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Measuring Biodiversity: Mathematical Applications in Ecology
TCM 2016
Who Goes There?
Estimating Ocean Populations in Chincoteague Bay
Who’s for lunch?
During an otter trawl in Chincoteague Bay, VA, a team of scientists collected the individual organisms
described in the table below. Using the information discussed in class about the food chain of sea life,
classify each species by trophic level, from 1 to 3. Use 1 for those species that exclusively feed on
phytoplankton, 2 for the small species that eat the ones classified as level 1 and 3 for larger species that
eat those classified as level 2.
The diagram below can help you understand the trophic levels. You may also want to do some research
on your own to help you classify the organisms by trophic level. For example, you may want to do an
internet search on "What do decorator crabs eat?" The answer to this question could help you learn more
about decorator crabs and determine the trophic level for this organism.
Food Chain Diagram of Ocean Life
Common Name
Hermit crab
Number
Observed
Trophic Level
137
1
Caprella
75
1
Sand Shrimp A
40
1
Sand Shrimp B
29
1
Sea Spider
3
1
Purple Sea Urchin
2
1
Caprella
2
1
Spiny Sea Urchin
1
1
Green Grass Shrimp
1
1
Clam worm
1
1
Common Shore Shrimp
1
1
Caprella with Brood
1
1
34
2
Lady crab
6
2
Mud crab
6
2
Decorator crab
4
2
Snails
4
2
Spider crab
3
2
Sea Cucumber
2
2
Starfish
1
2
Dusky Pipefish
1
2
Jonah Crab
1
2
Mantis Shrimp
1
2
Squid
1
2
Spotted Hake
4
3
Smooth Flounder
3
3
Bay Anchovy
Handouts: www.ncssm.edu/tcmconference/
http://www.chesapeakebay.net/discover/bayecosystem/foodwebs
Angela Teachey ([email protected])
Who Goes There?
Estimating Ocean Populations in Chincoteague Bay
The More, The Merrier
1) Based on the trophic classification from your classwork assignment, compute the following ratios:
Individuals in trophic level 2 to those in trophic level 1
Individuals in trophic level 3 to those in trophic level 2
We can use these as a measure of energy flow from level 1 to 2 and from level 2 to 3. Show your
calculations below.
2) Ecologists use Lindermann’s ten percent law for energy transfer between trophic levels. According to
this law, only 10-20% of the energy created at one trophic level is available to the next level.
a) Based on your results from part 1 above, does Lindermann’s rule apply to the sample collected?
Explain.
b) Considering that large predators (sharks, toothed whales, seals, dolphins) would be at a trophic
level 4, use Lindermann’s law to estimate the range in the number of large predators that you
would expect in Chincoteague Bay based on the sample given. Show your calculations.
Is Anybody Out There?
GPS data collected on board showed that the trawl sampled over 707 m in length with a surface area of 4
m2 (the trawl net’s opening is a 0.8 m x 5 m rectangle). The boat was travelling at an average speed of
3km/h. The total volume of water sampled was 2028 m3.
The total water volume of the Chincoteague Bay is estimated at 374.5 x 106 m3.
(http://www.mdcoastalbays.org/content/docs/Ch09_Chincoteague_040609.pdf)
1) Based on the data above, estimate a range for the number of individuals of trophic levels 1-3 in
Chincoteague Bay. Show your calculations below.
2) How many large predators would you expect to live in Chincoteague Bay based on your previous
results? Show your calculations.
3) Do your results above seem reasonable? What do you think affects the large predator population?
What is the role of humans in this system?
Activity Extension
Less is more!
Biological diversity or biodiversity refers to the variety of plants, animals and other living things in a
particular region. It is quantifiable in terms of the relative abundance of individuals of each species.
Simpson’s diversity index is a common measure of diversity that takes into account both species richness
and an evenness of abundance among the species present. The formula for calculating the index, D, is
given by:
where s is the number of species and pi is the proportion of individuals in the ith species. Below is a
diagram showing examples of various levels of biodiversity.
Picture by Jiang Long, “The Science Creative Quarterly” (scq.ubc.ca)
In this activity, we would like to get an idea of how Simpson’s Index can help us measure biodiversity for
a given ocean life region and then use the index to calculate the biodiversity of a region given some data.
Part I
1. What is the range of values for D? Explain.
2. Suppose we have 100 organisms and 5 difference species in sample. Consider various examples for the
proportions for each of these species and calculate Simpson’s Index for these examples. You will want to choose at
least 4 examples with varied proportions to help you see how the index reflects the biodiversity of a region. Fill the
tables in below with your example values and the diversity index for each set of examples.
# Species 1 =
# Species 1 =
# Species 1 =
# Species 1 =
# Species 2 =
# Species 2 =
# Species 2 =
# Species 2 =
# Species 3 =
# Species 3 =
# Species 3 =
# Species 3 =
# Species 4 =
# Species 4 =
# Species 4 =
# Species 4 =
# Species 5 =
# Species 5 =
# Species 5 =
# Species 5 =
Simpson’s Index =
Simpson’s Index =
Simpson’s Index =
Simpson’s Index =
3. What does a “large” value of D represent in terms of biodiversity? What does a small value of D represent?
4. In the formula for Simpson’s Index, we square the values of p and then
add these up. Why do you think we use the square of those values of p?
Explain some other possible alternatives to the formula and why you
think that squaring the p values might be a good choice for measuring
biodiversity. You may want to consider the graphs of 𝑦 = 𝑥, 𝑦 = 𝑥 2 , and
𝑦 = 𝑥 3 for 0 ≤ 𝑥 ≤ 1 as you discuss the answer to this question.
Part II
1. During an otter trawl in Chincoteague Bay, VA, a team of scientists
collected the individual organisms described in the table on page 1. Using
the data, calculate Simpson’s index.
2. Using your results, how diverse is the sea life in Chincoteague Bay?
Reflective Assignment
At the end of the biodiversity lab, each pod should prepare a narrative reflection that addresses the questions below.




Summarize the major results of the biodiversity investigation, including any interesting mathematical findings.
Feel free to incorporate graphs, pictures, tables, etc., to organize your summary, but be sure to indicate the
significance of these items within the narrative.
Discuss which aspect(s) of the lab your group found most challenging and how you handled the challenge.
What did you learn from the biodiversity lab?
Give suggestions for improving this lab activity.