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 (KK).
  • Exponential growth produces a J-shaped curve, while logistic growth produces an S-shaped curve.


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 77 years, without any deaths, we would have enough rabbits to cover the entire state of Rhode Island1,2,3^{1,2,3}. 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 11-foot layer in just 3636 hours4^4!
As you've probably noticed, there isn't a 11-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):
dNdT=rN\quad \quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\dfrac{dN}{dT} = rN
In this equation, dN/dTdN/dT is the growth rate of the population a given instant, NN is population size, TT is time, and rr 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 dN/dTdN/dT notation.)
If we assume no movement of individuals into or out of the population, rr is just a function of birth and death rates. You can learn more about the meaning and derivation of the equation here:
To start, let’s think about what a population’s growth rate really means. If we assume that no organisms enter or leave the population, we can define the population growth rate (change in population size over a given time interval) in a pretty straightforward way. It's the number of organisms that are born into the population, minus the number that die, in a given time period:
ΔNΔT=BD\dfrac {\Delta N}{\Delta T} = B - D
In this equation, NN is the population size, TT is time, BB is the number of births in our time period, and DD is the number of deaths in our time period.
For population modeling, researchers usually express birth rate and death rate in per capita terms. (This just means “per individual,” or literally, “per head”). So, we can rewrite the equation above to substitute per capita rates for population-level rates:
  • BB (population birth rate) = bNbN (the per capita birth rate bb multiplied by the number of individuals, NN)
  • DD (population death rate) = dNdN (the per capita death rate dd multiplied by the number of individuals, NN)
    ΔNΔT=bNdN=(bd)N\dfrac {\Delta N}{\Delta T} = bN - dN = (b-d)N
Finally, ecologists often want to calculate the growth rate of a population at a particular instant in time (over an infinitely small time interval), rather than over a long period. So, we can use differential calculus to represent the “instantaneous” growth rate of the population:
dNdT=bNdN=(bd)N\dfrac {dN}{dT} = bN - dN = (b-d)N
The dds on the left side of the equation refer to the derivative (in the calculus sense), not to the death rate (also called dd, but found on the right side of the equation). This is pretty sloppy of us, but it’ll only be an issue for a moment, because we’ll collapse the bdb-d term into a separate term (rr) in the next step.
Finally, we can simplify the relationship between birth and death rates by substituting the term rr (per capita growth rate): r=r = bd b-d. This gives us a single, compact equation:
dNdT=rN\dfrac {dN}{dT} = rN
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 (rr) takes the same positive value regardless of the population size, then we get exponential growth.
  • When the per capita rate of increase (rr) 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:
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 10001000 bacteria in a flask with an unlimited supply of nutrients.
  • After 11 hour: Each bacterium will divide, yielding 20002000 bacteria (an increase of 10001000 bacteria).
  • After 22 hours: Each of the 20002000 bacteria will divide, producing 40004000 (an increase of 20002000 bacteria).
  • After 33 hours: Each of the 40004000 bacteria will divide, producing 80008000 (an increase of 40004000 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 11 day (2424 cycles of division), our bacterial population would have grown from 10001000 to over 1616 billion! When population size, NN, is plotted over time, a J-shaped growth curve is made.
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 rr (the per capita rate of increase) for our population is positive and constant. While any positive, constant rr can lead to exponential growth, you will often see exponential growth represented with an rr of rmaxr_{max}.
rmaxr_{max} 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:
dNdT=rmaxN\quad \quad\quad\quad\quad\quad\quad\quad \quad\quad\dfrac{dN}{dT} = r_{max}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 KK.
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 rr (per capita growth rate) that depends on population size (NN) and how close it is to carrying capacity (KK). Assuming that the population has a base growth rate of rmaxr_{max} when it is very small, we can write the following equation:
dNdT=rmax(KN)KN\quad\quad\quad\quad\quad \quad\quad\quad\dfrac{dN}{dT} = r_{max}\dfrac{(K-N)}{K}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 KNK - N tells us how many more individuals can be added to the population before it hits carrying capacity. (KN)/K(K - N)/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 (KN)/K(K - N)/K term will reduce the growth rate.
When the population is tiny, NN is very small compared to KK. The (KN)/K(K - N)/K term becomes approximately (K/K)(K/K), or 11, 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 KK.

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).
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 numbers5^5. Since this program was shut down, seal populations have rebounded in a roughly logistic pattern6^6.
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 Laake5^5, as reported in Skalski et al6^6.
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.


  • 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:
    dNdT=rmaxN\quad\quad\quad\quad\quad\quad \quad\quad\quad\dfrac{dN}{dT} = r_{max}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(KK). It's represented by the equation:
    dNdT=rmax(KN)KN\quad\quad\quad\quad \quad\quad\quad\dfrac{dN}{dT} = r_{max}\dfrac{(K-N)}{K}N
    Logistic growth produces an S-shaped curve.


This article is a modified derivative of "Environmental limits to population growth," by OpenStax College, Biology, CC BY 4.0. Download the original article for free at http://cnx.org/contents/185cbf87-c72e-48f5-b51e-f14f21b5eabd@10.12.
The modified article is licensed under a CC BY-NC-SA 4.0 license.

Works cited:

  1. Krempels, D. (n.d.). Why spay or neuter my rabbit? Some scary numbers... In H. A. R. E.. Retrieved from http://www.bio.miami.edu/hare/scary.html.
  2. List of U.S. states and territories by area. (2016, May 11). Retrieved May 24, 2016 from Wikipedia: https://en.wikipedia.org/wiki/List_of_U.S._states_and_territories_by_area.
  3. Rabbit.(2016, May 24). Retrieved May 25, 2016 from Wikipedia: https://en.wikipedia.org/wiki/Rabbit.
    Calculation of rabbits covering Rhode Island approximates a rabbit as covering a 4040 cm x 88 cm area (based roughly on reference 33) and utilizes the rabbit reproduction numbers from reference 11 and the area of Rhode Island from reference 22. When the area of Rhode Island is divided by the number of rabbits multiplied by the average "rabbit area," it yields a value less than 11, indicating that the state would be covered by the rabbits.
  4. Pianka, E. R. (2007, July 7). Exponential population growth. In FS 301: The human overpopulation crisis. Retrieved from http://www.zo.utexas.edu/courses/THOC/ExponentialGrowth.htm.
  5. Huber, H. and Laake, J. (2002). Trends and status of harbor seals in Washington state: 1978-99. In AFSC Quarterly Report, 1-13. Retrieved from http://www.afsc.noaa.gov/Quarterly/ond2002/ond02feature.pdf.
  6. Skalski, J. R., Ryding, K. E., and Millspaugh, J. J. (2005). Figure 7.7. Coastal estuarine harbor seal abundance in Washington state, 1975-1999. In Wildlife demography: Analysis of sex, age, and count data. Burlington, MA: Elsevier Academic Press.

Additional references:

Carrying capacity. (2016, May 24). Retrieved May 24, 2016 from Wikipedia: https://en.wikipedia.org/wiki/Carrying_capacity.
In general, r represents the per capita birth rate minus the per capita death rate for a population (r=b-d). (2003). In Connecting concepts: Interactive lessons in biology. Retrieved from
Kopeny, M. (2002). Lecture #K5 - Population ecology, continued. In BIO 202. Retrieved from http://faculty.virginia.edu/bio202/lectures/LectureK5.pdf.
Population dynamics. (2016, May 5). Retrieved May 24, 2016 from Wikipedia: https://en.wikipedia.org/wiki/Population_dynamics.
Post CALC Project. (1999). The natural growth model. In Population growth models. Retrieved from https://services.math.duke.edu/education/postcalc/growth/growth2.html.
Purves, W. K., Sadava, D., Orians, G. H., and Heller, H. C. (2003). Fluctuations in population densities. In Life: The science of biology (7th ed., pp. 1042-1044). Sunderland, MA: Sinauer Associates, Inc.
Raven, P. H. and Johnson, G. B. (2002). Biotic potential. In Biology (6th ed., pp. 506-507). Boston, MA: McGraw-Hill.
Reece, J. B., Urry, L. A., Cain, M. L., Wasserman, S. A., Minorsky, P. V., and Jackson, R. B. (2011). The exponential model describes population growth in an idealized, unlimited environment. In Campbell biology (10th ed., pp. 1190-1192). San Francisco, CA: Pearson.
Rockwood, L. L. (2006). Density-independent growth. In Introduction to population ecology (pp. 5-32). Malden, MA: Blackwell.
Skalski, J. R., Ryding, K. E., and Millspaugh, J. J. (2005). Figure 7.7. Coastal estuarine harbor seal abundance in Washington state, 1975-1999. In Wildlife demography: Analysis of sex, age, and count data. Burlington, MA: Elsevier Academic Press.
Spooner, A. M. (2016). The environmental science of population growth models. In For dummies: Environmental. Retrieved from http://www.dummies.com/how-to/content/the-environmental-science-of-population-growth-mod.html.
Vandermeer, J. (2010). How populations grow: The exponential and logistic equations. Nature Education Knowledge, 3(10), 15. Retrieved from http://www.nature.com/scitable/knowledge/library/how-populations-grow-the-exponential-and-logistic-13240157.