Biology

HOW DO POPULATIONS GROW?

Student Guide

Thomas Austin was an Englishman who migrated to southern Australia to farm the land. His property, Barwon Park was located near Winchelsea, Victoria. In October of 1859, homesick for his homeland and the hunting he used to enjoy, Thomas enlisted his nephew, William Austin who still resided in England, to send two dozen wild English rabbits, which Thomas then released onto his land. Thomas dismissed the act as benign, not realizing the drastic consequences of his actions. Due to the well-known prolific nature of rabbits, and the suitability of the Australian climate, within 6 years, this population of 24 rabbits had increased to 22 million. By the 1930’s, Australia’s rabbit populations were estimated to exceed 750 million! How did the populations grow so large, so quickly? And what might the consequences be on the local ecosystem?

Procedure

1. Select a partner to work with and obtain 10 pennies. The pennies represent 10 individual rabbits in a population. Place the pennies in a container and shake them up. Pour them out onto a table. Each penny that lands with a tail showing represents a rabbit that gets to produce an offspring that is added to the original population of 10. [So chances are that approximately five individuals got to reproduce and your new population contains about 15 individuals (i.e., about 15 pennies)]. Now remove 10% of your population representing individuals that have died. Round down if the number is not an integer.

2. Repeat this procedure several times until the rabbit population exceeds 100 individuals. After each episode of births and deaths (i.e., after each “generation”), record the population size (i.e., the total number of pennies) in the chart below. Also record the Idealized population size (given that exactly half of your individuals reproduced each generation with NO ONE dying—go ahead and report the idealized numbers in decimals, but only keep two decimals. I’ve gotten it started for you.)

Flip/Generation Number	Experimental Population Size	Idealized Population Size (no death)
0	10	10
1		15
2		22.5
3		33.75
4
5
6
7
8
9
10

3. Using the graph on the next page, plot population size (on the y axis) versus flip/generation number (on the x axis). Use the data on your graph to determine the slope during each time interval (REMEMBER: slope is the change in y over the change in x, or rise over run; i.e., slope = (y/(x). Record the slopes below.

Interval

Slope

Between generations 0 and 1

Between generations 1 and 2

Between generations 2 and 3

Between generations 3 and 4

Between generations 4 and 5

Between generations 5 and 6

Between generations 6 and 7

Between generations 7 and 8

4. How do the slopes change? What do they tell you about how the rabbit population is growing? What if all the slopes were the same? What would that tell you about population growth? Does this suggest a reason why the rabbit population grew so rapidly in Australia? Explain.

5. How does this pattern of growth affect the rate at which the population is increasing? In other words, in each generation, what percent increase did the population experience (i.e., the number of new individuals added divided by the total number of individuals present)? Did the rate of increase also increase with each generation? See if you can find out by calculating some sort of growth rate for each time interval. To simplify the math, ignore death rates and use the idealized numbers (i.e., 10, 15, 22.5, 33.75, and so on). Record the rate of population increase for each of the first four time intervals in the table below:

Interval	Growth Rate
Between generations 0 and 1
Between generations 1 and 2
Between generations 2 and 3
Between generations 3 and 4

6. Based on your calculations, how does growth rate vary across time? Why?

7. How did you calculate growth rate (r)? Can you write a generic formula that could be used on any population?

r =

Do you suppose that this is a realistic pattern of growth for rabbits, and can it continue in this fashion indefinitely? Let’s look at some numbers from a hypothetical population of rabbits on the island of New Zealand, off the coast of Australia (Australia does not currently have a reasonable rabbit number map; currently, recording systems rely on individual reports). Use the initial generation interval to calculate a value for r. What value did you get?

Table 1. Initial Growth of a Rabbit Population (in thousands)

____________________________________

Generations # of rabbits

____________________________________

0 10

1 18

____________________________________

8. Can you predict the size of future generations? Use your r value to generate expected numbers for the next several time intervals – at least up to and including interval t = 8. Record your expected numbers in Table 2.

Table 2. Continued Growth of a Rabbit Population

Generations	Expected Numbers	Observed Numbers
0	10	10
1		18
2		29
3		47
4		71
5		119
6		174
7		257
8		351
9		441
10		513
11		560
12		595
13		629
14		641
15		651
16		656
17		660
18		662

9. How did you decide how many individuals would be added to each generation? Can you write a generic formula that could be used on any population, given that we know the growth rate, r?

10. Now plot a graph showing both the expected and observed numbers. How do they compare? If they do not match, can you suggest one or more reasons (alternative hypotheses) for the mismatch? List them below and be prepared to share them with the class.

11. Our equation for exponential population growth does not generate expected numbers that match the observed numbers in Table 2 very well, at least not after the first few time intervals. This means that we either have to throw out our equation (our mathematical model) and generate a better one, or modify the equation to make it better (in the sense that it will generate expected numbers that do in fact match the observed numbers). Because our initial equation seems to do a good job for the first few time intervals, let’s not throw it out but instead let’s see if we can modify it in some way that might make it work better for later time intervals. Read on.

12. How might the environment affect population growth? Is there a limit to how much it can hold, a carrying capacity? Let’s represent carrying capacity with a K and assume for the sake of argument we have a growing population with a carrying capacity of 100. With your partner try to come up with a way to write an equation to describe how close our growing population is to its carrying capacity of 100, i.e., how much “room” the population still has to grow. (See the figure below.)

a) For example, if the population size is 75, how much room does it still have to grow?

b) Can you write the population’s amount of “room to grow” as a percent?

c) Suppose another population growing in an environment with a carrying capacity of 600 is now at 250. How much room does it have to grow? Express this as a percent and show how you derived the percent.

d) Now, try to write a general expression for the relationship using the symbol K for carrying capacity and N for population size. The expression should describe how much room the population has to grow before it hits its environment’s carrying capacity. What is the range of values that your expression can have? When is the value largest? When is it smallest?

13. Recall our graph of the growing rabbit population. What appears to be the carrying capacity of this environment?

14. The population grows more slowly as it approaches K. In other words, with lots of room to grow, the population grows rapidly. And with little room to grow, the population grows slowly. Can you write an equation that uses this new “carrying capacity” issue to predict population growth? [Hint: Start with your exponential equation and add in our “carrying capacity” element.] Be prepared to discuss it with the class.

15. Now that you have your equation, let’s test its ability to model the rabbit population’s growth. First, use your equation to predict each generation of rabbits. How does your prediction compare now? Is this equation a better way to predict population growth?

16. When populations are growing in a limited environment, when is their growth the largest? (Hint, to figure this out, graph the amount of individuals added to the population with each generation.)

17. Around the same time that rabbits were introduced to Australia, the red fox was also introduced, for the same purpose of hunting. However, after seeing that it was an efficient predator to rabbits, additional red foxes were introduced after 1859 to help control rabbit populations. How do you suppose this will affect rabbit populations? Graph your prediction below.

18. What do you suppose will happen to the red fox population? Add a curve representing predator population to the graph above.

19. As red fox populations increase, what might happen to rabbit populations? As rabbit populations decrease, what might happen to red fox populations? Over time, what pattern do you predict will emerge? Graph it below:

Alfred Lotka and Vito Volterra independently derived a pair of equations to explain the relationships between predator and prey. The simultaneous non-linear, differential equations are now known as the Lotka-Volterra Predator-Prey Model. They are shown below:

Prey: N/t = rmaxN – aNP

Predator: N/t = bNP – rd P

Where rmax= growth rate of the prey

N = prey population size

P = predator population size

a = the probability of dying due to predation

rd = death rate of predator with or without prey present

b = the probability of capturing prey AND converting the usable energy into offspring

20. Notice that the first part of the prey equation (rmaxN) is the exponential growth equation. Explain how the addition of the “aNP” term limits the prey population. Is it a density-dependent or density-independent factor? How can you tell?

21. Your instructor will show you simulations of the Lotka-Volterra model using the computer program, Populus. We will set up a typical fox-rabbit predator-prey relationship using the following values:

rmax= 1.7

N = 500

P = 20

a = 0.05

rd = 0.10

b = 0.005

Graph the relationship below:

22. How would you simulate an increase in predator skill at capturing prey? What does this do to our model? Why?

23. How would you simulate a decrease in prey reproductive ability? What does this do to our model? Why?

24. How would you simulate a decrease in predator numbers due to a relocation program? What does this do to our model? Why?

25. Notice in our model that the “b” term is about 10% of the “a” term’s value. Why is that? Come up with some hypotheses and be prepared to share them with the class.