Learning Scenario - Bunny Hopping Model (Netlogo)

Basic Model:

Description

Two dimensional computer screens pose a problem for displaying slopes and heights of a landscape. Many mapmakers use topography indicators to describe the terrain depicted. The Bunny Hopping model takes this same approach by presenting a landscape with different shaded squares to represent the height of each section of land. Bunnies are randomly sprinkled across the hilly terrain as set by the user, from where they roll down the hills into the valleys. At the end of the simulation, bunnies will generally be clustered into groups in the valleys and will not be on the hillside. The purpose of the model is to demonstrate mathematical concepts in a visual manner. Local extrema are often seen when graphing functions, but the Bunny Hopping model shows the local (relative) and absolute extrema in relation to the place that the bunnies naturally congregate.

Background Information

Whether or not this demonstrates the actual behavior of bunnies on a hill, the model does introduce important mathematical concepts related to relative extrema. On a graph, a relative maximum is the highest point in relation to its neighboring points, while the absolute maximum is the highest point on the whole graph. The Bunny Hopping Model includes relative maxima and minima of the based on the elevation that the bunnies naturally congregate at. The model shades the absolute elevation height on the terrain blue and the absolute minimum elevation green. Bunnies may naturally congregate at either the absolute minimum or one of the relative minima. On the other hand, while some may start at the relative maximum, none will end at that point.

Science/Math

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

1. The user-set number of bunnies roll down the hills into valleys
2. The amount of bunnies within the relative minimum is calculated

The changes that the user makes to the population and terrain are not practical, but they do allow for the student to understand mathematical principles in relation to different real-world scenarios. The graphic representation of the population and their behavior provide students with a way to qualitatively understand these concepts.

Teaching Strategies

An effective way of introducing this model is to ask the students to come up with a series of hypotheses that can be tested when actually running the model. The following questions will provide students with a prompt that can help in the formation of hypotheses:

1. If bunnies were randomly scattered on a hilly surface, where do you expect they would eventually wind up? Why?
2. Will the bunnies tend to wind up in clusters or as spread out as possible?
3. The lowest point in the field is known as the absolute minimum. Would you expect there to be bunnies here? If, so, how many? Would there be bunnies anywhere else? What about the absolute maximum?

Students should compare their answers with their results after using the model.

Implementation:

How to use the model

The model includes two sliders that may be changed in order to affect the population and terrain depicted. Changing these variables will produce different results:

1. The "initwabbit" slider controls the number of bunnies that are initially placed on the map (from 0 to 100
2. The "smoothness" slider controls how hilly the surface is. A larger number will produce fewer, larger peaks and valleys, while a smaller number will produce a more broken terrain

These variables may be changed by sliding the respective sliders. The bunnies and terrain are entered by clicking the "setup" button. Since the rabbits tend to move very fast, it is recommended that the speed slider is set to very slow in order to track their movement.

The number of bunnies in the relative minimum is displayed on the graph "High Wabbits." This will allow students to see the trends in relation to relative and absolute minima. To run the simulation, press the "roll 'em" button. The simulation will run based on the variables set when the "setup" button was pressed. For more information about Netlogo, refer to the Netlogo tutorial: http://shodor.org/tutorials/NetLogo/Introduction.

Learning Objective

1. Understand the concept of relative extrema and how to interpret it visually

Objective 1

Bunnies tend to congregate at the lowest points that are close to where they start out. Sometimes, this might be at the absolute minimum, and sometimes it might be in one of the relative minimum areas. The following questions will guide the student to an understanding of extrema:

1. Start the simulation out with a smoothness of 40 and 50 initial rabbits. Where do the rabbits tend to congregate? Why?
2. Are all the rabbits congregated in one spot? Why could some not be in the same spot as the rest?
3. Change the smoothness to 20 and rerun the simulation. What trends do you see as the bunnies move around? What is different about these areas?
4. What is the difference between the spots that have a low elevation and the spots that have the lowest elevation (in green)?
5. The lowest spots, colored in green, are called absolute minima. These always have the lowest value elevation. The other low spots are called relative minima. Could you apply the idea of relative and absolute minima to a function? Explain.

Extensions:

1. Apply the idea of relative and absolute extrema to graphs of functions on an x-y plane
2. Extend the idea of population movement to population movement of humans.

Extension 1

The concept of relative and absolute extrema was applied to this model in order to teach students the mathematical concepts behind the population movement. For this extension, students are going backwards and applying what they know about extrema to the graph of a function. Using a graphing utility, display the graph of f(x)=sin(x)*cos(x2). There are many relative maximums and relative minimums. Restrict the domain to -2π to 2π. Ask the following questions:

1. How many relative maximums do you see within the domain [-2π, 2π]? Is one of these points the absolute maximum of the function? Can you know for sure?
2. In what way do the ideas of relative extrema from the Bunny Hopping Model apply to this function? Explain

Extension 2

The model can be extended to represent more than just elevation. Rabbits-and animals in general-respond to stimulus and change in their environment. This is known as taxis and kinesis. Picture the elevation in the Bunny Hopping model as a stimulus for the rabbits to move to a lower elevation. In this instance, the movement of the rabbits is somewhat directed, so it would be an example of taxis. Have the students connect the idea of taxis and kinesis to the model and answer the following questions:

1. If taxis is a response away from a specific stimulus and kinesis is a non-directional response, which do the bunnies represent in this model?
2. What would the green spots represent? How about the blue?

Related Models

Fields and Sheets

As in the Bunny Hopping model, geographers use different colors to represent the elevation of a particular landscape. This same method of differentiation can be applied to other fields as well. For example, in the Fields and Sheets model, the fields of different charges are represented by different colors based on their magnitude. This is particularly useful in studying the interaction of particles and magnetic fields, since the resulting sum of the charges change based on how close together the particles are. With the experience that students have from the Bunny Hopping model, they will easily understand how to interpret the visual elements of the Fields and Sheets model.

Bacterial Chemotaxis

The Bacterial Chemotaxis model represents a colony of bacteria which direct their movement according to the presence or absence of chemicals in their environment. If sugar is placed in the worksheet, the E. coli. will gravitate towards it, but if toxins are placed in the worksheet, they will move away from them. This connects to the Bunny Hopping model and Extension 2, since the bacteria-in their "semi-intelligence"-are showing taxis. There are many variables that can be manipulated in this model, such as the tendency for bacteria to move up and down the chemical gradient.

Chemotaxis Control Network

Similar in concept to the Bacterial Chemotaxis model, the Chemotaxis Control Network model allows for users to manipulate the variables in a controlled environment of bacteria. This specific scenario includes a membrane that carries chemicals through a membrane. Based on the bacterial interaction with chemicals, it will move in a different manner. This model again demonstrates taxis (chemotaxis), as the Bunny Hopping model does.