GEOG 110 – Lab #4 – Modeling Predator-Prey Dynamics
Due Date:
11:59 pm November 4, 2005
Objectives:
Familiarize yourself with the Lotka-Volterra model of predator-prey
dynamics by performing some Sensitivity analyses on the provided model.
Then, modify the model to include a third population in the model. Using
Sensitivity analysis, again vary some of the initial population sizes to see
how producer, prey, and predator population sizes effect all three
populations.
Background: Simulating the interactions of populations of organisms in an ecosystem is
another useful application of modeling to complex systems in the
environment. The Lotka-Volterra model of predator-prey dynamics
considers that the state of these populations is coupled, such that one
influences the other, and vice-versa. As was the case with the other
phenomena we have simulated, by constructing such a model for the
predator-prey system, experimenting with its variables can allow us to
reproduce trends in these populations that would be quite difficult to
observe in the environment, because it would be difficult or impossible to
hold all other factors constant while investigating the effect of a single
factor. Thus, simulating these sorts of systems gives an opportunity to
approach and understand phenomena that would be extremely difficult to
grasp purely through field studies. While field studies are tremendously
useful for obtaining the initial values needed to parameterize a model
realistically, the use of modeling allows us to go significantly further in
our exploration of the system dynamics of predator and prey populations.
Resources:
This lab exercise does not include the background and theory required for
you to understand how to study predator-prey dynamics. For that
information, you should look to material from the lectures and the entirety
of Chapter 4 (pp. 95-112) of the course text, paying special attention to
Section 4.2.3 onwards.
At this point, you should have already
successfully completed three labs, and you should be getting comfortable
with using STELLA. You can refer back to the course text’s background
material on Sensitivity analysis (Section 3.5.3, pp. 84-87) to help you
understand how to use Sensitivity analysis to better understand the system
dynamics of the predator-prey model. This lab continues the trend set by
the previous labs, where in each successive lab the procedure becomes less
step-by-step, and more open-ended. You are expected to use the
knowledge you have already acquired in previous labs to use STELLA in
these lab exercises in a less regimented fashion.
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Procedure:
Be sure to read Chapter 4 before attempting this lab, in order to familiarize
yourself with this model’s components and how they function.
1. Get STELLA started and open up the predator-prey dynamics model
provided for this exercise. The model (Chap4c.stm) will be in
j:/isis.unc.edu/html/courses/2005fall/geog/110/001/data (which you
can also access using a web browser as previously described).
2. Let’s take a few moments to familiarize ourselves with the many
components of this model. There are quite a few elements that it
includes that we have not seen previously:

You will notice that the model is enclosed in a pink outline,
and cannot be edited without clicking on the triangle in the top
right corner.

To the right of the model diagram, there is a purple box
containing a brief description of the model.
Below the elements mentioned above, we have a set of controls
that allow us to change some values in the model on the fly, as
the model is running. This model runs in what STELLA terms
flight simulator mode, meaning that if you move the controls as
the model is running, you can have an impact on what it does
(see Section 3.9.2 on p. 94 for a description).
The Dd (Deer death rate as a function of the Wolf population)
and Wb (Wolf birth rate as a function of the Deer population)
parameters are represented graphically in two controls that
allow you draw different functional relationships for them with
your mouse.


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
There are sliders for r (controls inflows to Deer stock), Kd
(death rate as a function of carrying capacity constraint), and
Wd (decline in Wolf population assuming there is no available
prey) that also allow numerical input by clicking on the number
shown. The dashed circle r is the “Ghost” of the solid circle r in
a different position as this makes the diagram looks better (it is
the same r).

There are two dials that control the initial values of the Wolf
and Deer populations
3. Leaving all of the values at their initial values, try running the model.
Click on the graph tools first to create a time series for the Deer
population and the Wolf population. Move the graph to the bottom of
the page and click on the pin to the upper left corner. First run the
model as with the default values as set and see how the two population
change with time. Copy the graph into your lab report and explain.
Then run the model a second time. As the model is running, you can
move the sliders to represent a change in those conditions as time
passes. Change the slider bar one at a time and see how the model
outputs change. You may see many different graphs here. Copy one of
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your favorite into your lab report, and explain what you did and how
what you did translate to what you see [note: You can go into the Run
Specs and change the Sim Speed if you’d like it to go slower or faster,
but do not change the DT as this will affect the models performance in
a way that we do not want to examine].
Now unpin your graph and then double click on it again, the graph
specification dialog comes out. Change the graph type from time series
to scatter [recall what a scatter plot is]. In the empty space to the right,
you will see X and Y. First click on the Deer in the window to the left
as X and then click on the Wolf in the window to the left as Y. Now
run the model with its default values and see what happens. Copy the
graph into your lab report and explain the graph.
In order to get a feel for how the initial sizes of the Wolf and Deer
populations affect the behavior of the model, we will perform a
Sensitivity analysis on these initial values. As usual, explore 3 values
for each of these parameters, using the given value as the mean value
and varying each by 50% in either direction (i.e. 3 runs per variable).
You can make use of the Sensi Specs utility to perform this analysis
(refer back to the instructions from Lab 3 to see how to do this). Copy
your Sensitivity analysis graphs (Deer population versus time and
Wolf population versus time) for the initial Deer population size and
the Wolf population size. Note, unfortunately, Sensi specs graphs do
not seem to work right with scatter plots like the phase space diagram,
nor with multi-page graphs, so you will have to do a Sensi run for each
quantity for which you wish to create a multi-run time series plot.
Now, you are going to need to put your model building skills to the
test, as well as your ability to reason out how to represent some
ecological relationships in STELLA by analogy. Just as the Wolf
population has relationships with the Deer population, in a real
ecosystem, the Deer population is also dependent on the availability of
its source of food. Adding a third population makes the model more
realistic in this sense. We will add a producer species (Grass), a
source of food for the Deer. What you will need to do is to modify the
structure of the model, such that you will have a producer species
(Grass) that has relationships with the Deer birth rate and defines
the depletion of Grass in terms of the number of Deer present
(hint: look the structure of the relationship between the Deer and
Wolves, and set up Grass to relate to the Deer as the Deer relate to the
Wolves). Begin by either returning all the values to their original
settings, or simply reopen the chap4c.stm model to get everything back
to the initial values.
To modify the model, click on the inverted triangle to unlock the
model diagram. When you do so, the outline will turn from pink to red
and open lock icon will appear in the top right corner of the red box.
Once you see this, you will be able to manipulate the model elements.
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8. Resize the red box to make a little bigger so you’ll have more room for
some new model elements. You do this by clicking in the upper part
where the title is; this will make pull-tabs visible at the corners, and
you can click and drag those to resize the box.
9. Select all the model elements and slide them down, to make room for
the Grass model elements that are ideally placed above the Deer
elements.
10. Construct the appropriate Grass reservoir and the various flows and
connections it needs to have with the Deer model elements for your
model to make sense and function realistically. You’ll need to have a
Grass reservoir, as well as inflows and outflows of Grass, and
converters that determine Grass growth and depletion (hint: Use our
basic structure for an exponential population model as a template, and
then make the appropriate connections between the Grass and Deer
portions of the model). For a more realistic model, we could limit the
Grass population in terms of carrying capacity, but this would add
more complexity than we really need to fulfill the goals of this lab.
11. Set the initial value of the Grass stock to 10000.
12. You’ll need to redefine the Deer birth rate (r) in terms of the Grass
population (setting up the dependence of Deer growth on the
availability of Grass). Set it to Grass/40000. Also, you MUST use the
dynamite to remove the r slider control from the canvas; otherwise this
will override the new definition you have entered (hint: This should
give you a good idea of one of the connectors that is required between
the Grass and Deer portions of the model).
13. Set your Grass growth rate converter to equal 0.03.
14. Set the Grass depletion rate converter to be a linear proportion of the
Deer population: Deer/10000 (setting up the dependence of Grass
death on the number of Deer present, again requiring a connector
between the two parts of the model).
15. Copy your new three-reservoir model into your lab report. Describe
the new model components added (lab report part 3 as described
below).
16. Create another Lotka-Volterra phase space diagram, this time with
Deer vs. Grass on the axes (as described in step 4 above). Also create
a time series graph that shows all three population sizes.
17. Run the model and copy the two new graphs into your lab report and
explain what you observe.
18. Perform sensitivity analyses on the initial size of the Grass reservoir
and the growth rate of Grass using the usual approach, and observe
what affect this has on the Grass, Deer and Wolf populations. Copy
the sensitivity analysis results into your lab report and explain your
results.
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What to do:
Follow the procedures described above, saving the specified version of
your three-population model in your course directory (e.g. mine would be
davidten-lab4.stm). Write a lab report that includes the elements specified
below, saving it in a Word document in your course directory, following
the same naming convention as the model (e.g. mine would be davidtenlab4.doc).
The following should be included in your lab report:
1. Objectives
2. Deer-Wolf Model
2.1 Model description
2.2 Model diagram
2.3 Model components:
a) Reservoirs:
b) Processes:
c) Converters:
d) Relationships:
Define
them
mathematically,
e.g.
Births=Birth_rate*Population
Rate Equation for each reservoir: identify the impact of each converter
3. Grass-Deer-Wolf Model
3.1 Model description
3.2 Model diagram
3.3 New model components:
e) Reservoirs:
f) Processes:
g) Converters:
h) Relationships:
Define
them
mathematically,
e.g.
Births=Birth_rate*Population
Rate Equation for each reservoir: identify the impact of each converter
on the rate equation.
3. Results: Put the graphs of your simulation as requested in your lab
instructions above and explain your results.
4. Discussion: How can model simulations like this one be helpful to
wildlife managers with regard to balancing wildlife resources?
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