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Exponential & logistic growth

How populations grow when they have unlimited resources (and how resource limits change that pattern).

Key points:

  • In exponential growth, a population's per capita (per individual) growth rate stays the same regardless of population size, making the population grow faster and faster as it gets larger.
  • In nature, populations may grow exponentially for some period, but they will ultimately be limited by resource availability.
  • In logistic growth, a population's per capita growth rate gets smaller and smaller as population size approaches a maximum imposed by limited resources in the environment, known as the carrying capacity (K).
  • Exponential growth produces a J-shaped curve, while logistic growth produces an S-shaped curve.

Introduction

In theory, any kind of organism could take over the Earth just by reproducing. For instance, imagine that we started with a single pair of male and female rabbits. If these rabbits and their descendants reproduced at top speed ("like bunnies") for 7 years, without any deaths, we would have enough rabbits to cover the entire state of Rhode Islandstart superscript, 1, comma, 2, comma, 3, end superscript. And that's not even so impressive – if we used E. coli bacteria instead, we could start with just one bacterium and have enough bacteria to cover the Earth with a 1-foot layer in just 36 hoursstart superscript, 4, end superscript!
As you've probably noticed, there isn't a 1-foot layer of bacteria covering the entire Earth (at least, not at my house), nor have bunnies taken possession of Rhode Island. Why, then, don't we see these populations getting as big as they theoretically could? E. coli, rabbits, and all living organisms need specific resources, such as nutrients and suitable environments, in order to survive and reproduce. These resources aren’t unlimited, and a population can only reach a size that match the availability of resources in its local environment.
Population ecologists use a variety of mathematical methods to model population dynamics (how populations change in size and composition over time). Some of these models represent growth without environmental constraints, while others include "ceilings" determined by limited resources. Mathematical models of populations can be used to accurately describe changes occurring in a population and, importantly, to predict future changes.

Modeling population growth rates

To understand the different models that are used to represent population dynamics, let's start by looking at a general equation for the population growth rate (change in number of individuals in a population over time):
start fraction, d, N, divided by, d, T, end fraction, equals, r, N
In this equation, d, N, slash, d, T is the growth rate of the population in a given instant, N is population size, T is time, and r is the per capita rate of increase –that is, how quickly the population grows per individual already in the population. (Check out the differential calculus topic for more about the d, N, slash, d, T notation.)
If we assume no movement of individuals into or out of the population, r is just a function of birth and death rates. You can learn more about the meaning and derivation of the equation here:
The equation above is very general, and we can make more specific forms of it to describe two different kinds of growth models: exponential and logistic.
  • When the per capita rate of increase (r) takes the same positive value regardless of the population size, then we get exponential growth.
  • When the per capita rate of increase (r) decreases as the population increases towards a maximum limit, then we get logistic growth.
Here's a sneak preview – don't worry if you don't understand all of it yet:
A diagram begins at the top with text reading d N over d t equals r N. r is highlighted in green and N is highlighted in red. Two arrows point downward from the text. The first arrow, labeled Exponential growth, points down and to the left from the original text to text reading Per capita growth rate (r) doesn't change, even if population gets very large. d N over d t equals r subscript max N. The r term is highlighted in green and N is highlighted in red. A line graph with population size on the y axis and time on the x axis shows a line beginning at the origin and curving up rapidly until it becomes nearly vertical. The second arrow, labeled logistic growth. points down and to the right from the original text to text reading Per capita growth rate (r) gets smaller as population approaches its maximum size. d N over d t equals r subscript max (K minus N over K) N. The r term is highlighted in green and N is highlighted in red. A line graph with population size on the y axis and time on the x axis shows a line beginning at the origin and forming an S-shaped curve. The curve reaches a maximum as it approaches a dashed horizontal line labeled K.
We'll explore exponential growth and logistic growth in more detail below.

Exponential growth

Bacteria grown in the lab provide an excellent example of exponential growth. In exponential growth, the population’s growth rate increases over time, in proportion to the size of the population.
Let’s take a look at how this works. Bacteria reproduce by binary fission (splitting in half), and the time between divisions is about an hour for many bacterial species. To see how this exponential growth, let's start by placing 1000 bacteria in a flask with an unlimited supply of nutrients.
  • After 1 hour: Each bacterium will divide, yielding 2000 bacteria (an increase of 1000 bacteria).
  • After 2 hours: Each of the 2000 bacteria will divide, producing 4000 (an increase of 2000 bacteria).
  • After 3 hours: Each of the 4000 bacteria will divide, producing 8000 (an increase of 4000 bacteria).
The key concept of exponential growth is that the population growth rate —the number of organisms added in each generation—increases as the population gets larger. And the results can be dramatic: after 1 day (24 cycles of division), our bacterial population would have grown from 1000 to over 16 billion! When population size, N, is plotted over time, a J-shaped growth curve is made.
A line graph titled Exponential growth with population size on the y axis and time on the x axis. The line on the graph begins at the origin and curves up rapidly until it becomes nearly vertical.
Image credit: "Environmental limits to population growth: Figure 1," by OpenStax College, Biology, CC BY 4.0.
How do we model the exponential growth of a population? As we mentioned briefly above, we get exponential growth when r (the per capita rate of increase) for our population is positive and constant. While any positive, constant r can lead to exponential growth, you will often see exponential growth represented with an r of r, start subscript, m, a, x, end subscript.
r, start subscript, m, a, x, end subscript is the maximum per capita rate of increase for a particular species under ideal conditions, and it varies from species to species. For instance, bacteria can reproduce much faster than humans, and would have a higher maximum per capita rate of increase. The maximum population growth rate for a species, sometimes called its biotic potential, is expressed in the following equation:
start fraction, d, N, divided by, d, T, end fraction, equals, r, start subscript, m, a, x, end subscript, N

Logistic growth

Exponential growth is not a very sustainable state of affairs, since it depends on infinite amounts of resources (which tend not to exist in the real world).
Exponential growth may happen for a while, if there are few individuals and many resources. But when the number of individuals gets large enough, resources start to get used up, slowing the growth rate. Eventually, the growth rate will plateau, or level off, making an S-shaped curve. The population size at which it levels off, which represents the maximum population size a particular environment can support, is called the carrying capacity, or K.
A line graph titled Logistic growth with population size on the y axis and time on the x axis. The line on the graph begins at the origin and forms an S-shaped curve. The curve reaches a maximum as it approaches a dashed horizontal line labeled Carrying capacity.
Image credit: "Environmental limits to population growth: Figure 1," by OpenStax College, Biology, CC BY 4.0.
We can mathematically model logistic growth by modifying our equation for exponential growth, using an r (per capita growth rate) that depends on population size (N) and how close it is to carrying capacity (K). Assuming that the population has a base growth rate of r, start subscript, m, a, x, end subscript when it is very small, we can write the following equation:
start fraction, d, N, divided by, d, T, end fraction, equals, r, start subscript, m, a, x, end subscript, start fraction, left parenthesis, K, minus, N, right parenthesis, divided by, K, end fraction, N
Let's take a minute to dissect this equation and see why it makes sense. At any given point in time during a population's growth, the expression K, minus, N tells us how many more individuals can be added to the population before it hits carrying capacity. left parenthesis, K, minus, N, right parenthesis, slash, K, then, is the fraction of the carrying capacity that has not yet been “used up.” The more carrying capacity that has been used up, the more the left parenthesis, K, minus, N, right parenthesis, slash, K term will reduce the growth rate.
When the population is tiny, N is very small compared to K. The left parenthesis, K, minus, N, right parenthesis, slash, K term becomes approximately left parenthesis, K, slash, K, right parenthesis, or 1, giving us back the exponential equation. This fits with our graph above: the population grows near-exponentially at first, but levels off more and more as it approaches K.

What factors determine carrying capacity?

Basically, any kind of resource important to a species’ survival can act as a limit. For plants, the water, sunlight, nutrients, and the space to grow are some key resources. For animals, important resources include food, water, shelter, and nesting space. Limited quantities of these resources results in competition between members of the same population, or intraspecific competition (intra- = within; -specific = species).
Intraspecific competition for resources may not affect populations that are well below their carrying capacity—resources are plentiful and all individuals can obtain what they need. However, as population size increases, the competition intensifies. In addition, the accumulation of waste products can reduce an environment’s carrying capacity.

Examples of logistic growth

Yeast, a microscopic fungus used to make bread and alcoholic beverages, can produce a classic S-shaped curve when grown in a test tube. In the graph shown below, yeast growth levels off as the population hits the limit of the available nutrients. (If we followed the population for longer, it would likely crash, since the test tube is a closed system – meaning that fuel sources would eventually run out and wastes might reach toxic levels).
A micrograph of round 3 yeast cells. Two of the cells are budding into new cells. Below the micrograph is a line graph with Amount of yeast on the y axis and Hours on the x axis. There is a red dashed horizontal line at y = 13. The line on the graph begins around 5 hours and rapidly increases until the curve reaches a maximum as it approaches the dashed horizontal line. The line is drawn to fit to several data points surrounding it.
Image credit: "Environmental limits to population growth: Figure 2," by OpenStax College, Biology, CC BY 4.0.
In the real world, there are variations on the “ideal” logistic curve. We can see one example in the graph below, which illustrates population growth in harbor seals in Washington State. In the early part of the 20th century, seals were actively hunted under a government program that viewed them as harmful predators, greatly reducing their numbersstart superscript, 5, end superscript. Since this program was shut down, seal populations have rebounded in a roughly logistic patternstart superscript, 6, end superscript.
A photo of a spotted seal. Below the photo is a line graph with Number of seals on the y axis and Year on the x axis. There is a red dashed horizontal line at y = 7500. The line on the graph shows a roughly S-shaped curve that flatlines and then decreases slightly as it nears a dashed horizontal line. The line is drawn to fit to several data points surrounding it.
Image credit: "Environmental limits to population growth: Figure 2," by OpenStax College, Biology, CC BY 4.0. Data in graph appears to be from Huber and Laakestart superscript, 5, end superscript, as reported in Skalski et alstart superscript, 6, end superscript.
A shown in the graph above, population size may bounce around a bit when it gets to carrying capacity, dipping below or jumping above this value. It’s common for real populations to oscillate (bounce back and forth) continually around carrying capacity, rather than forming a perfectly flat line.

Summary

  • Exponential growth takes place when a population's per capita growth rate stays the same, regardless of population size, making the population grow faster and faster as it gets larger. It's represented by the equation:
    start fraction, d, N, divided by, d, T, end fraction, equals, r, start subscript, m, a, x, end subscript, N
    Exponential growth produces a J-shaped curve.
  • Logistic growth takes place when a population's per capita growth rate decreases as population size approaches a maximum imposed by limited resources, the carrying capacity(K). It's represented by the equation:
    start fraction, d, N, divided by, d, T, end fraction, equals, r, start subscript, m, a, x, end subscript, start fraction, left parenthesis, K, minus, N, right parenthesis, divided by, K, end fraction, N
    Logistic growth produces an S-shaped curve.

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