PopulationDynamics-AfricanWildlifeCaseStudies_F20 (1)
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African Wildlife Case Studies
Population Dynamics
INTRODUCTION Mathematical models can be used to answer questions, solve problems, and make predictions about all kinds of populations. In this activity, you’ll use the exponential and logistic growth models in the Pop4rulation Dynamics
Click & Learn to investigate three different populations of African animals. The models and analyses you’ll use here can be used for many other types of populations as well. To use this software, you click on the linked text above then click on the box that says “Launch Interactive” in the upper left corner. We strongly recommend reading the introductory text. This is what we are covering in lecture and covered a little bit in the Mathbench module, but experience indicates that it takes a while for these terms to all make sense (hence all the activities!)
PART 1: Waterbuck
Africa is home to many different kinds of animals, including large antelope called waterbuck
that live near lakes and rivers. In certain areas, waterbuck populations are declining due to hunting and habitat loss.
1.
How could we use mathematical models to help waterbuck and other wildlife? Mathematical models serve as crucial instruments for comprehending the relationships between wildlife populations and their habitats, aiding in the development of conservation and management approaches grounded in evidence for species like waterbuck, which confront threats such as hunting and habitat degradation. These models enable the examination of population dynamics, habitat characterization, the ramifications of hunting, spatial ecology, disease prediction, and the influence of climate change.
Let’s investigate the waterbuck population in Gorongosa National Park, Mozambique. In the 1970s and 1980s, Mozambique experienced an intense civil war, and most of the waterbuck were killed to provide food and money for soldiers. After the war ended in 1992, many people worked together to rebuild the park. Scientists developed mathematical models to better understand how the park’s waterbuck population recovered afterward, and to help make decisions about managing this population in the future.
2.
What are the advantages of using a mathematical model to study a population rather than just observing the population? Employing a mathematical model to examine a population offers distinct advantages over solely observing the population. It furnishes a structured framework for scrutinizing population dynamics, forecasting future patterns, validating hypotheses, and refining management strategies.
An early model of the waterbuck population was based on the exponential growth model
, which is described in the “Exponential growth model” section of the Population Dynamics
Click & Learn. The population’s maximum per capita growth rate (
r
) was estimated as the difference between its per capita birth rate (
b
), the number of births per individual per unit time, and its per capita death rate
(
d
), the number of deaths per individual per unit
time:
r
=
b
−
d
Activity Modified from Population Dynamics: African Wildlife Case Studies; www.BioInteractive.org
Updated August 2020
Page 1 of 7
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Student Worksheet
Population Dynamics
3.
At the start of the recovery period, the waterbuck population contained only 140 individuals. The population
had 0.67 births per year per individual and 0.06 deaths per year per individual. a.
What is the maximum per capita growth rate (
r
) for this population? Include units in your answer. It is 0.61 or 61% per year.
b.
What is the initial population size (
N
0
) for this population? Include units in your answer.
The initial size is 140.
Go to the “Simulator”
section under the “Exponential growth model” tab in the Population Dynamics
Click & Learn.
Fill in the simulator settings based on your answers above. (
Note:
The simulator doesn’t show units for times or rates because many units are possible. In these examples, we’ll use “years” as our unit for time and “per year” as our units for per capita rates.)
4.
Using the simulator, fill in the following table with the population size (
N
) and population growth rate (
dN/dt
) at different time points (
t
, measured in years). Time (
t
)
5
10
15
20
25
Population size (
N
)
2956
62420
1,318,022
27,803,481
587,650,195
Population growth rate (
dN/dt
)
1803.25
38076.25
803,993.21
16,976,593.51
358,466,619.04
5.
Based on this model, how will the waterbuck population grow over time? Will the population ever stop growing or get smaller?
According to this model, the waterbuck population exhibits continuous growth over time, with no cessation or decline in size.
6.
Do you think this model reflects how the waterbuck population will grow in real life? Why or why not? In reality, I believe the population is improbable to exhibit such continuous growth due to factors such as food availability, habitat space, and disease prevalence that must be taken into account.
7.
Imagine that a decrease in the number of predators lowered the per capita death rate of the waterbuck to 0.04 deaths per year per individual.
a.
What would be the new maximum per capita growth rate (
r
)
for the waterbuck population? It would be 0.63 per year.
Activity Modified from Population Dynamics: African Wildlife Case Studies; www.BioInteractive.org
Updated August 2020
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Student Worksheet
Population Dynamics
b.
What would be the population size (
N
) after 20 years (
t
= 20)? Use the same N
0
as in Question 3. The population would be 41,518,199.
We originally estimated r
as the difference between the per capita birth rate (
b
) and the per capita death rate (
d
). However, r
is also affected by other processes, such as immigration (movement of individuals into
a population) and emigration (movement of individuals out
of a population). Let i represent the per capita immigration rate and m
represent the per capita emigration rate. The equation for r
can be updated to:
r
=(
b
−
d
)+(
i
−
m
)
8.
Imagine that new waterbuck immigrate into the park at a rate of 0.25 per year. Assume that there are no emigrations and that the rest of the population parameters are the same as in Question 3.
a.
What would be the population size after 20 years (
t
= 20)? The population would be 4,130,409,628.
b.
How does the size of the population with
immigration (your answer to Part A) compare to the size of the
population without
immigration (your result for t
= 20 in Table 1)? The population size including immigration surpasses that of the population without immigration. Minor alterations yield varying impacts on the exponential growth model.
PART 2: Kudu
Another type of African antelope is the kudu. Like waterbuck, many kudu have lost their habitat due to human activities. Male kudu are also hunted for their large spiraled horns, which are taken as trophies. As with waterbuck, developing population models for kudu can help us learn more about them.
Most populations, including those of the waterbuck and kudu, cannot grow forever. They are limited by factors such as food or space, which keep a population from getting too large.
9.
Besides food and space, what are two
other factors that could limit the size of a population? The heightened
incidence of diseases and parasitism in a large population may stem from factors such as pollution loss or natural disasters.
One model that includes the effect of limiting factors is the logistic growth model
, which is described in the “Logistic growth model” section of the Population Dynamics
Click & Learn. In this model, a population has an upper limit to its growth called the carrying capacity
(
K
), which is the largest size of a population that the environment can support in the long run.
Imagine a national park with an initial population of 10 kudu, which have a maximum per capita growth rate of 0.26 per year. The park can support a maximum of 100 kudu in the long run.
10.
What are the values of K
, r
, and N
0
for this kudu population?
Go to the “Simulator”
section under the “Logistic growth model” tab in the Population Dynamics
Click & Learn.
Fill in the simulator settings based on your answers above.
Activity Modified from Population Dynamics: African Wildlife Case Studies; www.BioInteractive.org
Updated August 2020
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11.
Based on this model, about how many years will it take the kudu population to reach the carrying capacity? (
Hint
: You may want to change the “Max” values for the axes on Plot 1 to get a better look at the curve.) It was around 29 years.
12.
What will happen to the population growth rate (
dN/dt
) as the population size (
N
) gets closer and closer to the carrying capacity? The growth will increase until it reaches 50, or half of the carrying capacity. The growth will decrease to 0 when the population gets closer to the carrying capacity, which is 100.
13.
Imagine that more land is added to the park, allowing it to support up to 250 kudu. How will the size of the kudu population change once this land is added? The population will be growing until it reaches the carrying capacity, which will be 250 kudu.
14.
Reset the model to the values you determined in Question 10. Now imagine that trophy hunters start killing kudu in the park, which decreases their maximum per capita growth rate to 0.15 per year. How would this impact the kudu’s population size over time? (
Hint:
Look at how many years it will take the population to reach its carrying capacity now.) The population will grow slowly after the trophy hunters kill the kudu. It will
reach its carrying capacity in about 29 years without the trophy hunter; it will not be reached in about 50 years with the hunter.
PART 3: Wildebeest The last type of antelope we’ll investigate is the wildebeest, which are found in eastern and southern Africa. Wildebeest live in giant herds that can contain over a million individuals! The wildebeest herd in the Serengeti region of Tanzania is one of the biggest populations of large herbivores in the world.
Before the 1960s, wildebeest and many other hoofed mammals in the Serengeti were killed by rinderpest, a virus related to the measles virus. In 1960, a campaign began to vaccinate domestic cattle, which were a major source of the virus. Over time, the campaign eliminated rinderpest and allowed many animal populations to recover. To learn more about this – you can check out this video: https://www.youtube.com/watch?
v=Sx3MTTtUw7Y
(it gives a lot of insight into the questions below and is quite well done)
Figure 4 shows the population sizes of two animals, wildebeest and zebras, before and after the rinderpest vaccination campaign.
Activity Modified from Population Dynamics: African Wildlife Case Studies; www.BioInteractive.org
Updated August 2020
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Population Dynamics
Figure 1.
Wildebeest and zebra populations in the Serengeti from the 1950s to 2010.
15.
Based on Figure 4, what kind of population growth model would you use to represent the Serengeti wildebeest population? Why? The population will follow the shape of the logistic growth model. The population size increases, then it becomes slow and supports the constant value of the carrying capacity.
16.
Was the wildebeest population at the carrying capacity in 1968? Why or why not? No, the population in 1968 was not in the carrying capacity because it grows until around the year 1980.
17.
Calculate the size of the wildebeest population in the year 1968, using the logistic model simulator with the following settings: K
= 1,245,000 wildebeest, r
= 0.2717 per year, and N
0
= 80,000 wildebeest in the year 1958. The population was 634,497 wildebeest. 18.
Imagine that the maximum per capita growth rate (
r
) for the wildebeest population increased to 0.4 per year
in 1958.
a.
Suggest a specific reason that r
could increase
for a population. It could be multiple factors that can affect r such as birth rate, death rate, immigration, and emigration. With an increase in birth rate or decrease in death rate, this could increase the r for a population. b.
Recalculate the population size in 1968 using the new r
. You can use the same values for the other settings as in Question 17. The population is 982,852.
Activity Modified from Population Dynamics: African Wildlife Case Studies; www.BioInteractive.org
Updated August 2020
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Population Dynamics
c.
Sketch or describe how the wildebeest population curve in Figure 4 might change as a result of the new r
. The population would be growing much faster and could possibly reach their capacity more quickly.
19.
Imagine that the carrying capacity (
K
) for the wildebeest population decreased to 1,000,000 wildebeest in 1958. a.
Suggest a specific reason that K
could decrease
for a population. The habitat could become smaller; this could be because of human expansion, or the less it rains it would decrease the food source of the animals.
b.
Recalculate the population size in 1968 using the new K
. You can use the same values for the other settings as in Question 17. The population is 568,235.
c.
Sketch or describe how the wildebeest population curve in Figure 4 might change as a result of the new K
. The population would be growing more slowly and would stop growing once it reaches the new smaller carrying capacity.
20.
Look at the size of the zebra population, which is shown as triangles in Figure 4, before and after the rinderpest vaccination campaign.
a.
What patterns or trends do you observe in the zebra population? The size of the zebra does not change, both during and after the rinderpest campaign.
b.
Based on your answer above, what effect does rinderpest have on zebras? There was no effect.
21.
Based on Figure 4, did the zebra population growth rate (
dN/dt
) differ in the years 1985 and 2003? Why or why not? (
Hint
: dN/dt
is equal to the slope of the curve showing population size, N,
over time, t
.) According to the figure, the curve remained relatively flat for both years, indicating that zebra population growth was minimal and exhibited little variation.
22.
Imagine that there is a large wildfire in the Serengeti in 2010. a.
How might the zebra and wildebeest populations change right after the wildfire? Fire would kill both of them so both populations would most likely decrease.
b.
How large do you think the zebra and wildebeest populations would be 50 years after the wildfire? Explain your answer, or what else you would want to know before making a prediction. Over time, assuming the habitat wasn't severely damaged, the population would likely revert to its previous state after the fire. However, this relies on understanding the impact of the fire on the habitat and considering other factors that could influence the population over the next 50 years.
Activity Modified from Population Dynamics: African Wildlife Case Studies; www.BioInteractive.org
Updated August 2020
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23.
We often design population models to answer certain questions. We may leave out other factors that are less relevant to our questions or that could overcomplicate our analysis.
a.
Propose one new
question about the waterbuck, kudu, or wildebeest populations that could be answered using the models you learned about in this activity. What effect would alterations in habitat quality, like the enlargement of water sources or the restoration of grazing areas, have on the carrying capacity of the waterbuck population?
b.
Propose one new
question about the waterbuck, kudu, or wildebeest populations that could not be answered using these models. What could you add to the models in order to answer your question? What role do social dynamics and interactions among kudus play in shaping population dynamics and distribution patterns?
Activity Modified from Population Dynamics: African Wildlife Case Studies; www.BioInteractive.org
Updated August 2020
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