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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/