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

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Presentation on theme: "Using Virtual Laboratories to Teach Mathematical Modeling Glenn Ledder University of Nebraska-Lincoln"— Presentation transcript:

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

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

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

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

5 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!

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

7 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.

8 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.

9 P. steadius Data

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

11 P. steadius Model

12 P. speedius Data*

13 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

14 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

15 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.

16 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 –

17 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 –

18 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 –

19 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

20 P. speedius Model

21 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.

22 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

23 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

24 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.

25 Finding the Growth Rate

26

27 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

28 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, ,


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