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Using Virtual Laboratories to Teach Mathematical Modeling Glenn Ledder University of Nebraska-Lincoln gledder@math.unl.edu

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

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

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

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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!

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Mathematical Models Independent Variable(s) Dependent Variable(s) Equations Narrow View Parameters Behavior Broad View (see Ledder, PRIMUS, Jan 2008)

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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.

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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.

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P. steadius Data

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Linear Regression On mechanistic grounds, the model is y = mx, not y = b + mx. Find m to minimize Solve by one-variable calculus.

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P. steadius Model

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P. speedius Data*

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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

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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

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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.

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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 – -------

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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 – -------

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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 – -------

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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

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P. speedius Model

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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.

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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

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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

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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.

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Finding the Growth Rate

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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

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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/

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