Using Virtual Laboratories to Teach Mathematical Modeling Glenn Ledder University of Nebraska-Lincoln

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 Conceptual Model Mathematical Model approximationderivation analysisvalidation A mathematical model represents a simplified view of the real world.

Mathematical Modeling Real World Conceptual Model Mathematical Model approximationderivation 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 Independent Variable(s) Dependent Variable(s) Equations Narrow View Parameters Behavior Broad View (see Ledder, PRIMUS, Jan 2008)

Presenting BUGBOX-predator, a real biology lab for a virtual world. 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. 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 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 rate s – search speed = · food total t space search t food space

Holling Type II Model Time is split between searching and feeding x – prey density y(x) – overall predation rate s – search speed = · · food total t search t total t space search t food space

Holling Type II Model Time is split between searching and feeding x – prey density y(x) – overall predation rate s – search speed = · · food total t search t total t space search t food space Each prey animal caught decreases the time for searching.

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 total t search t total t space search t food space search t total t feed t total t = 1 –

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 total t search t total t space search t food space search t total t feed t total t = 1 –

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 total t search t total t space search t food space search t total t feed t total t = 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. Semi-Linear Regression

P. speedius Model

Presenting BUGBOX-population, a real biology lab for a virtual world. 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 L t be the number of larvae at time t. Let P t be the number of juveniles at time t. Let A t be the number of adults at time t. L t +1 = + f A t P t +1 = 1 L t A t +1 = 1P t

Let L t be the number of larvae at time t. Let P t be the number of juveniles at time t. Let A t be the number of adults at time t. L t +1 = s L t + f A t P t +1 = p L t A t +1 = P t + a A t Final Boxbug model

Boxbug Computer Simulation A plot of X t / X t-1 shows that all variables tend to a constant growth rate λ The ratios L t :A t and P t :A t tend to constant values.

Finding the Growth Rate

Write as x t+1 = M x t. Run a simulation to see that x evolves to a fixed ratio independent of initial conditions. Obtain the problem M x t = λ x t. 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  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, ,