This is a more complicated population model that includes two different groups - predators and
prey. The population of prey in the model grows exponentially according to its birth fraction and
death proportionality constant, while the population of predators in the model grows or shrinks
based on the population of its prey alongside its birth rate and death proportionality constant.
The result is to create one of two states for the model. It is possible for the populations to
reach equilibrium where the number of predators and prey remains roughly stable over time, but it
is also possible for one or both species to die out. Users can set the initial populations, the
rules governing reproduction for both species, and the death rates of both species.

Background Information

Simple population models invariably involve exponential growth or decay, depending on how quickly
the members of the population are reproducing. In the real world, though, nothing can grow
exponentially for very long. Instead, population growth is constrained by a number of other
factors, including the carrying capacity of its environment and the existence of predators. This
model includes the concept of predators in a population model by creating a second population that
can only reproduce by eating the first. Aside from simply slowing down population growth,
predators also tend to help stabilize population levels long-term because the population of
predators grows or shrinks alongside the population of prey.

Science/Math

The fundamental principle behind this model is HAVE = HAD + CHANGE. Each time tick, the following
things happen:

Prey reproduce according to the player-set birth rate

Prey die according to the player-set death rate and the population of predators

Predators reproduce according to the player-set birth rate and the population of prey

Predators die according to the player-set death rate

As the above changes suggest, the CHANGE in this formula does not have a simple, obvious function
for its use. This is relatively common in the real world as well, since scientists rarely have
perfect access to all potential contributing factors to a population's growth or decline. Instead,
the relationship will have to be determined qualitatively and/or experimentally.

Teaching Strategies

An effective way of introducing this model is to ask students to brainstorm how adding a predator
changes the model as compared with simple population models. Ask the following questions:

How would you expect the introduction of a predator to affect population growth among the prey?
Would growth still be exponential, or would it follow some other formula?

Assuming that the population of predators can grow as well as their prey, what are the possible
"end states" of a model with predators and prey? Which of these is the most realistic? Why?

What do you think will happen to the population of prey as the population of predators
increases? Why?

What do you think will happen to the population of predators as the population of prey
increases? Why?

Which of the two do you think will increase "first" (in other words, on a graph of the
populations over time, which peak would come first, or would they come at the same time)? Why?

Have students write down hypotheses for each of these questions so that they can be tested when
the actual model is introduced.

Implementation:

How to use the Model

This relatively in-depth model has a number of parameters that can be manipulated to produce
different results:

The "Prey Population" and "Predator Population" parameters determine the number of hares and
lynx placed on the board at the start of the simulation

The "prey birth fraction" parameter determines the proportion of prey that reproduce each time
step

The "predator birth fraction" parameter determines the proportion of predators that reproduce
each time step

The "prey death proportionality constant" and "predator death proportionality constant"
parameters define the proportion of prey and predators that die each time step

All of the aforementioned parameters are manipulated by clicking and dragging their respective
sliders. The maximum, minimum, and step values for each parameter are pre-set. Any changes made to
the sliders take effect immediately with the exception of the initial values, which take effect
the next time the simulation is run. To run the simulation, click the "Run a Simulation" button.
The results from the simulation are displayed immediately in graphical form. To the right of the
model, two graphs display the populations of predators and prey. The top graph displays both
populations over time, allowing you to see how the populations changed from tick to tick. The
bottom graph displays the populations relative to each other, which creates a looping graph with
just one line. For more information on Vensim, reference the Vensim tutorial at:
http://shodor.org/tutorials/VensimIntroduction/Preliminaries.

Learning Objectives:

Understand the relationship between populations of predators and prey

Understand the effect of each parameter on the populations over time

Objective 1

To accomplish this objective, have students run the simulation with the default parameters, and
observe the graph. They should specifically pay attention to the peaks and valleys in the
populations of prey and predators. Ask the following questions:

From your observations, what happens to the population of predators when the population of prey
is high? Why do you think this happens?

What happens to the population of prey when the population of predators is high? Why?

Do you notice any patterns in the relative populations of the two species? If so, what types of
patterns are they?

Which population peak usually comes first, the prey or the predators? Why do you think this is?

What would happen if there were no predators, only prey? What would happen if there were no
prey, only predators?

Students should compare the hypotheses they made earlier to the results now, and discuss any
discrepancies.

Objective 2

Have students play around with each of the manipulable parameters to see the effect on the graph.
Ask the following questions to guide their exploration:

What changes do you notice in the graph if you change the initial number of prey and/or
predators? Are there any changes to the long-run behavior of the graph?

What changes when the prey birth rate is increased or decreased? Why do you think these occur?

What changes when the predator birth rate is increased or decreased? Why do you think these
occur?

Does lowering the death rate have the same effect as raising the birth rate for either
population? Why or why not?

What are some sets of parameters such that one or both species dies out? What are some sets of
parameters such that both species endure?

Extensions:

Extension 1

Ask students to discuss what happens to their simulations of the model in the long run. Nearly all
simulations will lead to both populations changing cyclically. Introduce the concept of a steady
state as it applies to the model - where neither population goes extinct even as their relative
populations fluctuate. Ask the following questions:

What does it mean to say that the populations are in a steady state?

Is it more realistic for the population of one or both species to go extinct, or for both
species to remain in a steady state? Why?

Extension 2

Ask students to go online and research the real-world populations of the Kaibab deer in Arizona in
the first half of the 20th century. During this time, the population was considered "endangered",
so the Forest Service attempted to rescue the deer by killing off their natural predators and
restricting hunting and environmental destruction. As a result, the population of deer shot up to
over 100,000. However, these policies backfired as tens of thousands of deer then starved to death
as they outstripped the carrying capacity of their environment. Ask the following questions:

Based on the model we've been using, what do you think happened when the deer's natural
predators were removed?

In our model, without predators the population of hares continues to increase forever. Why were
the results different in the real world?

What do you think would have happened to the deer populations if the Forest Service had not
intervened? Justify your answers

Extension 3

Have students consider the ways in which this model is not an accurate representation of the real
world. Ask the following questions:

What factors can you think of in the real world that this model leaves out?

In the real world, population graphs are not perfectly smooth curves like we see here. What are
some reasons that a population might not change exactly the way this model predicts? Can you
think of any ways to take these factors into account in this model?

What is a realistic growth rate for populations? Would changing the growth rates in this model
to the realistic totals have an effect on the outcomes? If so, what effect would you expect to
see?

This is a very similar model to the Vensim predator-prey model, but it adds one additional
variable - in order to reproduce, the prey must eat grass, which grows at a set rate. This
seemingly minor change has large ramifications on the model. In particular, it is much easier to
achieve a steady-state outcome, and populations cannot grow exponentially past a certain point.
Students should discuss why adding in a resource constraint causes these changes, and how it makes
the model more or less realistic.

This model takes the concept of an agent model of population interaction and moves it in a
different direction. Rather than merely moving about randomly, the predators and prey in this
model move intelligently and react to one another's presence or absence. Surprisingly enough, the
results of the model are almost exactly the same as models without this intelligence. Students
should be encouraged to discuss why some additions to a model that seem quite realistic don't
actually change the outcome. Brainstorm why certain realistic aspects are more important than
others.

This is the agent-model complement to the Vensim Predator-Prey systems model. Whereas the Vensim
model deals with populations as a whole, and has them change according to certain proportions, the
NetLogo model deals with individuals that move about and interact with one another according to
certain probabilities. Have students run simulations on both models with identical parameters, and
discuss the differences in results. In particular, it is much more difficult to create a
steady-state simulation in NetLogo, because random variation all but guarantees that one
population or the other will eventually go extinct. The randomness inherent in the model also
means that running the same model twice will not necessarily yield the same results both times.