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