TOPIC: Ecology: Biodiversity and species area relationships
TUTOR GUIDE
MODULE CONTENT: The major goal of this module is to allow students to
explore species area relationships using log-log graphs and power functions.
Students will use specific examples to explore the influence of logarithmic
transformations on relationships between species biodiversity and area.
Students will practice graphing data (by hand and using Excel) and interpreting
relationships. This module also extends the linear modeling skills they have used
in other modules (e.g., Introduction to Mathematical Modeling), if such modules
are used before this. Students will construct and interpret graphs of species area
curves using logarithmic scales and power functions. Graphing exercises are
done by hand and in Excel. The module includes a review of logarithms and the
use of quantitative models and graphs to explore the general relationship
between species numbers (biodiversity as measured by species richness) and
area. The final exercise extends these ideas to explore McArthur and Wilson's
theory of island biogeography.
The module is designed for a 60-minute classroom session with a preparatory
assignment in graph construction for students to complete before coming to class
- they will use the graphs constructed as homework during class to complete the
module. The module is designed for first-year biology majors in an introductory
biology course in ecology and evolution. Students often learn these concepts
without the opportunity to construct their own models of these relationships using
real data and often are confused about why we use logarithmic transformations
to evaluate relationships between variables.
TABLE OF CONTENTS
Alignment to HHMI Competencies for Entering Medical Students (Learning
Objectives).............................................................................................................2
Outline of concepts covered, module activities, and implementation……..……....2
Module: Worksheet for completion in class......................................................3 - 9
Pre-laboratory Exercises (mandatory)..........................................................10 - 15
Suggested Questions for Assessment.................................................................16
Guidelines for Implementation……………………………...............…...................16
Contact Information for Module Developers........................................................17
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Alignment to HHMI Competencies for Entering Medical Students:
Competency
E1. Apply quantitative reasoning
and appropriate mathematics to
describe or explain phenomena
in the natural world.
Learning Objective
E1.1. Demonstrate quantitative numeracy
and facility with the language of
mathematics.
Activity
Prelab 2
E1.2. Interpret data sets and communicate
those interpretations using visual and other
appropriate tools.
E1.3 Make statistical inferences from data
sets
E1.6. Apply algorithmic approaches and
principles of logic (including the distinction
between cause/effect and association) to
problem solving
3, 4,5
Accompanying material
- Excel spreadsheet file: "Plant species galapagos islands.xls"
Mathematical Concepts covered:
- logarithms
- power functions
In class activities:
- group discussion
- graphing in Excel and interpreting data
Components of module:
- preparatory assignment to complete and use in class
- in class worksheet:
- discussion questions
- plotting and interpreting data
- suggested assessment questions
- guidelines for implementation
Estimated time to complete in class worksheet
- 60 minutes
Targeted students:
- first year-biology majors in introductory biology course
Quantitative Skills Required:
- Basic arithmetic
- Logical reasoning
- Graph/Data Interpretation
2
Prelab 1,2
5,6,7
Worksheet: Biodiversity
Biodiversity is a term used to describe the diversity of life and living systems and
can measured at different hierarchical levels. In this module we will focus on
measuring and interpreting species diversity in communities with an eye toward
understanding factors that influence the biodiversity of communities.
Part I
To begin, form groups of 3 students. Discuss and write down the answers to the
following questions (individual answers should be turned in with your lab write
up):
1. The biodiversity of populations of a single species is genetic diversity. Why
do you think humans might be concerned with maintaining the genetic diversity of
populations of other species?
2. Extending this idea to communities, why do you think many scientists are
concerned with maintaining diversity within communities?
3
STUDENTS: TO RECEIVE FULL CREDIT FOR THIS LAB YOU WILL
ANSWER ALL QUESTIONS, COMPLETE ALL REQUESTED SKETCHES,
AND TURN IN THE EXCEL GRAPHS YOU MADE AT HOME (SEE
QUESTIONS 5 AND 6 IN THE PRELAB.
Part II - FOR ALL SKETCHES THAT YOU WILL DRAW BELOW, PLEASE
ONLY LABEL THE AXES AND SHOW THE TRENDLINE AND FORMULA FOR
THE LINE - DO NOT INCLUDE THE INDIVIDUAL DATA POINTS ON THE
GRAPHS YOU SKETCH
Get out the graphs that you did for the prelab for comparison with the graphs you
are about to make IN EXCEL - YOU WILL THEN SKETCH GRAPHS
PRODUCED USING THESE NEW EXCEL FILES.
3. Open the excel file entitled "Plant species galapagos islands". Now make and
sketch a plot of species vs. elevation (which should be labeled "The amount of
topographic relief (m)" of the island - this was measured for each island) similar
to the one you made for the species area relationship in the prelab (use log
scales on the x and y axes - please label the axes as well). Does there seem to
be a relationship between species number and elevation (topographic relief)? If
so, what is the relationship?
4. Now compare the plots of the species vs. elevation relationship and the
species vs. area relationship you made in the prelab. Knowing what you know
about fitting lines to data (think of the error sums of squares calculation you did in
the introduction to mathematical modeling module - no need to calculate, just
look at the spread of the data around the lines), which is a better predictor of
species number, area or elevation? Explain your answer.
5. Species area curves when plotted using untransformed data (not log
transformed) typically look like the figure shown below. Provide two hypotheses
about why such a relationship might exist? Include in your answer why you think
there is a relationship between species number and area AND why the
relationship should take the shape of a power function?
4
Data from http://www.qc.ec.gc.ca/faune/biodiv/en/methods/meth_invert_fish.html
Part III
McArthur and Wilson's classic theory of island biogeography (discussed in your
book pg. 1077) was developed to understand how the processes of immigration
and extinction led to stable numbers of species on islands of a given size. One
of the key features of their theory is that islands that are farther away from the
mainland (or other sources of new immigrants) will have a lower species
richness, compared with an island of the same size that is closer to the mainland
(or other source of immigrants). The prediction is given in the graph below:
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Now, let's explore this relationship with the Galapagos data set. Go to the
worksheet entitled "Sp. vs. Dist to Santa Cruz". Santa Cruz is one of the largest
islands and also is in the general center of the islands that were sampled.
6. Plot the relationship between proximity to Santa Cruz Island and Species
number (log-log plot). Is there a strong relationship between these variables?
Explain your answer, describing this relationship in words and sketching the
graph of this relationship below. Do the data match the predicted pattern of
species numbers on islands that are nearest vs. farthest from Santa Cruz as
predicted by the MacArthur and Wilson theory (shown on the above graph)?
Explain your answer.
6
Ok, one last graph. The second data set shown on the " Sp. vs. Dist to Santa
Cruz" (below the data set on the top of the page) is the same data set but sorted
by distance from Santa Cruz.
7. Make a new plot of the relationship between species and distance from Santa
Cruz, but this time only use data from islands that are within 2 km of Santa Cruz.
Fit a line to the data (power curve) as you did before (again using log
transformed x and y data). Now compare the relationship between species
number and distance to Santa Cruz on this graph with the one you plotted for 6
above.
a. What are the differences and why do you think there is a difference? Does this
tell you anything about the potential dispersal abilities of plants? If so, what?
b. How does this graph compare with the predictions of the MacArthur and
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Wilson theory of island biogeography?
MODULE FEEDBACK - Each year we work to improve the modules in the active
learning "discussion" sections. Please answer the following question with regard
to this module on this sheet and turn in your answer to the TA. You can do this
anonymously if you like by turning in this sheet separately from your module
answers.
How helpful was this module in helping you understand a fundamental
concept in biodiversity?
A = Extremely helpful
B= Very helpful
C= Moderately helpful
D= A little bit helpful
E = Not helpful at all
Module Rating ____________
Thank you!
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Pre-Laboratory Exercises – Ecology: Biodiversity and species area
relationships
Complete all exercises (graphs and problem sets) and bring the two graphs
you make to class.
The simplest way to characterize the diversity of any community is to count the
number of different species occurring there, a measure of diversity called the
Species Richness Index. The species richness index then will increase as the
number of species in a community increases.
For this exercise you will use the data contained in the excel spreadsheet entitled
"Plant species galapagos islands.xls". This file contains data from a paper by
Johnson and Raven (1973) that has information on the number of different plant
species found on 29 islands in the Galapagos. A map of the Galapagos Islands
(the larger ones) is included below.
The first sheet in your excel file contains all of the data. The additional
worksheets in this file have subsets of the data broken out with two of the
variables at a time, to help you to graph relationships between different variables
and the number of species.
(map used with permission from http://www.exoticbirding.com/galapagos-islands/about-birding.html)
Johnson, M.P., and Raven, P.H. (1973). Species number and endemism: The
Galapagos Archipelago revisited. Science, 179, 893-895.
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To receive credit for this exercise, use the above information
to answer the pre-lab questions. ALSO ***Use excel and the instructions
below to make the graphs and bring copies of these to class with you!
YOU WILL NEED THE GRAPHS YOU CREATE IN THIS EXERCISE TO GET
CREDIT FOR THE MODULE IN THE DISCUSSION SECTION!***
INSTRUCTIONS FOR MAKING THE GRAPHS TO BRING TO CLASS:
1. Open the excel spreadsheet entitled "Plant species galapagos islands".
Now make a scatterplot in excel of species (y-axis) vs. area (x axis). To
do this, to to the Species vs. Area worksheet and then highlight the
columns of data you wish to plot. Now go to the toolbar and click on Insert
tab and scroll over to the different types of charts available. Then find the
Chart heading called “Scatter” and follow the drop down box to insert a
scatter chart with only markers. ***NOTE: These are instructions for a PC.
If you are working on a Mac the method of creating a scatterplot will differ
slightly.***
- Once you have the graph made add a linear trend line (use the option
to display the formula on the graph). To do this click on the graph and
then click on an actual data point (all data points should be
highlighted). Then go to the “Layout” tab in the “Chart tools” box in the
top of the page. Scroll over to the “Trendline” option, and then use the
drop down box to pick the “More trend line options”. On this page click
the “Linear” and the option to “Display equation on Chart”. Based on
the graph. does it look like a “line” is a good fit to the data (if so, why or
if not, why not)? PRINT OUT THIS GRAPH WITH THE LINEAR
FORMULA AND BRING IT TO CLASS - YOU'LL NEED IT TO
COMPLETE THE MODULE IN CLASS!!!!!
2. Now make a new scatterplot (so create another graph, don’t just modify the
one you made above) of species vs. area but in this case, after the plot is
completed, convert the scales of each axis to logarithmic scales. To do this click
on the numbers on the y axis and check the box for logarithmic scale (base 10 is
fine, although any base should work). Repeat for the x axis. Now fit a trend line
to the data using the power curve option in excel (just as you did for the linear
option above). Also use the option for displaying the formula for the trend line on
the graph. Describe the new relationship in words. PRINT OUT THIS GRAPH
WITH THE FORMULA AND BRING IT TO CLASS - YOU'LL NEED IT TO
COMPLETE THE MODULE IN CLASS!!!!!
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Congratulations! If you completed 2 above you have just discovered a general
rule in ecology, the species area relationship. This species area relationship
usually best fits a power function with the following formula:
S = cAz
where S as the number of species, A as area, c is a constant (the number of
species in the smallest sampling area) and z as the slope of the species area
relationship when plotted on a log-log scale. As you have shown, when plotted
on log-log axes this function produces a straight line (which we call a log-linear
relationship). Bring these graphs to class!
** If you need to review what logarithms are go to:
http://en.wikipedia.org/wiki/Logarithm
http://www.math.utah.edu/~pa/math/log.html
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PRELAB REVIEW OF LOGARITHMIC FUNCTIONS:
Let us review the logarithmic function. For any positive number b not equal to 1,
the logarithm function base b is defined by y=logb x provided x=b^y. Recall the
following properties of the logarithm function:

logb x is defined only for x>0

logb 1 =0 because b0=1

logb (xy)= logb x + logb y

logb(x/y) = logb x – logb y (y not equal to 0)

logb (xy) = y logb x
Here are some practice problems using those formulas. Solve for x.
3. log3 27x – log3 3x =1
4. x=log3 9
5. x=log4 (1/16)
6. Let y=axm. Use the rules of logarithms to show that log10 y = log10 a + m
log10 x. Let v=log10 y and u=log10 x and rewrite the previous equation in terms
of a line.
7. Apply the previous argument to the equation S=cAz. Explain why power
relationships look linear on log-log plots.
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Suggested Questions for Formative Assessment
Learning Objective
E1.2. Interpret data sets and communicate
those interpretations using visual and other
appropriate tools.
E1.6. Apply algorithmic approaches and
principles of logic (including the distinction
between cause/effect and association) to
problem solving
Activity
4,5
6,7
Guide for implementation:
Pre-laboratory exercises need to be completed before class. Module should be
posted online one week prior to the module activity.
Before the module activities begin, TAs should provide a brief description of the
activities in the module, reviewing general concepts of species area relationships
if necessary.
Have students break up into groups of 3. Instruct the students to get out the
graphs they completed for their homework as they will need to use them to
complete the module. Have the students work together to complete the tasks. It
is best if every student has a computer with access to Excel so that all students
can get experience constructing the graphs.
Graphs need not be done by hand during the lab if a printer is available in the
classroom.
Note on Software:
This module contains instructions in graphing using Excel 2010 on the PC. The
instructions should be adjusted if Macs are used. In addition, as new versions of
Excel are released the instructions for graphing might also need to be modified.
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Module Developers:
Please contact us if you have comments/suggestions/corrections
Kathleen Hoffman
Department of Mathematics and Statistics
University of Maryland Baltimore County
khoffman@math.umbc.edu
Jeff Leips
Department of Biological Sciences
University of Baltimore County
leips@umbc.edu
Sarah Leupen
Department of Biological Sciences
University of Baltimore County
leupen@umbc.edu
Acknowledgments:
This module was developed as part of the National Experiment in Undergraduate
Science Education (NEXUS) through Grant No. 52007126 to the University of
Maryland, Baltimore County (UMBC) from the Howard Hughes Medical Institute.
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