, then levels off when the carrying capacity is reached, resulting in an S-shape.
Figure 1. When resources are unlimited, populations exhibit (a) exponential growth, shown in a J-shaped curve. When resources are limited, populations exhibit (b) logistic growth. In logistic growth, population expansion decreases as resources become scarce and levels off when the environment’s carrying capacity is reached. The logistic growth curve is S-shaped.

With limited resources, exponential growth cannot continue indefinitely in the real world. Exponential growth may occur in environments with few individuals and plentiful resources. However, when the number of individuals gets large enough, resources will be depleted, and the growth rate will slow down. Eventually, the growth rate will plateau or level off (Figure 1). This population size, which is determined by the maximum population size that a particular environment can sustain, is called the carrying capacity, symbolized as K. In real populations, a growing population often overshoots its carrying capacity, and the death rate increases beyond the birth rate, causing the population size to decline back to the carrying capacity or below it. Most populations usually fluctuate around the carrying capacity in an undulating fashion rather than existing right at it. A graph of logistic growth yields the S-shaped curve (Figure 1). It is a more realistic model of population growth than exponential growth. There are three different sections to an S-shaped curve. Initially, growth is exponential because few individuals and ample resources are available. Then, as resources begin to become limited, the growth rate decreases. Finally, the growth rate levels off at the environment’s carrying capacity, with little change in population number over time.

Examples of Logistic Growth

Yeast, a unicellular fungus used to make bread and alcoholic beverages, exhibits the classical S-shaped curve when grown in a test tube (Figure 2a). Its growth levels off as the population depletes the nutrients necessary. In the real world, however, there are variations to this idealized curve. Examples of wild populations include sheep and harbor seals (Figure 2b). In both examples, the population size exceeds the carrying capacity for short periods of time and then falls below the carrying capacity afterward. This fluctuation in population size continues to occur as the population oscillates around its carrying capacity. Still, even with this oscillation, the logistic model is confirmed.

Graph (a) plots amount of yeast versus time of growth in hours. The curve rises steeply, and then plateaus at the carrying capacity. Data points tightly follow the curve. Graph (b) plots the number of harbor seals versus time in years. Again, the curve rises steeply then plateaus at the carrying capacity, but this time there is much more scatter in the data. A micrograph of yeast cells, which are oval in shape, and a photo of a harbor seal are shown.
Figure 2. (a) Yeast grown in ideal conditions in a test tube shows a classical S-shaped logistic growth curve, whereas (b) a natural population of seals shows real-world fluctuation. The yeast is visualized using differential interference contrast light micrography. (credit a: scale-bar data from Matt Russell)

Population Dynamics and Regulation

While valid in many natural populations and useful, the logistic model of population growth simplifies real-world population dynamics. Implicit in the model is that the environment’s carrying capacity does not change, which is not the case. The carrying capacity varies annually. For example, some summers are hot and dry, whereas others are cold and wet; in many areas, the carrying capacity during winter is much lower than during summer. Also, natural events such as earthquakes, volcanoes, and fires can alter an environment and its carrying capacity. Additionally, populations do not usually exist in isolation. They share the environment with other species, competing with them for the same resources (interspecific competition). These factors are also important to understanding how a specific population will grow.

Why Did the Woolly Mammoth Go Extinct?

Image (a) shows a painting of mammoths walking in the snow. Photo (b) shows a stuffed mammoth sitting in a museum display case. Photo (c) shows a mummified baby mammoth, also in a display case.
Figure 3. The three images include (a) a 1916 mural of a mammoth herd from the American Museum of Natural History, (b) the only stuffed mammoth in the world is in the Museum of Zoology located in St. Petersburg, Russia, and (c) a one-month-old baby mammoth, named Lyuba, discovered in Siberia in 2007. (credit a: modification of work by Charles R. Knight; credit b: modification of work by “Tanapon”/Flickr; credit c: modification of work by Matt Howry)

Most populations of woolly mammoths went extinct about 10,000 years ago, soon after paleontologists believe humans began to colonize North America and northern Eurasia (Figure 3). A mammoth population survived on Wrangel Island in the East Siberian Sea and was isolated from human contact until as recently as 1700 BC. We know a lot about these animals from carcasses found frozen in the ice of Siberia and other northern regions. It is commonly thought that climate change and human hunting led to their extinction. A 2008 study estimated climate change reduced the mammoth’s range from 3,000,000 square miles 42,000 years ago to 310,000 square miles 6,000 years ago.2

Through archaeological evidence of kill sites, it is well documented that humans hunted these animals. A 2012 study concluded that no single factor was exclusively responsible for the extinction of these magnificent creatures.3 In addition to climate change and habitat reduction, scientists demonstrated another important factor in the mammoth’s extinction was the migration of human hunters across the Bering Strait to North America during the last ice age 20,000 years ago. Maintaining stable populations was and is very complex, with many interacting factors determining the outcome. It is important to remember that humans are also part of nature. Once, we contributed to a species’ decline using primitive hunting technology only.


Demographic-Based Population Models

Population ecologists have hypothesized that suites of characteristics may evolve in species that lead to particular adaptations to their environments. These adaptations impact the kind of population growth their species experience. Life history characteristics such as birth rates, age at first reproduction, the numbers of offspring, and even death rates evolve like anatomy or behavior, leading to adaptations that affect population growth. Population ecologists have described a continuum of life-history “strategies” with K-selected species on one end and r-selected species on the other. K-selected species are adapted to stable, predictable environments. Populations of K-selected species tend to exist close to their carrying capacity. These species tend to have larger but fewer offspring and contribute large amounts of resources to each offspring. Elephants would be an example of a K-selected species. r-selected species are adapted to unstable and unpredictable environments. They have large numbers of small offspring. R-selected animals do not provide many resources or parental care to offspring; the offspring are relatively self-sufficient at birth. Examples of r-selected species are marine invertebrates such as jellyfish and plants such as the dandelion. The two extreme strategies are at two ends of a continuum on which real species life histories will exist. In addition, life history strategies do not need to evolve as suites. Still, they can evolve independently, so each species may have some characteristics that trend toward one extreme or the other.

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Introduction to Environmental Sciences and Sustainability Copyright © 2023 by Emily P. Harris is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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