TITLE: Protecting Populations: From Ecological Models to Conservation
Action
Subtitle: Using Mathematical Models in Undergraduate Ecology Courses
Author: Kristin McCully, Ph.D. candidate, Ecology and Evolutionary Biology and Education, University
of California, Santa Cruz
Target audience: upper-division ecology (or ecology of specific ecosystems) courses
Field tested with: Ecology of Reefs, Mangroves, and Seagrasses course (30-70 students in
Winter 2012 and 2013) at University of California, Santa Cruz (UCSC)
Module Type: interactive lecture and computer inquiry
Duration: 2-hour lecture and 2-hour lab/discussion section
Key materials (per group of 2-4 students):
 PC laptops with Microsoft Excel
 Posterboard
 Colored markers
 Handouts
Concepts: mathematical model, conceptual model, population model, exponential growth,
logistic growth, structured population model, matrix algebra, population growth rate, elasticity,
reproductive value, stable stage distribution, simulation
Objectives:
• Learn how to use structured population models to assess and protect populations
• Appreciate the importance and uses of mathematical models in ecology and
conservation
Overview:
Both the biology research and education communities suggest that most undergraduate
biology curricula do not provide students adequate training in the quantitative skills they need
to obtain a deep understanding of biological phenomena and to contribute effectively to future
scientific inquiry. To help address those needs, we present a computer inquiry module using
ecology research literature to introduce structured population (matrix) models, one of the most
commonly-used types of ecological models. In this lesson, students
 Participate in an interactive lecture introducing the concepts and mechanics of structured
population models
 Read published research papers that apply structured population models to specific
populations and conservation questions
 Work through case studies in groups on computers using the MS Excel add-on PopTools
 Present the model with their own research question to the class
Navigate: Background
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Materials
Preparation
Timeline
Procedures
Supplemental
Ecological Models
Background for Teachers
Why this matters:
Both the biology research and education communities suggest that most undergraduate
biology curricula do not provide students adequate training in the quantitative skills they need
to obtain a deep understanding of biological phenomena and to contribute effectively to future
scientific inquiry. Students will be more interested in further studying quantitative biology and
using math in biology if they understand how biologists use math and apply math to real-world
questions, such as protecting populations in need of conservation.
In this module, students are introduced to one of the most common mathematical models used
in ecology: the structured, or matrix, population model. Students learn how the model is
developed and how it is used to answer the questions of wildlife managers, while focusing on
at least three separate case studies in preparatory reading, lecture, and lab. Students also
work together to develop the third case study’s model themselves, ask and answer their own
research question using a computer program, and present the case study and their own
research to the class in an oral presentation.
Assumed background:
Students should be familiar with equation, graph, and assumptions of exponential growth
population model. The interactive lecture briefly reviews peripheral nerves and isotopes, which
most students should have learned in general biology and chemistry courses (although these
are not key to the lecture).
Elements of module:
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Preparatory reading: classic structured population model paper on loggerhead sea
turtles with good explanations of all key terms and model development
Interactive lecture: demonstrates development and analysis of a structured population
model for lead poisoning of Laysan albatrosses at Midway Atoll, which shows how a
combination of field work, lab work, and a mathematical model resulted in conservation
action to protect a threatened species. The case study is based on research led by
Myra Finkelstein of University of California, Santa Cruz. The interactive lecture uses
key techniques of active learning, such as “Think-Pair-Share” and brainstorming.
Cooperative computer inquiry lab:
o Students prepare by choosing and reading a selected scientific paper
o Groups work through one of about five structured population models involving
very different species using the PopTools add-on to Microsoft Excel
o Groups ask and answer their own research question using PopTools
o Groups present their organism’s life cycle and their research to the class using a
poster they create
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Module Description
Materials:
Interactive Lecture:
 Projector
 Computer with Microsoft Powerpoint
 Lecture guide (photocopies or posted previously on course website)
 Dry erase board and markers, chalkboard and chalk, or document camera with pen and
paper to demonstrate problems
Computer Inquiry Lab (for each group of 2-4 students):
 Lab handout (1 per student)
 PC computer with Microsoft Excel
 Posterboard (or a page from an easel pad with adhesive on back)
 Colored markers
Preparation:
• Select and assign preparatory readings for lecture and lab. Post readings on course’s
website or course management system.
 In preparation for lecture, we recommend assigning the following classic and wellexplained paper on using structured population models for conservation and management:
Crouse, D. T., Crowder, L. B., & Caswell, H. (1987). A stage-based population model for
loggerhead sea turtles and implications for conservation. Ecology, 68(5), 1412-1423.
http://dx.doi.org/10.2307/1939225.
 You should choose several scientific papers using structured population models for
organisms with a variety of life histories. Check that they are well-explained and work
through the lab worksheet for each paper prior to assigning the papers.
 Because we originally used this module in a course on tropical marine ecosystems, the
key uses papers on groupers, mangroves, manatees, sharks, and corals.
 You can search for other appropriate papers by searching for papers that cite the classic
Crouse et al. (1987) paper or Caswell’s authoritative book Matrix Population Models:
Construction, Analysis and Interpretation (1989 or 2000).
 You may want to assign the readings with instructions to focus on the content of the paper,
rather than the math, by using these questions:
 What are the research questions the model is intended to answer?
 Sketch the life cycle diagram for the species, with arrows showing each transition or
contribution between stages that occurs in that species. If more than one model is
proposed in the paper for different species, populations, time periods, or scenarios,
focus on the simplest or basic model.
 Briefly, how did the authors estimate the demographic rates of survival, fecundity, and
growth for their model? Note: this usually includes tracking organisms in the wild
somehow; don’t worry about the math that may also be involved.
 Briefly, what did the authors conclude from the model? (i.e. their answers to their
research questions)
• Ask students to sign up for organism group in previous lab and assign appropriate readings.
• Arrange to use PC computers with Microsoft Excel to lab or ask students to bring their own
laptops. Make sure PopTools (available at http://poptools.org) is downloaded and installed
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Ecological Models
prior to lab.
• Make sure that lab instructor has at least skimmed all the groups’ papers and is comfortable
using PopTools.
Timeline:
Interactive lecture (2 hours)
1. Introduction to case study: Laysan albatrosses at Midway Atoll (20 minutes)
2. Introduction to mathematical models (15 minutes)
• Discussion: What is a model? (10 minutes)
• Review of classic population models (3 minutes)
• Steps to developing a mathematical model (2 minutes)
3. Structured population models (35 minutes)
• Assumptions (10 minutes)
• Model structure (8 minutes)
• 3 examples (15 minutes)
4. Mathematical model: Laysan albatrosses at Midway Atoll (40 minutes)
• Video: Midway’s Albatrosses (1 minute)
• Formulate math model (5 minutes)
• Use model to answer questions (25 minutes)
5. Discussion: Importance of models to ecology and conservation (5 minutes)
6. Introduction to lab activity (5 minutes)
Cooperative computer inquiry lab (2 hours)
7. Introduction to activity (5 minutes)
8. Brief review of lecture & introduction to PopTools (10 minutes)
9. Groupwork (about an hour)
10. Presentations (about 5 minutes per group)
11. Discussion: Importance of models to ecology and conservation (10 minutes)
Detailed Procedure:
Interactive Lecture
Note: Detailed script for lecture is available in Supplemental Material.
1. Introduction to case study: Laysan albatrosses at Midway Atoll (20 minutes)
• Introduce lecture.
• Introduce research of Myra Finkelstein.
• Discuss anthropogenic threats to albatrosses.
• Introduce Midway Atoll.
• Introduce “droopwing syndrome” and peripheral neuropathy.
• Discuss research Myra did in field and lab to show that albatross chicks obtain lead
from lead-based paint chips from buildings.
• Discuss necessity for mathematical model to show population consequences of lead
poisoning of chicks.
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Midway
Atoll
Laysan albatross chick exhibiting “droopwing syndrome” characteristic of lead poisoning
and a map of Midway Atoll in the Hawaiian Archipelago
2. Introduction to mathematical models (15 minutes)
• Discussion: What is a model? (10 minutes)
- Show examples of models
- Think-pair-share
• Review of classic population models (3 minutes)
• Steps to developing a mathematical model (2 minutes)
3. Structured population models (35 minutes)
• Assumptions (10 minutes)
- Have students brainstorm assumptions for exponential growth model
- Discuss how to change exponential growth assumptions for an organism with a
more complex life history
- Introduce term “vital rates”
- Introduce term “structured population model”
• Model structure (8 minutes)
- Structured population models are structured by …
- Introduce & explain transition matrix
• 3 examples (15 minutes)
- Ex1: Lionfish – build life cycle diagram, add vital rates, convert to transition matrix
- Ex2: Plant – add vital rates to life cycle diagram, convert to transition matrix
- Ex3 – convert transition matrix to life cycle diagram
4. Mathematical model: Laysan albatrosses at Midway Atoll (40 minutes)
• Review research questions and assumptions (2 minutes)
• Video: Midway’s Albatrosses (1 minute)
• Formulate math model (5 minutes)
E
1
2
3
4
5
6
7
8
Conceptual model (life cycle diagram) for Laysan albatrosses (Finkelstein et al. 2010)
•
Use model to answer questions (25 minutes)
- Population growth graph and rate
- Elasticity
- Reproductive value
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- Stable stage distribution
- Simulations
Note: If time is short, you may choose to omit stable stage distribution, elasticity, or
other parts of this section.
5. Discussion: Importance of models to ecology and conservation (5 minutes)
• Discuss briefly what we could do with the mathematical model in this case study and the
turtle case study (preparatory reading) that we couldn’t do otherwise
• If time is short, this can be delayed until the end of the lab.
6. Introduction to lab activity (5 minutes)
• Remind students that their group already chosen an organism and the assigned papers
are available for them to read
• Introduce PopTools and remind students to bring computers and install PopTools if
necessary.
Computer Inquiry Lab
7. Introduction to activity (5 minutes)
8. Brief review of lecture & introduction to PopTools (10 minutes)
• Review converting between life cycle diagram and transition matrix
• Review research questions and appropriate tools (population growth rate, elasticity,
reproductive value, stable stage distribution, simulation)
• Show how to use PopTools to calculate each tool (listed above). PopTools is very easy
to use, so if the instructor is comfortable with it, you may not need to provide
instructions on using PopTools (in Supplemental Materials).
9. Cooperative activity (about an hour)
• Students work through their group’s paper using the lab worksheet and PopTools.
• Students complete “Final Challenge” by asking and answering their own question about
their population using the paper’s model and PopTools. Usually, these questions focus
on some sort of simulation that is easy to test with PopTools, often based on a real
anthropogenic threat or conservation action, similar to “How will increasing the survival
of albatross chicks (stage E) by reducing ingestion of plastic impact the population
growth rate?”
• Students prepare their poster (usually with their life cycle diagram and research
question) and presentation.
• During the cooperative activity, the instructor should move between groups to answer
questions and make sure everyone understands what they’re doing.
10. Presentations (about 5 minutes per group)
• Each group presents their organism’s life cycle and their research question and answer.
• The instructor and students should ask the presenters questions about their model and
what they did. The instructor may want to emphasize similarities and differences
between these models and the previous models students have seen (turtle, lionfish, and
albatross models in preparatory reading and lecture).
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Student posters
11. Discussion: Importance of models to ecology and conservation (10 minutes)
• Ask students to take a minute to think about how we use and need mathematical
models in ecology, based on both your previous experience and this lab? What can we
do with mathematical models that we can’t do with field or lab work? Then discuss your
thoughts with your neighbor. [Think-Pair-Share]
Main points:
• Analyze and synthesize monitoring data
• Determine which types of data are most useful and important to collect in the future
• Identify key life stages or demographic processes as management targets
• Predict impacts of management actions
Soetaert & Herman 2008 A Practical Guide to Ecological Modeling:
• Generate testable predictions to guide experiments
• Analyze systems that are impossible or difficult to experiment with (when spatial or
temporal scales are too large, against regulations or morals, too expensive, dangerous,
take too long – think astronomy or endangered or invasive species!)
• Force scientists to think very clearly & be explicit about every assumption & term →
identifying gaps in knowledge that may give direction to research priorities & new
experiments
• Extrapolate or interpolate studies in time & space
• Quantify immeasurable processes (e.g., rate of oxygen flux across sediment-water
interface)
• Guide management by “performing experiments” that are inappropriate in the real
system and assessing consequences of our actions in advance
Gotelli 2008 A Primer of Ecology:
• Science of ecology is study of distribution and abundance, so we need tools of math &
statistics
• We need models because natural is so complex
• Mathematical models act of simplified road maps, giving us some direction and idea of
exactly what things we should be trying to measure in nature
• Models also generate testable predictions
• Highlight distinction between patterns we see in nature and the different mechanisms
that might cause those patterns
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Assessment Methods:
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Preparatory reading: You could ask students to answer questions in a online reading quiz
before coming to class.
Interactive lecture: Think-Pair-Shares, brainstorming, and questions in lecture are designed
to provide formative assessment to the instructor. Questions could easily be converted to
clicker questions.
Computer inquiry lab: The instructor should have a good understanding of students’
learning by talking to each group. The instructor will understand even more by observing
group presentations, listening to students’ answers to questions, and reading and grading
students’ lab worksheets.
Summative assessment: We usually include questions requiring students to interpret the
transition matrix and interpret provided data (reproductive value, population growth rate,
etc.) on the lab practical exam. Lecture exams could also include similar questions, as well
as questions on assumptions of structured population models and how models are used in
ecology and conservation.
Possible Pitfalls:
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Students often are confused between “survival rate” and “persistence,” because individuals
must survive in order to persist in the same stage or grow into another stage. Very few
students are able to extrapolate that mortality is 1 – persistence – growth.
Students are often confused when presented with a population growth graph with different
behavior in the first few years than in later years (after the population has settled into the
stable stage distribution).
Some students may have learned the term “elasticity” as a type of ecosystem resilience:
rate of recovery of a system following a disturbance (Westman and O’Leary 1986) and may
be confused with learning that elasticity is the effect that a change in each one of the vital
rates has on population growth rate λ.
Students often didn’t realize that transition elements are proportions and gave answers to a
question asking them to calculate mortality of stage 2 like N2(t)-(P2+G2).
Students sometimes confuse individual growth and population growth.
Glossary:
Terms for lead poisoning case study:
 Bycatch = organisms caught intentionally in a fishery while intending to catch other fish
(possibly because they are the wrong size or species), whether sold or discarded
 “Droopwing” syndrome = albatross chicks with paralyzed wings
 Peripheral neuropathy = damage to the peripheral nerves which run between the spinal
cord and the rest of the body, which can be due to many factors, including diabetes,
traumatic injuries, and exposure to toxins
 Isotopes = variants of chemical element with different numbers of neutrons
Terms for ecological model content:
Density dependence = a relationship between population growth (or vital rates) and population
density. Density-dependent models typically incorporate the idea that crowding leads to a
reduction in births and a decrease in deaths and thus a reduction in population growth.
Deterministic model = a model in which the parameters are constant and do not vary
unpredictably with time. If a model is run multiple times with the same initial conditions, a
deterministic model will always produce the same result. (Gotelli 2001)
Elasticity = effect that a change in each one of the vital rates has on population growth rate λ
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(calculated as the partial derivative of λ with respect to the transition probability aij, scaled
to both λ and aij) (from Crouse et al. 1987)
, where <> denotes the scalar product.
Fecundity = average number of offspring produced by an individual in the stage in one time
period
Growth [in a structured population model] = probability that an individual in the stage will
survive and grow into the next stage
Model = simplified representation of object or process which requires making assumptions
Parameter = terms in a model specified by the modeler, usually as a constant (for example, r
and K in the exponential and logistic growth models)
Persistence = probability that an individual in a stage will survive and remain in the same stage
Population growth rate (λ) = rate of increase of population during one time period which may
also be known more specifically as the finite rate of increase (calculated as ratio of
population size at one time to population size at previous time or dominant eigenvalue of
transition matrix)
Reproductive value = expected number of offspring that remain to be born to each individual of
a stage (calculated as left eigenvector of transition matrix)
Stable stage distribution = constant proportion of individuals represented in each stage of an
exponentially increasing (or decreasing) population if vital rates for each stage are constant
(calculated as right eigenvector of transition matrix). Once a population reaches SSD,
these proportions remain constant through time.
Stochastic model = a model in which some parameters vary unpredictably with time, reflecting
random or chance events in nature or complex phenomena that are too complicated to
model directly. If a model is run multiple times with the same initial conditions, a stochastic
model will generate somewhat different answers each time.
Transition matrix = a square n x n matrix whose elements aij give the probability of a transition
from state j (column) to state i (row) in a single time step. In demographic models, this
described changes between different ages or life history stages. Elements may be larger
than 1.0 only if they represent reproduction.
Transition probability = an element of the transition matrix that represents the probability than
an individual in one stage will contribute to another or the same stage. It is usually a vital
rate or a mathematical expression containing one or more vital rates.
Vital rates = rates of demographic processes, such as fecundity, survival rate, and growth rate
of individuals.
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Supplemental Materials:
Module Materials:
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PowerPoint Presentation for interactive lecture: “Lead Poisoning of Laysan Albatrosses at
Midway Atoll: From Ecological Model to Conservation Action”
Presentation notes (script for instructor)
Lecture guide (handout for students)
Lab worksheet (handout for students)
Lab worksheet key (in general and for tropical marine organisms)
Instructions for using PopTools (handout for students or for instructor only)
Videos
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Albatrosses at Midway Atoll (by Wayne Sentman):
http://www.youtube.com/watch?v=V57Q_g9ptd4
Albatross chicks learning to fly (by Chris Eckstrom):
http://www.youtube.com/watch?v=2H9PQ6jt9us
Adult Laysan albatross feeding chick (by Gina Ruttle):
http://www.youtube.com/watch?v=S83TP_iaEg8
Laysan albatross courtship (by Gina Ruttle):
http://www.youtube.com/watch?v=YvpHBALOCAI
Laysan albatross chicks with “droopwing” syndrome (by Myra Finkelstein):
http://www.youtube.com/watch?v=I3hrwPSlzbc
Resources
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Crouse, D. T., Crowder, L. B., & Caswell, H. (1987). A stage-based population model for
loggerhead sea turtles and implications for conservation. Ecology 68(5):1412-1423.
http://dx.doi.org/10.2307/1939225
Finkelstein, M. E., Doak, D. F., Nakagawa, M., Sievert, P. R., & Klavitter, J. (2010).
Assessment of demographic risk factors and management priorities: impacts on juveniles
substantially affect population viability of a long-lived seabird. Animal Conservation, 13(2),
148-156. http://dx.doi.org/10.1111/j.1469-1795.2009.00311.x
Gotelli, N. (Various) A Primer of Ecology. Chapter 3: Age-Structured Population Growth.
Sinauer.
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