Using Virtual
Laboratories to Teach
Mathematical
Modeling
Glenn Ledder
University 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 Modeling
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
A mathematical model represents a
simplified view of the real world.
Mathematical Modeling
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
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
Parameters
Independent
Variable(s)
Equations
Dependent
Variable(s)
Narrow View
Broad View
(see Ledder, PRIMUS, Jan 2008)
Behavior
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 Regression
On mechanistic grounds, the model is y = mx,
not y = b+mx.
Find m to minimize
F (m)   ( yi  mxi )
Solve by one-variable calculus.
2
P. steadius Model
P. speedius Data*
Holling Type II Model
• Time is split between searching and feeding
x – prey density y(x) – overall predation rate
s – search speed
space food
food
------- = --------- · ------total t search t space
y ( x)  s  x
Holling Type II Model
• Time is split between searching and feeding
x – prey density y(x) – overall predation rate
s – search speed
food search t space food
------= --------- · --------· ------total t
search
t
space
total t
y ( x)  ?  s  x
Holling Type II Model
• Time is split between searching and feeding
x – prey density y(x) – overall predation rate
s – search speed
food search t space food
------= --------- · --------· ------total t
search
t
space
total t
y ( x)  f ( y )  s  x
Each prey animal caught
decreases the time for
searching.
𝑓 0 = 1, 𝑓 ′ < 0,
𝑓 1 =0
Holling Type II Model
• Time is split between searching and feeding
x – prey density y(x) – overall predation rate
s – search speed
h – handling time
food search t space food
------= --------- · --------- · ------total t total t search t space
y ( x)  f ( y )  s  x
search t
feed t
--------= 1 – total
-------t
total t
𝑦 is prey / total time
ℎ is feed time per prey
Holling Type II Model
• Time is split between searching and feeding
x – prey density y(x) – overall predation rate
s – search speed
h – handling time
food search t space food
------= --------- · --------- · ------total t total t search t space
y ( x)  f ( y )  s  x
search t
feed t
--------= 1 – total
-------t
total t
f ( y )  1  hy
Holling Type II Model
• Time is split between searching and feeding
x – prey density y(x) – overall predation rate
s – search speed
h – handling time
food search t space food
------= --------- · --------- · ------total t total t search t space
search t
feed t
--------= 1 – total
-------t
total t
y ( x)  f ( y )  s  x
f ( y )  1  hy
1
sx
h x
qx
y
 1 1

1  shx s h  x a  x
Semi-Linear Regression
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)
G (a )  min
q
 ( y  qt ) 
2
i
4. Best a is the minimizer of G.
i
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.
+ fAt
Lt+1 =
Pt+1 = 1Lt
At+1 =
1Pt
Final Boxbug 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 = sLt
+ fAt
Pt+1 = pLt
At+1 =
Pt + aAt
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
𝐿 0 = 𝐿0 , 𝑃 0 = 𝑃0 , 𝐴 0 = 1
and growth rate 𝜆 for which 𝑁 1 = 𝜆𝑁(0).
𝐿 1 = 𝑠𝐿0 + 𝑓
𝑃 1 = 𝑝𝐿0
𝐿 1 = 𝜆𝐿0
𝑃 1 = 𝜆𝑃0
𝐴 1 = 𝑃0 + 𝑎
𝐴 1 =𝜆
Finding the Growth Rate
Eliminate 𝑃0 and 𝐿0 to get
𝜆 𝜆 − 𝑎 𝜆 − 𝑠 = 𝑝𝑓
This equation is already factored. There is a
unique solution larger than the maximum of 𝑎
and 𝑠.
Follow-up
• Write as xt+1 = Mxt .
• Run a simulation to see that x evolves to a fixed
ratio independent of initial conditions.
• Obtain the problem Mxt = λxt .
• 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.
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/