1. Introducing Endangered Birds to Ulva Island, NZ Modeling Exponential and Logistic Growth of the Yellowhead Population Because populations are numbers, we can model them mathematically. Given information about birth and death rates, we can predict the future size of a population size. Exponential growth occurs when resources like food are unlimited, and logistic growth occurs if there is a carrying capacity that limits the population Prepared for SSAC by Ben Steele – Colby-Sawyer College © The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2007 Core Quantitative issue Forward modeling SSAC2007.QH352.BS1.2 Supporting Quantitative concepts Rate of change XY scatter plots Exponential growth Logistic Growth Ulva

2. Slides 4-8 model exponential growth on Ulva. We will experiment with the effects of changing the initial population size or the reproductive rate. Slides 9-10 add in the effect of carrying capacity, creating logistic growth. Again we do experiments. Overview of Module Modeling populations with mathematics enables us to predict what a population will do in the future and also points out the concept of exponential growth. Any species that is capable of reproducing more than one individual per adult (all species) is capable of exponential population growth. However, exponential growth is rarely seen because most populations are already at their carrying capacity, the maximum number of individuals that an environment will support. One instance in which a species can demonstrate exponential growth is when a few individuals are introduced to a new environment. This has happened in New Zealand with endangered species. Birds in New Zealand evolved with no land predators, and consequently have no defenses against nest predation. Europeans introduced weasels, possums, domestic cats, and rats that decimated many bird populations on the main islands but many smaller islands remained predator free. Consequently, a strategy for preserving endangered birds is to relocate populations to remote islands. Hence the relocation of Yellowheads to Ulva. In this module we predict what will happen to this population. 2 source: http://en.wikipedia.org/wiki/Mohua Source: http://en.wikipedia.org/wiki/Brown_Rat

3. In 2001, 27 Yellowheads were relocated to Ulva Island. Rats had been eradicated and other predators never existed on the island. We will assume that Yellowheads have exclusive territories that are 0.1 hectares (ha). The island is 267 ha. We want to know when they will cover the island. Problem The question: As this population increases, when will it reach its maximum population? 3 Source: http://www2.nature.nps.gov/YearinReview/yir2003/06_C.html

4. When will the yellowhead population reach the maximum? Because Yellowheads do not like to fly over water, we will just consider birth (b) and death (d) rates, usually combined into a reproductive rate, r by: r = b – d To answer this problem we will need two things: 1.What is the maximum population? 2.How fast will the population grow? Task 1. If the island is 267 ha and each bird needs 0.1 ha, what is the maximum population? (A hectare is 100m by 100m) The simple model: exponential growth Task 2 -growth due to birth and immigration -decrease due to death and emigration Task 2. What can cause the population to grow or decrease 4

5. The exponential growth model is dN/dt = r N Where: r = reproductive rate (b-d) N = the number in the population, and dN/dt is the change in the population per unit time. If you have taken calculus you will recognize this format, but calculus is not required for this problem. We will replace dN/dt with I t, the increase during one generation at time t. The resulting equation is at the right. (But you should consider taking calculus next semester.) What happens to this increase when r is bigger? What happens when N is bigger? The simple model: exponential growth (cont.) I t = r N t Increase in population between generation t and generation t + 1 Reproductive rate (new individuals per existing individual each generation) The size of the population at generation t 5

6. When will the Yellowhead population reach the maximum? We will use an Excel spreadsheet to answer this question = cell with a number in it = cell with a formula in it Set up a spreadsheet that looks like this with the formula from the previous slide in Cell D4 and the formula from Slide 4 in Cell G4. C4 needs an equation too. It is the previous population (C4) plus the increase (D4). In this example, assume that birth rate (b) is 1.5 per individual in the population (3 birds survive from each nest tended by two adults), and death rate (d) is 1. These rates are per year. The simple model: exponential growth (cont.) To do this you will need to use autofill, enter equations. autofillenter equations and use absolute cell references. Click on these links if you do not know how to do these.absolute cell references 6 B CDEFG 2yearpopulationIncreaseBirth rateDeath rateReproductive rate 3t NIbdr 42001 27.0013.501.510.50 52002 40.5020.25 62003 60.7530.38 72004 91.1345.56 82005 136.6968.34 92006 205.03102.52 102007 307.55153.77 112008 461.32230.66 122009 691.98345.99 132010 1037.97518.99 142011 1556.96778.48 152012 2335.431167.72 162013 3503.151751.58 172014 5254.732627.36 182015 7882.093941.05

7. And finally: When will the Yellowhead population reach the maximum? Look back at Slide 4 if you forgot what you calculated for the maximum. Now graph the results and examine the graph. Look at this graph of exponential growth. 1.How would you describe a curve like this? It curves, but how? 2.When is growth rate the fastest? 3.When is it the lowest? 4.What is the difference between the growth rate in 2003 and 2008? Use an XY scatter graph so that each value is tied to a year Need help?Need help? The simple model: exponential growth (cont.) Note that you can get these answers by estimating from the graph or from reading values on the spreadsheet. 7 B CDEFG 2year populatio nIncreaseBirth rateDeath rateReproductive rate 3t NIbdr 42001 27.0013.501.510.50 52002 40.5020.25 62003 60.7530.38 72004 91.1345.56 82005 136.6968.34 92006 205.03102.52 102007 307.55153.77 112008 461.32230.66 122009 691.98345.99 132010 1037.97518.99 142011 1556.96778.48 152012 2335.431167.72 162013 3503.151751.58 172014 5254.732627.36 182015 7882.093941.05

8. Now we will see the value of this model. We can do modeling experiments. How long would it take to reach the maximum if they introduced only 10 birds? You can change the value in Cell C4. You should get this: The simple model: exponential growth (cont.) Now, what is the effect of increasing birth rate? Would it be a benefit to feed the birds so birth rate increased to 2.0? Do several experiments and write a general statement about how strongly birth rate affects final population compared to changing the initial population. Compare your experimental results with your predictions on Slide 5. 8 B CDEFG 2yearpopulationIncreaseBirth rateDeath rateReproductive rate 3t NIbdr 42001 10.005.001.510.50 52002 15.007.50 62003 22.5011.25 72004 33.7516.88 82005 50.6325.31 92006 75.9437.97 102007 113.9156.95 112008 170.8685.43 122009 256.29128.14 132010 384.43192.22 142011 576.65288.33 152012 864.98432.49 162013 1297.46648.73 172014 1946.20973.10 182015 2919.291459.65

9. Look carefully at what we added: (K-N)/K What happens to this quantity when N is very low (zero or almost zero). What is the effect on I? What happens when N is at or very close to K? What is the effect on I? But is the exponential growth model realistic? Our exponential growth model suggests that the population overshoots the maximum. What happens after that? If you expanded the model to more years, what would happen? A more complex model: Logistic growth Autofill the first three columns further down to answer the question. What will the population be in 2020? Is your answer reasonable? Possible? A better model might predict that as the population approaches carrying capacity, the growth would slow down. As birds become more crowded, there may be less food or nesting sites. The Logistic Equation (our difference version of it) is: I t = r N t (K-N t )/K Increase in population Reproductive rate The size of the population Carrying capacity 9

10. A more complex model: Logistic growth (cont.) To convert the model to a logistic model, we need to change the equation in the spreadsheet, Cell D4, to the equation on Slide 9 (and then fill it down). Make sure you get the parentheses correct. Then change the starting population back to 27. Use the maximum value (2670, right?) for carrying capacity (K). To make the graph look better, extend it out to 2020. It should look like this: 10 B CDEFG 2yearpopulationIncreaseBirth rateDeath rateReproductive rate 3t NIbdr 42001 27.0013.361.510.50 52002 40.3619.88 62003 60.2429.44 72004 89.6843.33 82005 133.0163.19 92006 196.2190.90 102007 287.10128.12 112008 415.22175.32 122009 590.54229.96 132010 820.51284.18 142011 1104.69323.82 152012 1428.51332.11 162013 1760.62299.83 172014 2060.44235.20 182015 2295.64160.93 192016 2456.5898.18 202017 2554.7655.13 212018 2609.8929.38 222019 2639.2715.19 232020 2654.467.73

11. End of Module Assignment Analysis Look at the graph or the spreadsheet. 1.When is the growth rate the greatest? 2.How do you identify this on the graph? 3.Why is growth rate low at the beginning? (Look at the equation and explain why.) 4.Why is growth rate low near the end? (Look at the equation and explain why.) 5.What do you think would happen if you extend the model out to 2050? The logistic growth model is much more realistic, right? Although we still are not considering factors such as fluctuating food supplies, changes in weather, disease, competing species. However, we can do analyses and numerical experiments. 11

12. End of Module Assignment (cont.) Experiments Change the variables. 1.What is the effect of raising the birth rate to 2.0? 2.What is the effect of raising the birth rate to 3.0? How would you describe these population changes? 3.What is the effect of raising the birth rate to 4.0? If this high birth rate were possible for the Yellowhead by some sort of management, would it be a good idea for preserving the species? 4.Change b and d back to 1.5 and 1. What would be the effect of introducing 10 Yellowheads rather than 27? How about 2? 5.Now vary the carrying capacity. What if each bird used 1 ha (K = 267)? When would K be reached? How about if K = 6000? Note that to answer the last question you will need to change the value of K in both places in Cell D4, hit “enter”, and then fill that column down to the bottom 12

13. 13 You will often want to fill a whole column with the same function. However, typing a function over and over and changing the cell reference is inefficient and time- consuming. Excel Lesson: Autofill Excel has a built-in feature that accounts for this and recognizes patterns in functions and numbers. Simply select the cell containing the function you wish to copy. Click the small black box in the lower-right corner of the highlighted cell (your cursor should turn into a thin black cross when you’ve reached the right spot) and drag down until the box extends to the desired row. When you release the mouse button, your function will be copied in the selected cells and the cell references adjusted as necessary. The same feature can be used to fill in a column of numbers in a sequence. For example, the temperatures (F) in the spreadsheet at right can be created by typing 0.0 and 2.0 in the first two boxes and dragging down as above. Return to the module Or Learn about entering equations

14. 14 Excel Lesson: Entering Equations Formatting simple functions is like typing them in a scientific calculator. The equation f(x) =(2x6)+3 would be altered to = (2*6)+3 (replacing “x” with “*” to avoid confusion of “x” as a multiplication symbol with “x” as a variable). When this function is typed into an Excel cell... Excel uses a specific formatting style to distinguish text or numbers from equations. The primary feature of the Excel equation is the equal symbol (=). All functions are preceded with this symbol.... the function disappears from the cell and is replaced with the value 15. Return to the module Or Learn more about entering equations

15. 15 Excel can also calculate equations using values in other cells. In this example, temperature values in Fahrenheit were entered in Column A. Excel Lesson: More about Equations To convert these values into degrees Celsius, first you need the conversion equation, then you need to translate that into “Excel-speak”. Beginning with 0° F (Cell A2) the conversion equation, turns into =(A2-32)*(5/9) Remember that order of operations is important and parentheses should be used accordingly. When this equation is dragged down in column B, the reference A2 changes appropriately. Return to the module Or Learn about absolute cell references

16. 16 Excel Lesson: Absolute cell references 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. 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 before. 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. Absolute Cell references Return to the module

17. 17 Excel Lesson: Scatter 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 Graphs If you have Office 2007, It will be a little different. Select the “Insert” ribbon and find the scatter graph. Select the one with a line between the points.

18. Which of these graphs is 1. Linear growth A B 2. Logistic growth 3. Exponential growth 4. None of the above C D 5. In each of the diagrams above describe where the greatest rate of change occurs. Which of the following is a 6. Line graph 7. XY scatter graph 8. Column graph 9. Pie graph Pre and Post test 18

19. Pre and Post test (cont.) 19 10.A model is (circle all that are correct) A.A woman who shows people new clothing styles by wearing them B.A rate of flow in a river C.A series of equations that predict the behavior of a system D.A small airplane made out of balsa wood E.Y= 35 x + 102 F.Y= 35x + 102, where x is time an auto repair takes and Y is the total cost. G.Mice that are used for testing cancer drugs H.The area of a circle 11.What kind of growth (linear, exponential or logistic) would you expect in A.Compound interest in a savings account B.Height as a person grows from a baby into an adult C.Distance traveled as you proceed along a highway (at constant speed) D.Your speed as you accelerate up to the speed limit from a stop light