ICE
Inquiry-based Curriculum Enhancement
Lesson Plan: Population Dynamics
General Description
In this activity, students explore exponential and logistic growth of populations by working
with examples and with the mathematical formulas that describe these types of growth.
The basic pattern of this activity is 1) work through an example about population growth, 2)
have students create some predictions about population growth, and 3) describe the
patterns represented in words and formulas. The inquiry lesson in this activity is making
predictions using mathematics.
Objectives
1. Students will learn the basic formulas used by ecologists to describe population
growth.
2. Students will explore mathematical models in a conceptual manner using figures of
population growth.
3. Students will learn vocabulary associated with population ecology.
4. Students will differentiate between exponential and logistic growth.
Concepts
populations, exponential growth, logistic growth, density-dependent population regulation,
mathematical models of growth.
Time: about 50 minutes assuming that the activity is started promptly and the majority of
the time is devoted to student work rather than introductory lectures.
Prerequisite Skills
None
Materials
2 pages of blank tables for students to work on during the activity (r only and r & K)
Overhead package including four figures, one full page of grid, many half pages of grid and
growth figures
Individual Accountability for each student (optional)
Take Home Worksheet for each student (optional)
NOTE: Calculators are required for this activity – remind students to bring one to class.
ICE
Inquiry-based Curriculum Enhancement
UTI Instructions: Population Dynamics
Introduction:
In this activity, students will explore exponential and logistic growth of populations by
working with examples and with the mathematical formulas that describe these types of
growth. The basic pattern of this activity is 1) work through an example about population
growth, 2) have students create some predictions about population growth, and 3) describe
the patterns represented in words and formulas. The inquiry lesson in this activity is
making predictions – these components are indicated by *. Focus students’ attention on
these particular components of the activity.
Procedure:
[3-5 minutes] If the professor for your course has assigned the pre-activity worksheet,
have students warm-up to the activity by reviewing their answers with one or two partners.
Encourage students to focus on creating the best possible answers to the questions by
combining the answers of their group members.
[1 minute] Begin the activity with a short introduction stating that students will explore
population dynamics and factors that can influence populations using both words and
mathematical formulas.
PART 1: EXPONENTIAL GROWTH [TOTAL TIME 19-26 minutes]
[5-8 minutes] Begin the first example of population growth. Assign students to different
jobs. Several students need to be responsible for calculator work – this group checks each
other’s work so that no mistakes are introduced to the example. One student is responsible
for graphing population growth on the overhead (allow this student to make all the decisions
about graphing, even if initial attempts are unsuccessful; suggest consulting a partner).
This is the example:
The kelp population has a starting size of five individuals. Each individual
has a 1 in 2 (50% or 0.50) chance of reproducing. Each individual has a 1 in
4 (25% or 0.25) chance of dying. How many individuals will be in each
successive generation for 15 generations?
As a class, work through the calculations and graphing. This example is the baseline that
students will work with for the rest of the activity.
[3-4 minutes] Describe this pattern as exponential growth, and write the formula they
used on the board:
Nt+1 = Nt + bNt – dNt which simplifies to Nt+1 = Nt + (b-d)Nt
Divide students into teams, and distribute the exponential growth overheads. Assign each
team one alteration to the above scenario – increasing or decreasing birth or death rate.
Some groups will have the same alteration. Have students predict what will happen to
population growth under the altered condition by drawing the new figure on the overhead
(*). The teams must be able to explain their prediction.
[3-4 minutes] Choose four teams representing each possible alteration in initial conditions
to VERY BREIFLY report to the group. Two sentences of explanation are all that are
necessary.
[8-10 minutes] Have the teams write down general rules describing the patterns they see
in the figure they created. For example “increasing the birth rate without changing the
death rate causes the population to grow faster”. Introduce r as the conceptual way that
ecologists represent growth rate (rate is always per capita per unit time). r is the intrinsic
growth rate, and is mathematically b-d (r=b-d). Have students modify their general rules
into the form of r. Also ask students to identify assumptions of exponential growth. Solicit
the rules and assumptions of two or three groups to make sure that the class is on the right
track. Answer any questions. The equation used in the examples now can be written as:
Nt+1 = Nt + rNt
PART 2: LOGISTIC GROWTH [TOTAL TIME 24-29 minutes]
[4-5 minutes] Pose the question: Obviously, populations don’t grow exponentially. Why
not? Allow students to brainstorm a list of suggestions; record all appropriate suggestions
on the board. Hopefully one of the suggestions will be space (nest sites, suitable habitat,
etc. – this factor is the basis for the example of logistic growth. Confirm their ideas by
naming them as density-dependent factors. Briefly summarize what happens when densitydependent factors are regulating population growth.
[4-6 minutes] In their teams, have the students make predictions about the new outcome
of population growth by drawing a figure (*). Many students will probably be familiar with
the logistic growth model, especially if it has been covered in class. Encourage them to
think in conceptual terms. Your job during this time is to find the group that has figured out
what will happen if density-dependent factors are influencing the population. Have this
team report to the class. The new equation is:
Nt+1 = Nt + rNt((K-Nt)/K)
where K represents the carrying capacity of some limiting resource.
[3-4 minutes] Reassign the roles of grapher and calculaters. This is the example:
The kelp population has a starting size of five individuals. Each individual
has a 1 in 2 (50% or 0.50) chance of reproducing. Each individual has a 1 in
4 (25% or 0.25) chance of dying. There are only 50 possible attachment
spots for kelp. How many individuals will be in each successive generation
for 30 generations?
Work through the example, beginning at generation 16 (the previous generations are
already calculated in order to speed up the process). You should find that the population
size is initially similar to the exponential growth example from above, but that population
size starts to lag behind the exponential curve after only a few generations.
[3-4 minutes] Have students predict what will happen if there is a change in the limiting
resource (*). Repeat the example modifying only the number of possible attachment spots
(an example with 100 attachment sites is partially worked out and included). Students
should detect the pattern after this one example. Have one student describe the influence
of number of possible attachment spots on population growth. Record the general rule on
the board.
[10 minutes] Have the teams work out different examples of changing r and K so that all
possible combinations are covered (*). You should create the values for r and K based on
the alterations created above. The overhead package contains overhead half sheets with
the figure of the original example that you should distribute to each team. They should
draw the figure of their particular alteration using their conceptual understanding rather
than perform multiple calculations. After the teams report to the group (drawing their
predictions on the master overhead), answer any questions.
Give students the individual accountability question(s) to finish the class.
Example of K=100 through Generation 15:
gen
N
rN
(K-N)/K
Nt+1
1
5
1
0.95
6
2
6
2
0.94
8
3
8
2
0.92
10
4
10
3
0.90
13
5
13
3
0.87
16
6
16
4
0.84
19
7
19
5
0.81
23
8
23
6
0.77
28
9
28
7
0.72
33
10
33
8
0.67
38
11
38
10
0.62
44
12
44
11
0.56
50
13
50
13
0.50
57
14
57
14
0.43
63
15
63
16
0.37
69
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
160
140
120
population size
100
80
60
40
20
b=0.50, d=0.25
0
1
2
3
4
5
6
7
8
9
generation
10
11
12
13
14
15
60
logistic r=0.25, K=50
50
population size
40
30
20
10
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
generation
160
exponential r=0.25
140
logistic r=0.25, K=50
120
population size
100
80
60
40
20
0
1
2
3
4
5
6
7
8
9
generation
10
11
12
13
14
15
4500
exponential
4000
logistic
3500
population size
3000
2500
2000
1500
1000
500
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
generations
160
140
120
population size
100
80
60
40
20
b=0.50, d=0.25
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
generation
160
140
120
population size
100
80
60
40
20
b=0.50, d=0.25
0
1
2
3
4
5
6
7
8
9
generation
10
11
12
13
14
15
60
logistic r=0.25, K=50
50
population size
40
30
20
10
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
generation
60
logistic r=0.25, K=50
50
population size
40
30
20
10
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
generation
ICE
Inquiry-based Curriculum Enhancement
Pre-Activity Worksheet: Population Dynamics
General Description
In the activity you will do this week during your learning/discussion group, you will be
examining the growth of populations, both mathematically and conceptually. In order to be
prepared for this activity, complete this worksheet.
Reading
Browse the “Population Ecology” chapter in your text. Pay particular attention to figures
53.18, 53.19, and 53.20. Carefully read the section on Population Growth beginning on pg.
1150.
Definitions
Write a definition of the following words. Use your text, textbook glossary, and your
previous knowledge to create the best definition possible. Remember to connect your
definitions to population ecology.
1) population
2) per capita
3) exponential
4) growth rate
5) density dependence
Questions
Answer the following questions. You will explore your answers to these questions in-depth
during learning/discussion group.
1) What are the numerical limits of b (birth rate)? For example, one would represent the
age limits of IU students registered full time in Fall 2002 as 14≤X≤75.
2) What are the numerical limits of d (death rate)?
3) Name three density-independent factors and three density-dependent factors that could
influence population growth.
ICE
Inquiry-based Curriculum Enhancement
Population Dynamics
Generation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
N
b=
d=
B
D
Nt+1
r=0.25
K=50
r=
K=
gen
N
rN
(K-N)/K
Nt+1
gen
1
5
1
0.90
6
1
2
6
2
0.88
8
2
3
8
2
0.84
10
3
4
10
3
0.80
12
4
5
12
3
0.76
14
5
6
14
4
0.72
17
6
7
17
4
0.66
20
7
8
20
5
0.60
23
8
9
23
6
0.54
26
9
10
26
7
0.48
29
10
11
29
7
0.42
32
11
12
32
8
0.36
35
12
13
35
9
0.30
38
13
14
38
10
0.24
40
14
15
40
10
0.20
42
15
16
16
17
17
18
18
19
19
20
20
21
21
22
22
23
23
24
24
25
25
26
26
27
27
28
28
29
29
30
30
N
rN
(K-N)/K
Nt+1
ICE
Inquiry-based Curriculum Enhancement
Individual Accountability: Population Dynamics
Demonstrate your new understanding of population dynamics by answering the following
questions:
(1) Which population will grow faster?
Population 1
Population 2
N=20
N=20
b=1.05
b=0.46
d=0.92
d=0.30
Defend your answer in three or four sentences.
(2) Assume the number of nest sites in this population is 100. Which population will reach
maximum capacity first? Why?
ICE
Inquiry-based Curriculum Enhancement
Take Home Worksheet: Population Dynamics
To practice with the concepts you learned in class today, answer the following questions.
One good strategy for working with these questions is to answer them on your own to the
best of your ability, then compare your answers with a fellow student. Together you will be
able to create good answers to the questions. In all cases, be prepared to explain your
reasoning clearly and succinctly.
1. What does r=1.0 mean, in mathematical and conceptual terms, and in terms of the kelp
example?
2. What are the assumptions of r? What are the assumptions of a population that is
growing at r?
3. What is the maximum value of r?
4. Describe zero population growth in terms of r. If a population is at ZPG, how many
individuals will be in the next generation? How many individuals died in a given year?
5. Examine figure 52.8. Assuming that the births rates are the same for the two
populations, which population had more deaths? Assuming that the death rates are the
same for the two populations, which population had more births?
6. Name three events or factors that are consider density-dependent influences on
population growth. Of these, and the other factors named in class, which do you think is
the most important regulator of populations?
7. Some events can affect a population regardless of its size (density-independent factors).
Name three such events. Draw the population growth of a population in which 50% of
individuals die due to a density-independent event in the 3rd, 8th, and 13th generations.
Do the same thing again, but assume 90% of individuals die. Repeat once more, but in
this case only 10% of individuals die.
8. Which type of regulators on populations is most important – density-dependent or
density-independent? What are the assumptions of your answer?