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 byBen Steele – Colby-Sawyer College

© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2007

Core Quantitative issueForward modeling

SSAC2007.QH352.BS1.2

Supporting Quantitative conceptsRate of changeXY scatter plotsExponential growthLogistic Growth

Ulva

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

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?

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Source: http://www2.nature.nps.gov/YearinReview/yir2003/06_C.html

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

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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 It, 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.)

It = r Nt

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

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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.

and use absolute cell references. Click on these links if you do not

know how to do these.

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  B C D E F G

2 year population Increase Birth rate Death rate Reproductive rate

3 t N I b d r4 2001 27.00 13.50 1.5 1 0.505 2002 40.50 20.25      6 2003 60.75 30.38      7 2004 91.13 45.56      8 2005 136.69 68.34      

9 2006 205.03 102.52      10 2007 307.55 153.77      11 2008 461.32 230.66      12 2009 691.98 345.99      13 2010 1037.97 518.99      14 2011 1556.96 778.48      15 2012 2335.43 1167.72      16 2013 3503.15 1751.58      17 2014 5254.73 2627.36      18 2015 7882.09 3941.05      

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?

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.

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  B C D E F G

2 yearpopulatio

n Increase Birth rate Death rate Reproductive rate

3 t N I b d r4 2001 27.00 13.50 1.5 1 0.505 2002 40.50 20.25      6 2003 60.75 30.38      7 2004 91.13 45.56      8 2005 136.69 68.34      9 2006 205.03 102.52      10 2007 307.55 153.77      11 2008 461.32 230.66      12 2009 691.98 345.99      13 2010 1037.97 518.99      14 2011 1556.96 778.48      15 2012 2335.43 1167.72      16 2013 3503.15 1751.58      17 2014 5254.73 2627.36      18 2015 7882.09 3941.05      

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.

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  B C D E F G

2 year population Increase Birth rate Death rate Reproductive rate

3 t N I b d r4 2001 10.00 5.00 1.5 1 0.505 2002 15.00 7.50      6 2003 22.50 11.25      7 2004 33.75 16.88      8 2005 50.63 25.31      9 2006 75.94 37.97      10 2007 113.91 56.95      11 2008 170.86 85.43      12 2009 256.29 128.14      13 2010 384.43 192.22      14 2011 576.65 288.33      15 2012 864.98 432.49      16 2013 1297.46 648.73      17 2014 1946.20 973.10      18 2015 2919.29 1459.65      

Look carefully at what we added: (K-N)/KWhat 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:

It = r Nt (K-Nt)/K

Increase in population Reproductive rate

The size of the population Carrying capacityCarrying capacity

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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:

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  B C D E F G

2 year population Increase Birth rate Death rate Reproductive rate

3 t N I b d r4 2001 27.00 13.36 1.5 1 0.505 2002 40.36 19.88      6 2003 60.24 29.44      7 2004 89.68 43.33      8 2005 133.01 63.19      9 2006 196.21 90.90      

10 2007 287.10 128.12      11 2008 415.22 175.32      12 2009 590.54 229.96      13 2010 820.51 284.18      14 2011 1104.69 323.82      15 2012 1428.51 332.11      16 2013 1760.62 299.83      17 2014 2060.44 235.20      18 2015 2295.64 160.93      19 2016 2456.58 98.18      20 2017 2554.76 55.13      21 2018 2609.89 29.38      22 2019 2639.27 15.19      23 2020 2654.46 7.73      

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.

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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

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Which of these graphs is

1. Linear growth A B2. Logistic growth3. Exponential growth4. 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 a6. Line graph7. XY scatter graph8. Column graph9. Pie graph

Pre and Post test

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Pre and Post test (cont.)

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10. A model is (circle all that are correct)A. A woman who shows people new clothing styles by wearing themB. A rate of flow in a riverC. A series of equations that predict the behavior of a systemD. A small airplane made out of balsa woodE. Y= 35 x + 102F. 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 drugsH. The area of a circle

11. What kind of growth (linear, exponential or logistic) would you expect inA. Compound interest in a savings account B. Height as a person grows from a baby into an adultC. 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