*based on an activity by Street and Laubach, American Biology Teacher, v75, no. 4, April 2013
(from Kathy Vanhoeck: York Community High School (Elmhurst, IL)
“We watch fields turn from a sickly brown to a vibrant green every spring season, and then return to this brown hue in subsequent seasons. Consider how this change occurs: plants first sprout new leaves and shoots, grow very rapidly, and expand across an area until the whole location is covered in new plant life. Then these plants slow down until they stop growing at all.”
Consider these questions:
1. Why do these plants slow their growth?
2. What mechanisms control how fast the population grows?
Define these terms:
Birth rate
Death rate
Exponential growth
Logistic growth
Growth rate (r, or rmax)
Carrying capacity (K)
Limiting factors
Density dependent vs. density independent growth
Procedure: In this activity, you will model logistic growth, manipulating the parameters N, K and r. You will use a prepared model using an excel spreadsheet. As you manipulate the variables, the information displayed in the graph will change. Note that the default settings for N0 (population size at time 0), r and K are 2, 0.5 and 300 respectively. So, there are 2 individuals at the start, the growth rate is 50%, ie. one half the current population size is added with each generation.
Part 1: Review the Logistic Growth equation
For the equation below, write out in words what each part of the equation represents.
Logistic Growth= ΔN/ Δt= rmaxN (K-N)/K
How does (K-N)/K apply to the diagram to the right? What do the empty “slots” represent? What does the entire egg carton represent? Do the math for the picture.
What happens to the equation when the carton is full? Do the math!
Part 2: How Does Growth Rate (r) Affect Population Size?
· Vary the growth rate. Set the population size to 1, K to 1000, and r to a value greater than 0.5, but less than 1.0. Describe the effect on the graph. What would happen if you set r=0? What if you set r=2?
· In nature, when is exponential growth observed?
· Why can’t it continue forever?
· Why do we multiply by rmax? What does this represent?
Part 3: How Does Carrying Capacity (K) Affect Population Size?
In this part, you will model the effect of carrying capacity. Leave the other variables at their default settings. Describe/draw graphs of what happens when you vary K, and the effect this has on the growth curves.
Part 4: Your turn!
a. Imagine 20 rats of a new species have first arrived in San Francisco on a boat from Asia. The sewers of San Francisco have enough food to support a population of 100,000 rats at most. Use the model to determine the approximate number of years it will take the population to reach carrying capacity if the population doubles its size every year.
r=____________________ K=_______________________ N0=_____________________
Once the population reaches carrying capacity, what happens in the population over time from that point on? (ie. is K a straight line?) Explain.
b. The spreadsheet models logistic growth. On the AP exam, you will need to do the math on your calculator. In 3 places from the spreadsheet, leaving the numbers for r and K at their default settings, change N and plug the numbers manually into the logistic growth equation on your calculator. Show your work below. Look at the graph in the approximate places you have chosen your numbers. Remember, you are calculating the rate of growth, or the number of individuals added per unit time. So, explain why you are seeing the values you have calculated.
0-10: 10-20: 20-30:
c. Return to the scenario provided at the beginning. Answer the two questions asked using the knowledge you have gained in this activity.