Lemmings of Norway Modeling Exponential Growth

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1 Lemmings of Norway Modeling Exponential Growth Lemmings are small rodents. They are found in tundra biomes. The lemmings of Norway reproduce so quickly that their population fluctuates. Lemmings are small creatures weighing only about 1 to 4 ounces, and 3 to 6 inches long. They resemble a gerbil in size and fur. They are herbivores but their incisors enable them to feed on tough roots and bulbs. They do not hibernate even in the tough northern winters of Norway. Therefore, they must burrow through the snow to find food or store it ahead of time. They are basically loners, but they do have a high reproductive rate and breed rapidly when there is enough food. When their populations become too dense, they will search for a new habitat. In particular, the lemmings of Norway may migrate in large groups searching for a new place only to reach a cliff that overlooks an ocean. When they reach this cliff they may be pushed off due to the crowd and/or jump into the ocean. Here they will swim until they are too exhausted to continue their journey. The idea that they are committing suicide is a myth, however lemming suicide is a metaphor referring to people who go along mindlessly with whatever is popular at the time. In this experiment you will simulate a population of lemmings. Traditionally, this activity is done with dice. Each die represents a single individual and the outcome of a dice roll determines the life event, such as a birth or death, for that individual. For example, with a starting population of 20 (20 dice), every three or six rolled represents the birth of an offspring, thereby adding an individual to your initial population. Each one is a death, decreasing your initial population by one. Each roll of the dice represents one year in the population. The size of subsequent populations is determined by adding the numbers of births and subtracting the number of deaths from your initial population. You will be calculating populations over a series of years, adding births, and subtracting deaths from your initial population, until you finally reach a final population of 500 individuals. The twist here is that we will be using the random number generator on a graphing calculator instead of rolling hundreds of dice. PURPOSE In this activity you will use the graphing calculator to model exponential growth of a lemming population. MATERIALS TI-83 or 84 Plus graphing calculator PROCEDURE 1. Do some calculator housekeeping by performing the following steps: a. Clear all lists by turning the calculator on and press repeatedly to clear the screen followed by to clear any data that currently resides in the lists. The following screen indicates you have been successful. If Done does not appear, try again! b. Clear your Y= by pressing and until any and all equations have been cleared. Press to return to the home screen if necessary. Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 1

2 c. Prepare the random number generator by providing it a seed value. This seed value can be any number, but to keep things random within this classroom, let s all use a long and different number, like YOUR birthday. For example, if your birthday were November 14, 1992 you would enter the 8 digits and then store ( ) this value to the random generator. The random number generator is located in the MATH menu and the PRB submenu. The key strokes for this example would be: to get to PRB (probability) then press to seed the random generator. If you are successful, your birth date will be displayed after pressing. If your birth date were November 14, 1992, your screen would look like Figure 1. Figure 1 d. Press to clear your screen once more. 2. Your starting population is given as 20 and would be represented by 20 individual dice if we were doing this the old-fashioned way. A die has six sides, so we want to set the generator to select integers ranging from one to six, twenty times. Press to get to PRB then to select the randint( command. Now set the number range and repeat by entering. Pressing will execute the random generator function and run it 20 times. This is exactly like rolling twenty dice at once (or a single die 20 times). If successful, your screen looks like Figure 2 (except, perhaps your numbers are different!). Figure 2 3. Rather than counting all the 1s, 3s and 6s to see how the population size changes during this generator, we can let the calculator do that work for us by creating a histogram from our list of 20 random numbers. a. Navigate to your lists by pressing. b. Position your cursor at the very top of L1 by using the key, so that L1 is highlighted. c. With your cursor atop the L1 re-run the RandInt command so that the results will be posted in the L1 column. To do this, with the cursor atop L1, press Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 2

3 . If successful, your screen will look similar to Figure 3. Figure 3 4. Set up STAT PLOT for a histogram. This way, we can graph the data and count the number of 1s (deaths), 3s (births) and 6s (also births) faster. Press and use your arrow and ENTER keys to set up Plot 1 like the one shown below. Make sure that Plot 2 and Plot 3 are both OFF. The screen should match Figure 4. Figure 4 5. To ensure that each bar represents an integer from 1 to 6 and to size the height of the bar such that it is readable on the calculator screen, we must adjust the window settings of the graph before we plot. (This is similar to deciding how to scale your x and y axis on paper so as to fully use the graph area.) a. Press. Set the parameters as follows: Xmin = 1 Xmax = 7 Xscl = 1 Ymin = 0 Ymax = 25 Yscl = 1 Xres = 1 b. As your population size increases, the number of entries for any given bar will eventually exceed 25 (YMax). Simply come back to this screen and increase the value of YMax accordingly until you can see the top of each bar when graphed. 6. To see your histogram, press. Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 3

4 7. To collect the number of hits for each integer, 1 through 6, press to see the specifics for each bar. Of the three values displayed at the bottom of the screen for each cursor position, you are interested in two of them: the min= value and the n= value. The min= value indicates which integer 1 6 you are quantifying and the n= value indicates how many hits that number received in the random generated list of numbers. Be sure to record those in your data table. Your numbers may be different, but your screen should look similar to Figure 5. Figure 5 In Figure 5, the 1s column (min=1) only got one hit (n=1), so we would record one death for this generation in our data table. Press repeatedly to hop from bar one to other bars. Record the number of births from both bar 3 and bar 6. There would be a total of 7 births in this run (two 3s + five 6s) as shown in Figure 6. Figure 6 8. Throw the dice again for your next generation, but consider that your population size has changed by the number of births and deaths from the first generation. Return to the lists,, and position your cursor atop L1. Re-run the RandInt function in L1 again, BUT this time use the new population size instead of 20 (in our example the new population would be =26). With the cursor atop L1, the keystrokes would be. Be sure to use YOUR numbers. 9. Repeat steps 6 through 8 for each generation, being careful to change the 20 each time with your new final population numbers until you have a total of at least 500 individuals. As the height of each bar grows in your histogram, you may need to change the YMax setting for your window to allow for taller columns. Follow the procedure in step 5 for making this adjustment. 10. Graph your results on the paper provided. 11. Complete the ANALYSIS section and the CONCLUSION questions on your student answer page. Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 4

5 Year Lemmings of Norway Modeling Exponential Growth Data Table Initial Population Births Deaths Change in Population Final Population Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 5

6 ANALYSIS Part I: Unrestricted Exponential Growth of the Lemming Population Use your data table to plot a graph demonstrating the unrestricted exponential growth of the lemming population. Label the lag phase and exponential phase of your population. Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 6

7 CONCLUSION QUESTIONS Part I: Unrestricted Exponential Growth of the Lemming Population 1. What is the ratio of births to deaths in this model population? 2. How many years did it take you to reach a population of 100? 3. After you reached a population of 100, how many more years did it take to reach a population of 200? How many more years to reach 300? 400? 500? 4. Using this experiment, define exponential (or geometric) growth. 5. In what way do exponential (or geometric) growth rates differ from arithmetic growth rates? 6. Some populations reach a carrying capacity where the population becomes stable. What are some conditions that would limit exponential growth and therefore allow carrying capacity to be reached? Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 7

8 Part II: Predator-Prey Relationships 1. Identify the predator and the prey for this graph. 2. Give an example of a predator/prey relationship. 3. Predict the effects of a drought on the predator population size. Explain your reasoning. 4. Why is there a time delay between the increase in prey and the increase in predators? 5. What is the ecological benefit of the predator-prey relationship? 6. How might introducing a predator such as the Red-tailed hawk influence the exponential growth rate of the lemmings? Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 8

9 EXTENSION In 2006, the U.S. Fish and Wildlife Service estimated the Yellowstone National Park grizzly population at 500 animals with the population predicted to increase 4-7% annually. It has recommended that the grizzly be reintroduced into Idaho, Montana and Wyoming. These three states oppose the reintroduction effort citing the risk to ranchers and their livestock. In addition, the U.S. Fish and Wildlife Service recommended that the grizzly be removed from the endangered species list. You have received an invitation to speak before Congress on the reintroduction effort. As a member of the Bozeman Montana Ranchers Association and a supporter of the Y2Y (Yukon to Yellowstone) Grizzly Bear Conservation Strategy whose goal is to make sure that grizzly bears have adequate core habitats to sustain viable populations and that bears and other wildlife can move safely between core habitats, you support the reintroduction effort. Write a one page statement in defense of the reintroduction effort and include why the grizzly should remain on the endangered species list. Copyright 2010 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 9