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Logistic Population Growth: Equation, Definition & Graph

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  • 0:00 Logistic Population Growth
  • 0:50 Graphing & Equation for Growth
  • 2:15 Below Carrying Capacity
  • 3:05 Near Carrying Capacity
  • 3:35 Larger Than Carrying Capacity
  • 4:05 Lesson Summary
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Lesson Transcript
Instructor: Angela Lynn Swafford

Lynn has a BS and MS in biology and has taught many college biology courses.

Populations tend to get larger until there is no longer enough food or space to support so many individuals. This type of growth is called logistic population growth, and you can learn more about it in this lesson.

What Is Logistic Population Growth?

A group of individuals of the same species living in the same area is called a population. The measurement of how the size of a population changes over time is called the population growth rate, and it depends upon the population size, birth rate and death rate. As long as there are enough resources available, there will be an increase in the number of individuals in a population over time, or a positive growth rate. However, most populations cannot continue to grow forever because they will eventually run out of water, food, sunlight, space or other resources. As these resources begin to run out, population growth will start to slow down. When the growth rate of a population decreases as the number of individuals increases, this is called logistic population growth.

Graphing Logistic Population Growth

Logistic Growth

If we look at a graph of a population undergoing logistic population growth, it will have a characteristic S-shaped curve. The population grows in size slowly when there are only a few individuals. Then the population grows faster when there are more individuals. Finally, having lots of individuals in the population causes growth to slow because resources are limited. In logistic growth, a population will continue to grow until it reaches carrying capacity, which is the maximum number of individuals the environment can support.

Equation for Logistic Population Growth

We can also look at logistic growth as a mathematical equation. Population growth rate is measured in number of individuals in a population (N) over time (t). The term for population growth rate is written as (dN/dt). The d just means change. K represents the carrying capacity, and r is the maximum per capita growth rate for a population. Per capita means per individual, and the per capita growth rate involves the number of births and deaths in a population. The logistic growth equation assumes that K and r do not change over time in a population.

Logistic Growth Equation

Let's see what happens to the population growth rate as N changes from being smaller than K, close or equal to K and larger than K. We will use a simple example where r = 0.5 and K = 100.

Populations Size Smaller Than Carrying Capacity

If N is very small compared to K, then the population growth rate will be a small positive number. This means the population is slowly getting larger because there are a few more births than deaths. For example, if N = 2, the population growth rate is 0.98. (Remember the units are individuals per time. We didn't specify time in this example because it depends upon the species, but it is often measured in years or generation times.)

Logistic Growth Equation When N=2

For a while, as N increases, so does the growth rate of the population. If N = 50, then the growth rate has increased to 12.5. This means the population is rapidly getting larger. However, remember in logistic growth the population does not continue to grow forever.

Logistic Growth Equation When N=50

Population Size Near Carrying Capacity

As N gets closer to K, the population growth rate decreases and approaches zero. In our example, if N = 98, then the growth rate has decreased to 0.98 again, which means the population is still getting larger but not as quickly.

Logistic Growth Equation When N=98

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