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SSAC2007.QH352.BS1.2 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 Core Quantitative issue Forward modeling Supporting Quantitative concepts Rate of change XY scatter plots Exponential growth Logistic Growth Prepared for SSAC by Ben Steele – Colby-Sawyer College © The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2007 Ulva 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. 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. source: http://en.wikipedia.org/wiki/Mohua Source: http://en.wikipedia.org/wiki/Brown_Rat 2 Problem 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. Source: http://www2.nature.nps.gov/YearinReview/yir2003/06_C.html The question: As this population increases, when will it reach its maximum population? 3 The simple model: exponential growth When will the yellowhead population reach the maximum? To answer this problem we will need two things: 1. What is the maximum population? 2. How fast will the population grow? Task 2. What can cause the population to grow or decrease 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) Task 2 -growth due to birth and immigration -decrease due to death and emigration 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 267/.1 = 2670 birds/.1ha 4 The simple model: exponential growth (cont.) The exponential growth model is dN/dt = r N Where: r = reproductive rate (b-d) Increase in population between generation t and generation t + 1 Reproductive rate (new individuals per existing individual each generation) 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 It, the increase during one generation at time t. The resulting equation is at the right. (But you should consider taking calculus next semester.) It = r Nt The size of the population at generation t What happens to this increase when r is bigger? What happens when N is bigger? 5 The simple model: exponential growth (cont.) When will the Yellowhead population reach the maximum? We will use an Excel spreadsheet to answer this question C B 2 year 3 t 4 2001 5 2002 6 2003 7 2004 8 2005 9 2006 10 2007 11 2008 12 2009 13 2010 14 2011 15 2012 16 2013 17 2014 18 2015 D E F G population Increase Birth rate Death rate Reproductive rate N 27.00 40.50 60.75 91.13 136.69 I 13.50 20.25 30.38 45.56 68.34 205.03 307.55 461.32 691.98 1037.97 1556.96 2335.43 3503.15 5254.73 7882.09 102.52 153.77 230.66 345.99 518.99 778.48 1167.72 1751.58 2627.36 3941.05 b 1.5 d 1 = cell with a number in it r 0.50 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). To do this you will need to use autofill, enter equations. and use absolute cell references. Click on these links if you do not know how to do these. 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. = cell with a formula in it r = b – d = .5 rN = .5 X 27 = 13.5 6 2002pop = 27 + 13.5 = 40.5 Max in 2012 The simple model: exponential growth (cont.) Now graph the results and examine the graph. 2 year 3 t 4 2001 5 2002 6 2003 7 2004 8 2005 9 2006 10 2007 11 2008 12 2009 13 2010 14 2011 15 2012 16 2013 17 2014 18 2015 C D E F Look at this graph of exponential growth. G populatio n Increase Birth rate Death rate Reproductive rate N 27.00 40.50 60.75 91.13 136.69 205.03 307.55 461.32 691.98 1037.97 1556.96 2335.43 3503.15 5254.73 7882.09 I 13.50 20.25 30.38 45.56 68.34 102.52 153.77 230.66 345.99 518.99 778.48 1167.72 1751.58 2627.36 3941.05 1. How would you describe a curve like this? It curves, but how? 2. When is growth rate the fastest? 7000 3. When is it the lowest? 6000 4. What is the difference between the growth rate in 2003 and 2008? b 1.5 d 1 r 0.50 9000 8000 Population B Use an XY scatter graph so that each value is tied to a year Need help? 5000 4000 3000 2000 1000 0 2000 2005 2010 2015 2020 Year Note that you can get these answers by estimating from the graph or from reading values on the spreadsheet. And finally: When will the Yellowhead population reach the maximum? Look back at Slide 4 if you forgot what you calculated for the maximum. dN/dt = rmaxN 2003 = 60.75 x .5 = 30.4 2008 = 230.66 7 max = 2670 The simple model: exponential growth (cont.) 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? 2 year 3 t 4 2001 5 2002 6 2003 7 2004 8 2005 9 2006 10 2007 11 2008 12 2009 13 2010 14 2011 15 2012 16 2013 17 2014 18 2015 C D E F G population Increase Birth rate Death rate Reproductive rate N 10.00 15.00 22.50 33.75 50.63 75.94 113.91 170.86 256.29 384.43 576.65 864.98 1297.46 1946.20 2919.29 I 5.00 7.50 11.25 16.88 25.31 37.97 56.95 85.43 128.14 192.22 288.33 432.49 648.73 973.10 1459.65 b 1.5 d 1 r 0.50 3500 3000 2500 Population B You can change the value in Cell C4. You should get this: 2000 1500 1000 500 0 2000 2002 2004 2006 2008 2010 2012 2014 2016 Year 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 A more complex model: Logistic growth 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? What will the population be in 2020? Is your answer reasonable? Possible? Autofill the first three columns further down to answer the question. 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: Reproductive rate Increase in population It = r Nt (K-Nt)/K The size of the population Carrying capacity 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? 9 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: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 year t 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 C D E F G population Increase Birth rate Death rate Reproductive rate N 27.00 40.36 60.24 89.68 133.01 196.21 287.10 415.22 590.54 820.51 1104.69 1428.51 1760.62 2060.44 2295.64 2456.58 2554.76 2609.89 2639.27 2654.46 I 13.36 19.88 29.44 43.33 63.19 90.90 128.12 175.32 229.96 284.18 323.82 332.11 299.83 235.20 160.93 98.18 55.13 29.38 15.19 7.73 b 1.5 d 1 r 0.50 3000 2500 Population B 2000 1500 1000 500 0 2000 2005 2010 2015 2020 2025 Year 10 End of Module Assignment 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. 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? 3000 Population 2500 2000 1500 1000 500 0 2000 2005 2010 2015 2020 2025 Year 11 PRACTICE PROBLEMS 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 Pre and Post test Which of these graphs is 1. 2. 3. 4. Linear growth Logistic growth Exponential growth None of the above A B C D 5. In each of the diagrams above describe where the greatest rate of change occurs. Earnings 15000 10000 5000 0 2001 2004 2005 2006 Year 2001 14000 12000 10000 8000 6000 4000 2000 0 2000 2004 2005 2006 Earnings Which of the following is a 6. Line graph 7. XY scatter graph 8. Column graph 9. Pie graph 14000 12000 10000 8000 6000 4000 2000 0 2002 2004 Year 2006 2008 18 Pre and Post test (cont.) 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 19