Learning Scenario - Bunny Comparison (Vensim)

Basic Model:

Description

This is a system model of the growth of a population of bunnies. It tracks the number of bunnies that are alive at each time step, and plots the results on a graph. Parameters in this model - birth rate, carrying capacity, and competition - determine how quickly the bunnies reproduce and their maximum population. When the model is run, it will display either exponential or asymptotic growth, depending on the parameters chosen. A graph displays the size of both populations of bunnies at each time step.

Background Information

In the absence of constraints, most species on Earth will grow exponentially and without end. However, exponential growth is not sustainable in the real world for a number of different reasons, including resource, space, and energy constraints. This model simulates two of the most common and important constraints on a population size - the carrying capacity of the environment and the competition between members of the species for mates and resources.

The carrying capacity of the environment is essentially the total potential supply of food, water, air, and other resources necessary for life. In the absence of other constraints, a population will approach the carrying capacity, but it is impossible for a population to stay above the carrying capacity for an indefinite time. Competition is related to carrying capacity and represents the ways in which individual members of the species prevent one another from reproducing by fighting over territory, mates, or food. Like resource constraints, competition becomes more intense as the population size increases, so it too tends to produce a steady state population.

Science/Math

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

1. Rabbits breed according to the "birf rate" in both populations

   birf rate * bunnies
   birf rate 0 * bunnies 0

2. The "bunnies" population is multiplied by the following function based on its carrying capacity

   1 - bunnies/carrying capacity

3. The "bunnies 0" population is reduced by its competition factor multiplied by the number pairs of bunnies

   competition * bunnies 0 * (bunnies 0 - 1)/2

4. The new values for both populations are plotted on the graph

Teaching Strategies

An effective way of introducing this model is to ask students to brainstorm why there are limits on the population of animals that can exist in a certain area. Then, discuss how different factors can affect population size in different ways. Ask the following questions:

1. The average litter of rabbits contains anywhere from 4 to 9 bunnies, and rabbits can breed several times a year. Why don't rabbit populations explode to trillions of members?
2. What are some factors that might limit population growth for rabbits?
3. If students mention space or resource constraints, ask: What happens if the population of rabbits is too high for the available space? What happens if there isn't enough food/water for all the rabbits?
4. In the long run, what do you expect will happen to the population of bunnies if it faces the limiting factors we discussed?

Implementation:

How to use the model

This is a relatively basic model, yet still has a number of interesting parameters to manipulate:

1. The "bunnies" parameter's initial value determines the number of rabbits present in the population limited by carrying capacity when the model is run
2. The "bunnies 0" parameter's initial value determines the number of rabbits present in the population limited by competition when the model is run
3. The "birf rate" parameter determines the proportion of rabbits in the first group that reproduce each timestep
4. The "birf rate 0" parameter determines the proportion of rabbits in the second group that reproduce each timestep
5. The "carrying capacity" parameter determines the maximum carrying capacity of the environment in which the first group of rabbits is placed
6. The "competition" parameter determines the effects of inter-rabbit competition on population growth rates in the second population

All of the aforementioned parameters are manipulated before the model is run by right clicking (control clicking on a Mac) on their labels in the diagram. When the model is run, parameters 3-6 can also be 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, a graph depicts the populations of rabbits in both groups, with bunnies indicated by the blue line and bunnies 0 indicated by the red line. For more information on Vensim, reference the Vensim tutorial here here.

For a complete tutorial on how to use Vensim, please go to the following link here.

Learning Objectives

1. Understand the relationship between carrying capacity and steady-state population
2. Understand the relationship between competition and steady-state population

Objective 1

To accomplish this objective, have students run the simulation with several different values for carrying capacity, and observe how the graph of the bunny population changes each time. They should specifically pay attention to the asymptotic steady state. Ask the following questions:

1. From your observations, what is the relationship between the carrying capacity and the eventual steady state population?
2. The bunnification equation is "birf rate*bunnies*(1-bunnies/carrying capacity)". What happens when bunnies = carrying capacity? Does this make sense with what you observed?
3. Now change the birf rate and re-run the simulation a few times. What is the relationship between birf rate and bunny population? Does the steady state population change? Why or why not?

Objective 2

To accomplish this objective, have students run the simulation with several different values for competition, and observe how the graph of the bunny population changes each time. They should specifically pay attention to the asymptotic steady state. Ask the following questions:

1. From your observations, what is the relationship between the competition factor and the eventual steady state population?
2. The asymptotic steady state for "bunnies 0" with default parameters is a population of exactly 5001. What is the relationship, if any, between this and the competition factor?
3. Now modify the birf rate 0 and re-run the simulation a few times. What is the relationship between birf rate 0 and bunny 0 population? Does the steady state population change? Why or why not?
4. How does the relationship between the competition factor and the asymptotic steady-state population change when the birf rate 0 changes? What does that suggest about the differences between competition and carrying capacity?

Extensions:

1. Understand the mathematics behind the equation for the competition factor
2. Discuss which constraint is more realistic, and what qualities both parts of the model lack when compared with the real world

Extension 1

Have students solve the equation in bunnification 0 for the population of bunnies 0. The final equation should be in terms of birf rate 0 and competition. Ask the following questions:

1. Why is the competition factor multiplied by "bunnies 0 * (bunnies 0 - 1)/2" instead of just "bunnies 0"? What does that function measure in the real world?
2. If the population of bunnies 0 were to double, what would happen to the term multiplied by the competition factor? Does this make sense in the real world?
3. How does the solved version of the equation depend on the competition factor? How does it depend on the birf rate?

Extension 2

Discuss with students the real-world implications of both types of constraints. Ask the following questions:

1. Set the parameters in both functions so that they have the same asymptotic steady state. How does the graph of the populations differ in the time before they reach the steady state?
2. Now increase the birf rate and birf rate 0 by the same amount. How does the graph of the populations change for each group?
3. The carrying capacity equation suggests that the birf rate of the population determines how quickly it reaches its maximum, but does not change the maximum itself. Is this realistic? Why or why not?
4. This equation suggests that the maximum number of bunnies that can sustainably live in an area depends, in part, on how quickly they can reproduce. Is this realistic? Why or why not?
5. Does either of these population functions ever decrease? Is this realistic? What real-world factors might account for a temporarily decreasing population that is not captured by these models?

Related Models

Rabbits and Wolves

This is an agent model roughly analogous to the Bunny Comparison model. Like Bunny Comparison, this models the population of rabbits as they reproduce and die; however, it approximates both carrying capacity and competition by requiring the rabbits to eat grass in order to survive. Grass grows at a set rate, so if there are too many rabbits, they will run out of grass and start to starve to death. As an additional factor, this model introduces the concept of a predator - wolves. These reproduce alongside rabbits and, as the rabbits eat the grass, the wolves eat the rabbits. As an agent model, this introduces an element of randomness to the population graph, and as such, it is an excellent way to explain why real-world populations don't asymptotically grow towards a steady state.

Predator-Prey (NetLogo)

This is another agent model, but this model forgoes resource constraints in favor of predators as the only constraint on the prey population. This is a very different constraint than either carrying capacity or competition, as it is inherently much less stable. The relative reproduction rates of predators and prey tend to lead to cycles rather than a steady state: when the population of prey increases, the population of predators will also increase, causing the population of prey to decrease, which in turn causes the population of predators to decrease. The danger is that if the population of predators ever gets too large, they could eat all of the prey before they have time to start reproducing. Conversely, it is possible that all the predators will starve to death while some prey still remain, leading to an explosion in the prey population.

Supplemental Materials:

• Tutorial