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