teaching mathematics to biologists and biology to mathematicians
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Teaching Mathematics to Biologists and Biology to Mathematicians. Gretchen A. Koch Goucher College MathFest 2007. Introduction. Who: Undergraduate students and faculty What: Improving quantitative skills of students through combination of biology and mathematics - PowerPoint PPT PresentationTRANSCRIPT
Teaching Mathematics to Biologists and Biology to Mathematicians
Gretchen A. Koch
Goucher College
MathFest 2007
Introduction
Who: Undergraduate students and faculty What: Improving quantitative skills of
students through combination of biology and mathematics
When: Any biology or mathematics course Simple examples interspersed throughout
semester Common example as theme for entire semester
How??
Communication is key Talk with colleagues in natural sciences Use the same language Make the connections obvious
Example: Why is Calculus I required for many biology and chemistry majors??
Case studies, ESTEEM, and the BioQUEST way
Have an open mind and be creative
What is a case study?
Imaginative story to introduce idea Self-discovery with focus
Ask meaningful questions Build on students’ previous knowledge Students expand knowledge through research
and discussion. Assessment
Case Studies – Beware of…
Clear objectives = easier assessment Clear rubric = easier assessment Focused questions = easier assessment Too much focus = students look for the “right”
answer Provide some starting resources
Continue building your database Have clear expectations (Communication!) Be flexible
Where do I start???http://bioquest.org/icbl/
C3: Cal, Crabs, and the Chesapeake Cal, a Chesapeake crabber, was sitting at the
end of the dock, looking forlorn. I approached him and asked, “What’s the matter, Cal?” He replied, “Hon – it’s just not the same anymore. There are fewer and fewer blue crabs in the traps each day. I just don’t know how much longer I can keep the business going. You’re a mathematician – and you always say math is everywhere…where’s the math in this???”
Case Analysis – Use for Discussion What is this case about? What could be causing the blue crab
population to decrease? Can we predict what the blue crab population
will do? Can we find data to show historic trends in
the blue crab population? How will you answer these questions?
A Good Starting Place for Students
What do you know? What do you need to know?
Learning Objectives - Mathematics Use different mathematical models to explore
the population dynamics Linear, exponential, and logistic growth models
Precalculus level Continuous Growth ESTEEM Module
Predator-prey model Calculus, Differential Equations, Numerical Methods Two Species ESTEEM Module
SIR Model Calculus, Differential Equations, Numerical Methods SIR ESTEEM Module
Learning Objectives - Biology
Explore the reasons for the decrease in the crab population Habitat Predators Food Sources Parasites Invasive species
In field experiments Journal reviews of ongoing experiments
Assessment and Evaluation Plan Homework questions to demonstrate
understanding of use of ESTEEM modules Homework questions to demonstrate
comprehension of topics presented in ESTEEM modules
Group presentations of background information
Exam questions to demonstrate synthesis of mathematical concepts using different examples
Sources
Blue Crab Chesapeake Blue Crab Assessment 2005 Maryland Sea Grant The Living Chesapeake
Coast, Bay & Watershed Issues Blue Crabs
Blue Crab:http://www.chesapeakebay.net/blue_crab.htm
But – what’s the answer??
Assessments and objectives vary Knowledge of tools and structure Adopt and adapt
Continuous Growth Models Module
First Growth Model
Suppose you ask Cal to keep track of the number of crabs he catches for 10 days. He gives you the following:
Do you see a pattern?
Day 1 2 3 4 5 6 7 8 9 10
Number of Crabs
30 65 47 145 163 185 245 234 122 140
Linear Growth Model
Simplest model:
where C is the number of crabs on day t, and D is some constant number.
Questions to ask: What is D? Can you describe it in your own words? What’s another form for this model? Describe what this model means in terms of the crabs. Does this model fit the data? Why or why not? Is this model realistic?
( 1) ( )C t C t D
ESTEEM Time!
Continuous Growth Module
Summary of Manipulations
Entered data in yellow areas Clicked on “Plots-Size” tab
Manipulated parameters using sliders until fit looked “right”
Asked questions about what makes it right
Exponential Growth Model
Simplest model:
where C is the number of crabs on day t, and r is some constant number.
Questions to ask: What is r? Can you describe it in your own words? What’s another form for this model? Describe what this model means in terms of the crabs. Does this model fit the data? Why or why not? Is this model realistic?
( 1) ( ) * ( )C t C t r C t
ESTEEM Time!
Documentation Continuous Growth Module
Compare the two models…
Why can the initial population be zero in the linear growth model, but not in the exponential growth model?
Why do such small changes in r make such a big difference, but it takes large changes in D to show a difference?
What do these models predict will happen to the number of crabs that Cal catches in the future?
Logistic Growth Model Canonical model:
where C is the number of crabs on day t, and r and K are constants.
Questions to ask: What are r and K? Can you describe them in your own
words? Describe what this model means in terms of the crabs. Does this model fit the data? Why or why not? Is this model realistic?
* ( )*( ( ))( 1) ( )
r C t K C tC t C t
K
Further Analysis
What does the initial population need to be for each of the three models to fit the data well?
Why is the logistic model more realistic? How did the parameters (D, r, K) affect the
models? What does each model say about the total
capacity of Cal’s traps? Do these models give an accurate prediction
of the future of the crab population?
Let’s kick it up a notch!
How do we model the entire crab population? According to
http://www.chesapeakebay.net/blue_crab.htm, blue crabs are predators of bivalves.
Cannibalism is correlated to the bivalve population.
Predator-Prey Equations
Canonical example (Edelstein-Keshet):
Assumptions (pg 218): Unlimited prey growth without predation Predators only food source is prey. Predator and prey will encounter each other.
dx dy
ax bxy cy dxydt dt
Put it into context!
B B
C C
dBB BC
dtdC
C BCdt
dxax bxy
dtdy
cy dxydt
is the birth rate, and is the death rate for species .
( ) is the population of crabs, while ( ) is the population
of bivalves at time .
What does this mean for the variables in the canonical
i i i
C t B t
t
example? What do the terms mean?
Why does multiplication give likelihood of an encounter ?? Law of Mass Action (Neuhauser)
Given the following chemical reaction
the rate at which the product AB is produced by colliding molecules of A and B is proportional to the concentrations of the reactants.
Translation to mathematics Rates = derivatives, k is a number What about [A] and [B]?
,A B AB
rate = [ ][ ]k A B
Another version
Cushing:
What are the variables? Put them into context.
What’s the extra term? Did the assumptions change?
1 11 12
2 21
' ( )
' ( )
x x r a x a y
y y r a x
ESTEEM Two-Species Model
Isolation (discrete time):
What kind of growth? What are the terms and variables?
N1(t1) N1( t) r1 N1(t)[K1 N1(t)]
K1
N2(t 1) N2(t) r2 N2(t)[K2 N2(t)]
K2
ESTEEM Two-Species Model
Discussion Questions What do the terms mean? Which species is the predator, which is the prey? What other situations could these equations describe? Why discrete time? For what values of the rate constants does one species
inhibit the other? Have no effect? Have a positive effect? Can we derive the continuous analogs?
N1(t1) N1( t) r1 N1(t)[K1 N1(t) N2(t)]
K1
N2(t 1) N2(t) r2 N2(t)[K2 N2( t) N2 (t)]
K2
ESTEEM Time!
Documentation Two-Species Module
Summary of Manipulations
Use sliders to change values of parameters. Examine all graphs. Columns B and C have formulas for
numerical method.
Discussion Questions
How did one species affect the other? What did the different graphs represent? Did one species become extinct? How can you have 1.25 crabs? What would happen if there was a third
species? Write a general set of equations (cases as relevant).
Can you determine the numerical method used?
Simple SIR Model
Yeargers:
Susceptible, Infected, Recovered Given the above equations, explain the
assumptions, variables, and terms.
( ) ( )
( ) ( ) ( )
( )
dSr S t I t
dtdI
r S t I t a I tdtdR
a I tdt
Connections to Case Study and Beyond Possible ideas for research projects
Parasites and crabs Is there a disease affecting the crab population? Pick an epidemic, research it, and model it.
Analytical or numerical solutions Make teams of biology majors and math majors.
ESTEEM module…
SIR ESTEEM Module - Equations
(1) S(t 1) b[S(t) R(t)] S(t) (1 d) [1 jV j(t)j
],
(2) I j (t 1) b I j (t) I j( t) (1 d k j rj) S(t) (1 d)[ jV j( t)],
(3) R (t 1) R( t) (1 d) rjI j( t)j
,
(4) U (t1) b [U (t) V j(t)j
] U ( t) (1 d ) [1 jI j (t)j
],
(5) V j( t1) V j (t) (1 d ) U (t) (1 d ) [ jI j (t)].
Hosts (S, I, R) are infected by vectors (U, V) that can carry one of three strains of the virus (i=1, 2, 3).
ESTEEM Time!
Documentation SIR ESTEEM Module
Red boxes are for user entry.
SIR Module Discussion
Can you draw a diagram representing the SIR model?
What are all of the variables and parameters in the SIR model?
Can you find the continuous analog for the system? Can you rewrite the system in matrix form? What numerical method was used? Why did some values of the parameters work, while
others did not?
Conclusion
Many, many ways to bring biology into the classroom
Build on students’ intuition and knowledge Make obvious connections between ideas Don’t be afraid to try something new. Experiment and experiment some more! Have fun!
Works Consulted/CitedTexts:Cushing, J.M. (2004) Differential Equations: An Applied Approach. Pearson Prentice Hall.
Edelstein-Keshet, L. (1988) Mathematical Models in Biology. Birkhäuser.
Neuhauser, C. (2004) Calculus for Biology and Medicine. 2 ed. Pearson Education.
Yeargers, E.K., Shonkwiler, R.W., and J.V. Herod. (1996) An Introduction to the Mathematics of Biology with Computer Algebra Models. Birkhäuser.
Online Sources:BioQUEST SourcesBioQUEST: http://bioquest.org
ICBL: Investigative Case Based Learning: http://bioquest.org/icbl/
ESTEEM Module Documentation Continuous Growth Models Documentation (John R. Jungck, Tia Johnson, Anton E. Weisstein, and Joshua Tusin):
http://www.bioquest.org/products/files/197_Growth_models.pdf
SIR Model Documentation (Tony Weisstein): http://www.bioquest.org/products/files/196_sirmodel.doc
Two Species Documentation (Tony Weisstein, Rene Salinas, John Jungck): http://www.bioquest.org/products/files/203_TwoSpecies_Model.doc
Blue Crab ResourcesBlue Crab: http://www.chesapeakebay.net/blue_crab.htm
Chesapeake Blue Crab Assessment 2005: http://hjort.cbl.umces.edu/crabs/Assessment05.html
Maryland Sea Grant The Living Chesapeake Coast, Bay & Watershed Issues Blue Crabs: http://www.mdsg.umd.edu/issues/chesapeake/blue_crabs/index.html