Explore logistic growth. Understand the logistic population growth. Use the logistic growth equation and logistic growth graph to predict the population growth.
Updated: 11/21/2023
Logistic Growth | Definition, Equation & Model
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Logistic Population Growth: Activities
Study Prompt 1:
Make a set of flashcards that list all of the keywords (those in bold) from the lesson and their definitions. The keywords should be on one side and the definition should be on the other. That way, with a partner or in small groups, you can quiz each other to make sure you understand the key terms from the lesson. Tip: It may help to also illustrate the definition on the flashcard in addition to writing it out.
Poster Prompt 1:
Create a poster or other type of graphic organizer or visual aid that depicts population growth rate and the factors that contribute to a positive growth rate.
Example: Illustrating factors that sustain a population, such as food, space, water, and economic opportunities will help you show the elements of a positive growth rate.
Graphing or Equation Prompt 1:
In the lesson, logistic population growth was presented in an S-curved graph, as well as in a mathematical equation. Which one makes more sense to you? In an essay of a couple of paragraphs, explain which representation makes more sense to you and why. Then, create your own graph or mathematical equation that depicts logistic population growth.
Example: You could state in your essay that you are good at math, so equations naturally make sense to you, and they give you the opportunity to plug in more numbers that more precisely reflect logistic population growth. Or, you could explain that you are a visual learner so the graph is more appealing and gives you an overall idea of logistic population growth without making you feel bogged down in the numbers.
What is the logistic population growth model?
The logistic population growth model shows the gradual increase in population at the beginning, followed by a period of rapid growth. Eventually, the model will display a decrease in the growth rate as the population meets or exceeds the carrying capacity.
What is the difference between exponential and logistic population growth?
Exponential population growth shows rapid population growth if resources were not a factor. Logistic population growth shows population growth and the decrease of the growth as the population meets the carrying capacity or sustainable resource limit.
What is an example of a population that shows logistic growth?
A population of 10 people living on an island will grow slowly and then rapidly grow as the population expands and more people give birth. Eventually, assuming resources are limited, and no major event is introduced, the population will slow or decrease because the island will have limited resources.
Table of Contents
ShowThe logistic growth definition in population models refers to the gradual growth of the population in the beginning and then increases when the number of people grows. When does logistic growth occur? According to the logistic growth model, the population grows the most when a high number of people use the efficient number of resources available to sustain that population. In this way, logistic growth is a type of exponential growth model. The logistic growth model also shows a decrease or reduced rate of population growth as the population begins to exceed the necessary number of resources needed. This is known as exceeding the carrying capacity. Some people may not use exponential growth models because the models would not show an accurate growth rate due to the cap on resources and the exceeding of the carrying capacity.
Logistic Growth Curve
The logistic growth curve represents the logistic population growth rate. The logistic growth curve on a line graph is S-shaped to show the slow increase, rapid population growth, and finally, the reduction in the growth.
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That graph shows that the exponential growth curve surpasses the carrying capacity while the logistic growth curve flattens when the population meets the carrying capacity line.
The Y-axis shows the total population increase or decreases going up and down on the graph.
The X-axis shows the advancement of time moving to the right of the graph.
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Economists, mathematicians, government officials, and others would more than likely choose the logistic growth model over other population models because of the accuracy the model displays. Logistic population growth can only expand so much before the population experiences the limit of resources that can help sustain the growing population.
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The logistic growth graph is created by plotting points found from the calculations involved in the logistic growth equation. The logistic growth equation components are:
- dN - Change in population
- dt - Change in time
- rN - r is the maximum per capita growth rate for a population. Per capita means per person, while the per capita growth rate is the number of births vs. death.
- K - Carrying capacity
- N - Population size
- In the logistic growth equation, the K and R values do not change over time in a population
The logistic growth equation is dN/dt=rN((K-N)/K).
A different equation can be used when an event occurs that negatively affects the population.
This equation is: f(x) = c/(1+ae^{-bx}).
Each component of this equation is:
- f(x) - population over time
- c - carrying capacity
- 1 - the starting point of the population
- a - population target or affected event
- -bx - the constant rate of growth and the time of the target event (which is usually shown as a negative because of the population decreasing
Example
Below are examples using both logistic growth equations to find the logistic growth model.
f(x) = c/(1+ae^{-bx}) Example
A city of 100,000 people was infected with the coronavirus. The logistic growth rate (b) for the city is 0.8090. The city council knows that at least one person is infected and has read reports from other cities that the virus spreads quickly. The city council is interested in the number of people that will be infected after 15 days (x). The limiting factor of the city, c, is 100,000 since everyone in the city can get infected. The city council also knows that there has to be at least one case at the beginning which means the rest of the population will be represented by (a) = 99,999.
The steps for the equation would be f(x) = c/(1+ae^{-bx}) :
- f(x) = 100,000 / 1+99,999e^-0.8090 x 15
- -0.8090 x 15 = -12.135
- 1+99,999e^-12.135 = 1.54
- 100,000 / 1.54 = 64,935
- The end result is that by day fifteen, 64,935 would be infected.
- In the long term, only a few more days after day 15, all 100,000 people will be infected.
dN/dt=rN((K-N)/K) Example
In this example, the equation will show issues when certain components carry different numerical values and how they can affect the logistic population growth.
- If population (N) is larger than the carrying capacity (K), the population will begin to drop because of the loss of people due to limited resources to sustain the population. This will occur until the population gets back below the carrying capacity.
- If a population is decreasing, its growth rate becomes a negative number (dN/dt).
- If (N) is less than (K), the population can continue to grow at a steady or sometimes exponential rate.
- When the population is experiencing growth, the growth rate (dN/dt) becomes a positive number.
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The logistic growth model is a population model that shows a gradual increase in the population at the beginning, followed by a period of large growth, and finishes with a decrease in growth rate. The logistic growth model is a type of exponential growth model. The logistic growth model is more accurate than other models in determining population growth because of the effect of the carrying capacity. The carrying capacity is the concept that resources are always limited because the environment can only support a certain number of individuals in a population. The logistic growth curve is S-shaped to represent the gradual growth stage, then the rapid growth phase, followed by the slow growth rate at the end.
The logistic growth graph is created by plotting points using the logistic growth equation. The logistic growth equation is dN/dt=rN((K-N)/K). If the population size (N) is less than the carrying capacity (K), the population will continue to grow. When a population grows, its growth rate (dN/dt) is a positive number. If the population size (N) is larger than the carrying capacity (K), the population will lose individuals until it gets back to carrying capacity. If a population is shrinking in size, its growth rate is a negative number.
The equation f(x) = c/(1+ae^{-bx}) can be used if there is an event that occurs that causes the population to decrease, such as a virus infecting a city.
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Video Transcript
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
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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.
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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.)
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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.
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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.
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A growth rate of zero means that the population is not growing, which is what happens at carrying capacity because the birth rate usually equals the death rate. When N is equal to K, a population has reached carrying capacity.
Population Size Larger Than Carrying Capacity
If N happens to be higher than K, then the population will lose individuals until N is equal to K. Population growth will be negative during this time because there will be more deaths than births. If N = 105 (the carrying capacity is 100), then the growth rate will be -2.6. The growth rate will continue to be negative until the population has decreased in size to carrying capacity. Then remember that at carrying capacity, the population growth rate will be zero.
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Lesson Summary
Let's review. Logistic population growth occurs when the growth rate decreases as the population reaches carrying capacity. Carrying capacity is the maximum number of individuals in a population that the environment can support. A graph of logistic growth is shaped like an S. Early in time, if the population is small, then the growth rate will increase. When the population approaches carrying capacity, its growth rate will start to slow. Finally, at carrying capacity, the population will no longer increase in size over time.
Learning Outcomes
After watching this lesson on logistic population growth, measure your ability to:
- Contrast logistic population growth and carrying capacity
- Interpret a graph of logistic growth
- Use the equation for logistic population growth to explain the relationship between population size and carrying capacity
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Logistic Growth | Definition, Equation & Model
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