1. Exponential and Logistic Population growth
SSAC2007.QH540.BS1.2
Exponential and Logistic Population growth
Modeling the population of an endangered bird on an Island in New Zealand
Core Quantitative issue
Forward modeling
Because populations are numbers, we can model them with math. Given what we know about birth and death rates, we can predict what a population size will be in the future. Exponential growth happens when resources like food are unlimited and logistic growth happens if there is a carrying capacity which limits the population
Supporting Quantitative concepts
Rate of change
XY scatter plots
Exponential function
Prepared for SSAC by
Ben Steele – Colby-Sawyer College
© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. 2007

2. Overview of Module
Modeling populations with math enables us to predict what a population will do in the future and also points out the principle of exponential growth. Any species that is capable of reproducing more than one individual per adult (all of them) 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 what is called logistic growth. Again we experiment with changing the parameters.
source:
Source: 2

3. Problem
In 2001, 27 Yellowheads were relocated to Ulva Island, on which rats had been eradicated and other predators never existed. These birds have exclusive territories that are .1 hectares (ha). The island is 267 ha so we need to know when they will cover the entire island.
Source:
The question: As this population increases, when will it reach its maximum population? 3

4. When will the yellowhead population reach the maximum?
The simple model: exponential growth
When will the yellowhead population reach the maximum?
To answer this problem we will need to things:
What is the maximum population?
How fast will the population grow?
Task 1. If the island is 267 ha and each bird needs .1 ha, what is the maximum population? (a hectare is 100m by 100m)
Task 2. This involves answering the following question: What can cause the population to grow or decrease
Task 2
-growth due to birth and immigration
-decrease due to death and emigration
Since 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

5. It = r x Nt The simple model: exponential growth (cont.)
The exponential growth model is
dN/dt = r x 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 if you have not, don’t panic. We will simplify dN/dt to It, the increase during one generation at time t. The resulting equation is at the right. (But you should consider taking calculus next semester.)
Reproductive rate (new individuals per existing individual each generation)
Increase in population between generation t and generation t + 1
It = r x Nt
The size of the population at generation t
What happens to this increase when r is bigger? What happens when N is bigger?

6. When will the yellowhead population reach the maximum?
The simple model: exponential growth (cont.)
When will the yellowhead population reach the maximum?
We will use an excel spreadsheet to answer this question
Set up an 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). B C D E F G 2
year
population
Increase
Birth rate
Death rate
Reproductive rate 3 t N I b d r 4
2001
27.00
13.50
1.5 1
0.50 5
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
518.99 14
2011
778.48 15
2012 16
2013 17
2014 18
2015
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 number in it
= cell with a formula in it

7. The simple model: exponential growth (cont.)
Now graph the results and examine the graph.
Use and XY scatter graph so that each value is tied to a year Need help? B C D E F G 2
year
population
Increase
Birth rate
Death rate
Reproductive rate 3 t N I b d r 4
2001
27.00
13.50
1.5 1
0.50 5
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
518.99 14
2011
778.48 15
2012 16
2013 17
2014 18
2015
Look at this graph of exponential growth.
How would you describe a curve like this. It curves, but how?
When is growth rate the fastest?
When is it the lowest?
What much bigger is the growth rate in 2003 compared to 2008?
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.

8. You can change the value in cell C4. You should get this:
The simple model: exponential growth (cont.)
Now we will see the value of this model. We can do experiments.
How long would it take to reach the maximum if they only introduced 10 birds?
You can change the value in cell C4. You should get this: B C D E F G 2
year
population
Increase
Birth rate
Death rate
Reproductive rate 3 t N I b d r 4
2001
10.00
5.00
1.5 1
0.50 5
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
648.73 17
2014
973.10 18
2015
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.

9. It = r x Nt x (K-Nt)/K A more complex model: Logistic growth
But is this realistic? Our 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 this reasonable? Possible?
Auto fill the first three columns further down to answer this
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:
Increase in population
Reproductive rate
It = r x Nt x (K-Nt)/K
The size of the population
Carrying capacity
Carrying capacity
Look carefully at what we added: (K-N)/K What happens to this amount 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?

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 this look better, extend it out to It should look like this: B C D E F G 2
year
population
Increase
Birth rate
Death rate
Reproductive rate 3 t N I b d r 4
2001
27.00
13.36
1.5 1
0.50 5
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
323.82 15
2012
332.11 16
2013
299.83 17
2014
235.20 18
2015
160.93 19
2016
98.18 20
2017
55.13 21
2018
29.38 22
2019
15.19 23
2020
7.73

11. End of Module Assignment
This is a much more realistic model, right? Although we still are not considering things like fluctuating food supplies, changes in weather, disease, competing species, etc. However, we can do analysis and experiments.
Analysis Look at the graph or the spreadsheet.
When is the growth rate the greatest?
How do you identify this on the graph?
Why is growth rate low at the beginning? (look at the equation and explain why)
Why is growth rate low near the end? (look at the equation and explain why)
What do you think would happen if you extend the model out to 2050?

12. End of Module Assignment (cont)
Experiments Change the variables
What is the effect of raising the birth rate to 2.0?
What is the effect of raising the birth rate to 3.0? How would you describe these dynamics?
What is the effect of raising the birth rate to 4.0? If this were possible for the yellowhead by some sort of management, would it be a good idea for preserving the species?
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?
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

13. Pre and Post test
Which of these graphs is Linear growth A B 2. Logistic growth 3. Exponential growth 4. None of the above C D 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

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