Using Virtual Laboratories to Teach

Mathematical Modeling

Glenn LedderUniversity of Nebraska-Lincoln

gledder@math.unl.edu

Mathematical modeling is much more than “applications of mathematics.”

Mathematical modeling is much more than “applications of mathematics.”

“Mathematical modeling is the tendon that connects the muscle of mathematics to the bones of science.”

GL

Mathematical ModelingReal

WorldConceptual

ModelMathematical

Modelapproximation derivation

analysisvalidation

A mathematical model represents a simplified view of the real world.

Mathematical ModelingReal

WorldConceptual

ModelMathematical

Modelapproximation derivation

analysisvalidation

A mathematical model represents a simplified view of the real world.

• We want answers for the real world.

• But there is no guarantee that a model will give the right answers!

Mathematical Models

IndependentVariable(s)

DependentVariable(s)

Equations

Narrow View

Parameters Behavior

Broad View

(see Ledder, PRIMUS, Jan 2008)

Presenting BUGBOX-predator, a real biology lab for a virtual world.

http://www.math.unl.edu/~gledder1/BUGBOX/

The BUGBOX insect system is simple:– The prey don’t move.– The world is two-dimensional and homogeneous.– There is no place to hide.– Experiment speed can be manipulated.– No confounding behaviors.– Simple search strategy.

Presenting BUGBOX-predator, a real biology lab for a virtual world.

http://www.math.unl.edu/~gledder1/BUGBOX/

The BUGBOX insect system is simple:– The prey don’t move.– The world is two-dimensional and homogeneous.– There is no place to hide.– Experiment speed can be manipulated.– No confounding behaviors.– Simple search strategy.

But it’s not too simple:– Randomly distributed prey.– “Realistic” predation behavior, including random movement.

P. steadius Data

Linear RegressionOn mechanistic grounds, the model is y = mx, not y = b + mx.

Find m to minimize

2)()( ii mxymF

Solve by one-variable calculus.

P. steadius Model

P. speedius Data*

Holling Type II Model• Time is split between searching and feeding

x – prey density y(x) – overall predation rates – search speed

------- = --------- · ------- foodtotal t

spacesearch t

foodspace

xsxy )(

Holling Type II Model• Time is split between searching and feeding

x – prey density y(x) – overall predation rates – search speed

------- = --------- · --------- · ------- foodtotal t

search ttotal t

spacesearch t

foodspace

xsxy ?)(

Holling Type II Model• Time is split between searching and feeding

x – prey density y(x) – overall predation rates – search speed

------- = --------- · --------- · ------- foodtotal t

search ttotal t

spacesearch t

foodspace

Each prey animal caught decreases the time for searching.

,

xsyfxy )()(

Holling Type II Model• Time is split between searching and feeding

x – prey density y(x) – overall predation rates – search speed h – handling time

------- = --------- · --------- · ------- foodtotal t

search ttotal t

spacesearch t

foodspace

xsyfxy )()(

search ttotal t

feed ttotal t--------- = 1 – -------

is prey / total timeis feed time per prey

Holling Type II Model• Time is split between searching and feeding

x – prey density y(x) – overall predation rates – search speed h – handling time

------- = --------- · --------- · ------- foodtotal t

search ttotal t

spacesearch t

foodspace

xsyfxy )()(

search ttotal t

feed ttotal t--------- = 1 – -------

hyyf 1)(

Holling Type II Model• Time is split between searching and feeding

x – prey density y(x) – overall predation rates – search speed h – handling time

------- = --------- · --------- · ------- foodtotal t

search ttotal t

spacesearch t

foodspace

search ttotal t

feed ttotal t--------- = 1 – -------

xaqx

xhsxh

shxsxy

11

1

1

xsyfxy )()( hyyf 1)(

Fitting y = q f ( x; a):1. Let t = f (x ; a) for any given a.

2. Then y = qt, with data for t and y.

3. Define G(a) by (linear regression sum)

4. Best a is the minimizer of G.

2)(min)( iiqqtyaG

Semi-Linear Regression

P. speedius Model

Presenting BUGBOX-population, a real biology lab for a virtual world.

http://www.math.unl.edu/~gledder1/BUGBOX/

Boxbugs are simpler than real insects: They don’t move. Each life stage has a distinctive appearance.

larva pupa adult

Boxbugs progress from larva to pupa to adult. All boxbugs are female. Larva are born adjacent to their mother.

Boxbug Species 1 Model*

Let Lt be the number of larvae at time t.

Let Pt be the number of juveniles at time t.

Let At be the number of adults at time t.

Lt+1 = + f At

Pt+1 = 1 Lt

At+1 = 1Pt

Let Lt be the number of larvae at time t.

Let Pt be the number of juveniles at time t.

Let At be the number of adults at time t.

Lt+1 = s Lt + f At

Pt+1 = p Lt

At+1 = Pt + a At

Final Boxbug model

Boxbug Computer Simulation

A plot of Xt/Xt-1 shows that all variables tend to a constant growth rate λ

The ratios Lt:At and Pt:At tend to constant values.

Finding the Growth Rate• Find the initial condition

, and growth rate for which .

Finding the Growth RateEliminate and to get

This equation is already factored. There is a unique solution larger than the maximum of and .

• Write as xt+1 = M xt .• Run a simulation to see that x evolves to a fixed

ratio independent of initial conditions.• Obtain the problem M xt = λ xt .• Develop eigenvalues and eigenvectors.• Show that the term with largest |λ| dominates

and note that the largest eigenvalue is always positive.

• Note the significance of the largest eigenvalue.• Use the model to predict long-term behavior and

discuss its shortcomings.

Follow-up

Online Resources• www.math.unl.edu/~gledder1/MathBioEd/

G.Ledder, Mathematics for the Life Sciences: Calculus, Modeling, Probability, and Dynamical Systems, Springer (2013?) [Preface, TOC]

G.Ledder, J.Carpenter, T. Comar, ed., Undergraduate Mathematics for the Life Science: Models, Processes, & Directions, MAA (2013?) [Preface, annotated TOC]

G.Ledder, An experimental approach to mathematical modeling in biology. PRIMUS 18, 119-138, 2008.

• www.math.unl.edu/~gledder1/Talks/