1. 1 Predator - Prey Interactions Modeling the number of fishers and porcupines in New Hampshire When a predator preys on another species, it affects the number of that other species. Furthermore, the number of prey available affects the number of predators. The combination of these two interactions creates changes in the population of both species. Prepared for SSAC by Ben Steele – Colby-Sawyer College, New London NH © The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2006 Supporting Quantitative Concepts and Skills Analysis – interpreting equations Analysis – difference equations Graphs – XY scatter plots SSAC2006.QH540.BS1.1 Source :http://yotophoto.comSource :http://commons.wikimedia.org Core Quantitative Issue Forward modeling

2. 2 Understanding predator - prey interactions is important to explain changes in endangered species and in pest species as well as just to understand the natural world. Every species is a potential predator or prey or both. We will examine a New England example: the porcupine and its only effective predator, the fisher (not the fisher cat – that is a baseball team). Fishers were nearly eliminated in New Hampshire by the fur trade in the 19th century. In the 20 th century, porcupines became a major bane to foresters, as they can kill valuable timber trees. Meanwhile, fishers became a protected species, and, with protection, the fisher population grew. As a result, dogs with a mouth full of porcupine quills have become less common. Recently, however, porcupines have increased in number. What caused the changes in the populations of these two species? We can model the interaction between fishers and porcupines if we make some simplifying assumptions. We will construct a model to make mathematical predictions of population changes that result from the interaction of predator and prey. Problem Question 1: How does a predator affect the population of a prey species? Question 2: How does the prey, as food, affect the predator population? Question 3: How do they affect each other? Fisher tracks leading to a porcupine carcass Bob Arnebeck http://www.geocities.com/bobarnebeck/ppi nes.htm

3. 3 Slides 4-5 present the Lotka-Volterra predator-prey equations and their variables. Slides 6-8 will have you create a spreadsheet using the equations to see how a constant predator population affects the number of prey. You will graph the results to see the effects of changing the variables. Slides 9-10 model how a constant prey population affects growth of the predator population. We will explore the effects of changing variables here too. In slides 11-14, we put the two above interactions together and see what factors can cause population cycles of both species, equilibrium, extinction of the predator, the prey or both. Slides 15-16 contain questions for you to answer. Overview of Module We will model predator-prey interactions using a simplified version of the Lotka-Volterra equations. In order to understand the interaction you need to understand what the equations mean and how changes in the variables change the results. You will also learn how graphs show changes over time and how to interpret a phase-plane graph. You may already know how a single population can grow exponentially and how it then may level off as the population approaches carrying capacity. In the model of this module, we will consider the mutual effects of two interacting populations: the effect on the prey population of being killed by predators and the effect on the predator population of the availability of prey. We will model each of the interactions separately, then put them together.

4. 4 What causes prey populations to change? List the factors that cause increase in population and those that cause decrease. The equation for the porcupine population Your answer should have included: 1.Increase due to reproduction. 2.Decrease due to predation. 3.Decrease due to other causes (disease or old age) The Lotka Volterra equations that we will use express these this way: dR/dt = r R – p R P Where dR/dt = the time rate of change in the prey population r = reproductive rate for the prey (this value includes non-predator losses.) R = prey population p = predation rate P = predator population Note: This is a differential equation! Don’t worry if you have not had calculus. We will simplify dR/dt to I R, the increase between generations. This effectively replaces differentials with differences, which we can handle with Excel. (But do think about taking calculus.) In other words, the change in prey population increases because of reproduction, and decreases because of encounters with predators times the rate of predation. dR/dt = r R – p R P Reproductive rate (includes births minus natural deaths) Change in prey (dR) Per unit time (dt) Prey population Predation rate Encounters between prey and predators are affected by density of each, hence P×R

5. 5 What causes the predator population to change? List the factors that cause increase in population and those that cause decrease The equation for the fisher population Mortality rate of predators Your answer should have included: 1.Increase due to reproduction, which is higher when there are more prey to eat. 2. Decrease due to death (mortality). The equation that we will use expresses these this way: I P = c p R P – m P Where: I P = Increase in prey population for one generation c = conversion of prey that are eaten into predator offspring (This is really the effect of food on reproductive rate) p = predation rate R = prey population P = predator population m = mortality rate of predators I P = c p R P – m P Conversion of prey eaten to new predators Increase in predators per generation Predation rate times encounter rate Predator population In other words, the change in predator population increases because of the amount of food they eat converted into offspring and decreases because of death.

6. 6 We will put the equation from Slide 4 into a spreadsheet, choose some variables, and graph the results. We will start with a given number of porcupines and fishers, calculate the increase in population for a generation from the equation, add that value to the original population, then repeat 17 times. For this first problem, we set the fisher population at 50, and we will not change it throughout the simulation. We will start with 100 porcupines, use a reproductive rate (r) of 0.5 for the porcupines, and a predation rate (p) of 0.005. We will assume that a generation is one year. Open Excel and recreate your own spreadsheet like this one. The equation should come from I R = r R – p R P converted into an Excel expression with the information above inserted for the constants. If you would like help entering columns of numbers and formulas, click here. You will also need to use absolute and relative cell references. For help with these, look here,For help with these, look here, Question 1: How does a fixed predator population affect the population of the prey ? = cell with a number in it = cell with a formula in it

7. 7 What happens if you change the number of predators? First, graph the population change over 17 years. What type of graph is most appropriate here? Now, what do you predict will happen if we decrease the number of predators from 50 to 40? Answer this question three ways: 1.In the real world, what would happen to porcupines if fishers increased? 2.Look at the equation. If you decrease P, what happens to I R ? 3.Change the number in the spreadsheet. Question 1: How does a predator population affect the population of the prey ? (Cont.) Hint, what do you want on the x axis? Does the graph look just the same? If so, change it back to 50 and look closely. Something has changed. For help with graphs, click here

8. 8 What happens if you change the predation rate or the prey reproductive rate? Again, predict what will happen if we increase the predation rate? Again, think (1) real world, (2) the equation, and (3) test your predictions by changing the values on the spreadsheet. Can you get the prey to go extinct? What predation rate creates this? Now change prey reproductive rate. What happens? Question 1: How does a predator population affect the population of the prey ? (Cont.) What does this number mean? Are porcupines extinct?

9. 9 Now we will look at the reverse effect: the effect of prey (food supply) on predator population. We start with the simple situation of fixed prey population. Predict what will happen to the predator population if we increase the predation rate? Again, think (1) real world, (2) the equation, and (3) test your predictions by changing the values on the spreadsheet. But first, you need to add a column for I P (increase in predator population, calculated by I P = c p R P – m P from Slide 5), a mortality rate, and the constant c which is a conversion from prey eaten to predator offspring. Fill down the predator columns and graph the predator population. Your spreadsheet should look like this: Question 2: How does a prey population affect the population of the predator?

10. 10 Do some more experiments. Write the question, the prediction, and the result. For example, if you double the population of prey (which is constant in this simulation), how much higher do you think the predator population will be? Twice as high? If it is not twice as high, propose a reason for why. Question 2: How does a prey population affect the population of the predator? (Cont.) Now we are ready to put the two parts of the model together, so that the changing prey population affects the predator increase and predator population affects the prey increase. This is simply a matter of changing absolute cell references to a relative ones, but which cells in which equations?

11. 11 Make this change and do not forget to fill the new equation down the entire column. Also change the variables (yellow boxes) to the values shown below. Now add the data for Prey population to the graph. Question 3: How do both predators and prey affect each other? Right! In both the equations for increase in population (I R and I P ) you should change $C$4 to C4. For a hint on how to add another column of data to a graph, click here. You should get this. This looks quite interesting! The prey population goes up, then down. Predators go up slightly, then level off. We better see what happens after that.

12. 12 How would you describe these changes in words? There are some real world examples with changes like this. Look at the Lynx-hare interaction.Lynx-hare interaction. Question 3: How do both predators and prey affect each other? (Cont.) If it does not look like this, check your equations, check what is being graphed, check the variables. Checking for errors is a valuable Excel skill. The last rows should look like this: So now, fill down all the columns to 100 years. Then extend the graph to cover 100 years. It should look like this:

13. 13 Question 3: How do both predators and prey affect each other? (Cont.) There is another way to graph these changes that shows some other features. It is called a phase-plane plot and is useful in many situations. It has prey as one axis and predators as the other. Each point represents the population of each in a different year. Produce this graph (look at the axes to figure out how to do it). Identify the point where our simulation starts. Remember that we started with 100 prey and 20 predators. Follow the trajectory through the 100 years. What is happening to the prey population when the line is moving to the right? To the left? What is happening to the predator population when the line is moving up? Down? What does it mean when the points are close together? Far apart? What happens at year 91? Check the values of predator and prey on your spreadsheet for year 91. What does.82 prey really mean? Why does the predator population decline so sharply after this? (That is, does this make sense biologically?) Year 91

14. 14 Question 3: How do both predators and prey affect each other? (Cont.) Now you can do some interesting experiments. See what happens when you change the variables one by one. Are the results predictable? See if you can get the phase-plane trajectory to spiral inwards instead of spiraling outwards. ( I was not able to do this). Do you think there is an equilibrium point here? That would be a point that after both predator and prey have those values, there are no further changes. It should be in the center of the spiral. See if you can find it by changing the starting populations. Close, but not quite at equilibrium.

15. 15 Concept 1. Interpreting equations - What does an algebraic expression mean? Concept 2. Graphs – X Y scatter plots 1. In the equation for increases in predator population (below), what would happen to the predator population if you increased the death rate of the predator (d)? (You can check your answer by using the spreadsheet you created). I P = c p R P – d P 2.Here is an equation for the net caloric value of a food item, in which V = net value of the item C = caloric value of the food item F = calories spent finding the item and, H = calories spent handling the item V = C – (F + H) What happens to the value if you increase C? What happens if you increase F? What happens if you increase H? Express this equation as a sentence. 3.Write an equation that shows how the value of a food item (V) changes with the percent of sugar (S) in the item, its total weight (W), and the percent of tannin (T, which inhibits digestion). End of module assignment

16. 16 4. Print out the graphs from Slides 12 and 13. In the phase-plane graph, identify the part of the graph where the prey population increases. Circle this area in red. Circle the part where the prey decreases in blue. Now, find the part where the predator increases and circle it with a dotted line. Circle the area in which the predator decreases with a dashed line. Label each of these. a. Mark one of the points in the phase-plane graph where both predators and prey are increasing and label it 1. Mark this same point in time with a 1 on the time series graph for the predator and prey. Find a second point in both graphs where predators are increasing and prey are decreasing and label it with a 2 in both graphs. Find a third where predators are decreasing and prey are increasing. Label this 3 and label one where both are decreasing with a 4. b.Explain the biological reasons why the changes are occurring at each of the four points you labeled. Why is each species changing in the direction described? End of module assignment (Cont.)

17. 17 Using a Spreadsheet – Data Input and formulas For absolute cell references and graphs, click here. To return to the module, click here. Filling numbers down. The numbers in Cells B3 through B7 can just be typed in. As an alternative, Excel can do this for you. Type in the first three values then highlight them (B3 through B5) and place the cursor at the bottom right of the last highlighted cell until you see a small cross. Hold down the left mouse button, drag the pattern through as many cells as you want, and release the button to fill the cells. Excel recognizes the pattern from the first three cells and copies it. Writing equations If you want to multiply each of these numbers by 2, you can create a recurrence formula to perform this task. In Cell C3, type the formula as shown. (All formulas begin with =.) Filling down equations You can copy the formula by clicking on Cell C3 and placing the cursor on the bottom right-hand corner of the cell until you see a small cross. Then drag the cursor down the column, and your results will be displayed.

18. 18 Using a Spreadsheet: Equations, absolute cell references, and graphs When the formula in Cell D3 is copied, the cell referenced in the numerator of the formula will adjust row by row, but the cell referenced in the denominator remains fixed. Absolute Cell references Suppose you always want to divide the numbers in Column C by the same number – let’s use 10 for an example. You could create a formula for the first cell in Column C, =C3/10, and drag the formula down Column D as described in Slide 5. Suppose, though, that you might want to divide by a value in a particular cell. So that you don’t have to change the formula for each value in Column C, you can reference the cell (here, C9) in the formula. In your formula, you refer to this cell as an “absolute” (or “fixed”) cell whose position doesn’t change when you copy the formula. To indicate that this cell is absolute, precede both the column and the row number with a dollar sign. Graphs You can make a graph by highlighting a range of data (here, from B3 to C7) and then clicking on the chart wizard button: Select a graph type (in this case, an X-Y scatter plot connected by a smooth line) and follow the prompts. Voila! A graph! Return to module

19. 19 Using a Spreadsheet – Adding another column of data to a graph Adding another column to a graph Click on the graph to select it. A “Chart” menu will appear at the top of the page next to “Tools”. From “Chart”, select “Add data”. From the dialog box that appears, select the cells to add ( the column for I P ) and hit OK. Return to module