PopulationDynamics-AfricanWildlifeCaseStudies_F20 (1)
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Click & Learn
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
Click & Learn
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
Page 2 of 7
Click & Learn
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
Page 3 of 7
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