An “Experimental”
Interdisciplinary Course in
Mathematical Ecology*
Glenn Ledder† and Brigitte Tenhumberg‡†
†Department of Mathematics
‡School of Biological Sciences
University of Nebraska-Lincoln
gledder@math.unl.edu
*funded by NSF grant DUE 0531920
Overview
DESIGN
IMPLEMENTATION
• Goal
• Laboratory Activities
• Design Issues
• “Lecture” Activities
• Pedagogical Principles • A Virtual Laboratory
• The Research Topic
• Results
Goals
• Project Goal: Prepare a cadre of young
scientists who can do interdisciplinary
research in mathematics/biology
• Course Goal: Introduce interdisciplinary
research in mathematics/biology to
talented students at an early stage in
their careers.
Design Issues
• The course must be self-contained.
– We cannot assume knowledge of calculus,
statistics, or any specific biology topic.
– We cannot assume laboratory experience.
Design Issues
• The course must be self-contained.
– We cannot assume knowledge of calculus,
statistics, or any specific biology topic.
– We cannot assume laboratory experience.
• The course must be integrated at
different levels.
– Math and biology
– Theory and experiment
– Research design, conduct, and dissemination
Pedagogical Principles
• The experiments must all contribute to
a coherent body of theory.
• The math must always be motivated by
the experiments.
• Background material must be presented
when needed.
• Students must be asked to assemble
the pieces into an integrated whole.
The Research Topic
• General Theme: biological pest control
The Research Topic
• General Theme: biological pest control
• Specific Research Questions:
– How does an aphid population grow?
• The theory must incorporate the aphid life cycle.
• The theory must be formulated as a mathematical
model.
• The theory must be quantitatively realistic.
The Research Topic
• General Theme: biological pest control
• Specific Research Questions:
– How does an aphid population grow?
• The theory must incorporate the aphid life cycle.
• The theory must be formulated as a mathematical
model.
• The theory must be quantitatively realistic.
– How does predation by lady beetles affect
aphid populations?
• The theory must be formulated as a mathematical
model.
The Greenbug Aphid: S. graminum
– Many greenbug colonies consist of viviparous
(born live) females that reproduce asexually.
Sexual reproduction occurs only when
conditions require overwintering.
The Greenbug Aphid: S. graminum
– Many greenbug colonies consist of viviparous
(born live) females that reproduce asexually.
Sexual reproduction occurs only when
conditions require overwintering.
– Greenbugs have 5 developmental stages: 4
instars of juveniles plus adult.
The Greenbug Aphid: S. graminum
– Many greenbug colonies consist of viviparous
(born live) females that reproduce asexually.
Sexual reproduction occurs only when
conditions require overwintering.
– Greenbugs have 5 developmental stages: 4
instars of juveniles plus adult.
– Plant damage results from toxic saliva rather
than loss of nutrients. Severe crop damage
occurs before the aphids become short of
food. (we can ignore density effects)
Theoretical Population Dynamics
• Discrete linear stage-structured model:
xt+1 = Mxt, where x is a vector giving the
populations of the different stages and M is a
matrix of parameters
Theoretical Population Dynamics
• Discrete linear stage-structured model:
xt+1 = Mxt, where x is a vector giving the
populations of the different stages and M is a
matrix of parameters
• Research tasks:
– construct the model
– estimate the parameters
– predict population growth
– test the predictions
Laboratory Activities
• Clip-cage experiments:
– Put 1 newborn aphid in a clip-cage on a live
sorghum leaf.
– Check its development stage daily by looking
for exuvies.
– After it becomes an adult, count its daily
offspring.
• Population growth experiments:
– Put 1 adult aphid in a large cage with a
sorghum plant.
– Count the aphids each day.
“Lecture” Activities
1. Mathematical modeling
a. Develop model
b. Determine parameter values from
clip-cage experiments
c. Run Matlab simulation with model
d. Compare simulation results with
population growth experiment data
“Lecture” Activities
1. Mathematical modeling
a. Develop model
b. Determine parameter values from
clip-cage experiments
c. Run Matlab simulation with model
d. Compare simulation results with
population growth experiment data
2. Discovery of model behavior
“Lecture” Activities
1. Mathematical modeling
a. Develop model
b. Determine parameter values from
clip-cage experiments
c. Run Matlab simulation with model
d. Compare simulation results with
population growth experiment data
2. Discovery of model behavior
3. Model analysis
a. Find eigenvalues and eigenvectors
b. Determine growth rate and proportions
Teaching Modeling Skills
•
Modeling process:
a.
b.
c.
d.
•
Develop model
Determine parameter values
Run Matlab simulation
Check simulation results
Issues:
– Data takes a long time to collect.
– The real model is 5-dimensional.
Presenting Bugbox, a simple computer
simulation for structured population dynamics!
Because Bugbox is a simulation, its behavior
doesn’t necessarily match any real insect
population. It functions as a biology lab for a
virtual world.
Boxbugs are simpler than real insects:
– They don’t move.
– Development rate is chosen by the experimenter.
– Each life stage has a distinctive appearance.
larva
pupa
adult