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title. An Inverse Problem :. Trapper Drove Hare to Eat Lynx. Bo Deng UNL. B. Blaslus , et al Nature 1999. Mark O’Donoghue , et al Ecology 1998. An empirical data of a physical process P is a set of observation time and quantities: with

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  1. title An Inverse Problem : Trapper Drove Hare to Eat Lynx Bo Deng UNL

  2. B. Blaslus, et al Nature 1999

  3. Mark O’Donoghue, et al Ecology 1998

  4. An empirical data of a physical process P is a set of • observation time and quantities: • with • The aim of mathematical modeling is to fit a mathematical form to • the data by one of two ways: • 1. phenomenologically without a conceptual model • 2. mechanistically with a conceptual model • We will consider only mathematical models of differential equations: • with thaving the same time dimension as t i j, x the state • variables, and p the parameters.

  5. Inverse Problem : • Let be the predicted states by the model to the • observed states, Then the inverse problem is to fit the model • to the data with the least dimensionless error between the predicted • and the observed: • The least error of the model for the process is • with the minimizer being the best fit of the model to • the data. • The best model for the process F satisfies • for all proposed models G .

  6. Gradient Search Method for Local Minimizers: In the parameter and initial state space , a search path satisfies the gradient search equation: A local minimizer is found as • My belief: The fewer the local minima, the better the model.

  7. Dimensional Analysis by the Buckingham Theorem: Old Dimension = m + n New Dimension = (m – n – 1) + n + l+ 1 = n + m –( n –l ) Degree of Freedom for the Best Fit = Old Dimension –New Dimension = n –l • A best fit by the dimensionless model corresponds to a (n –l)-dimensional surface of the same least error fit, i.e., best fit in general is not unique.

  8. Example: Logistic equation with Holling Type II harvesting where n = 1, m = 4, and m – n – 1 = 2. With best fit to l = 1 data set, there is zero, n – l = 0, degree of freedom.

  9. Basic Models Holling’s Type II Form (Can. Ent. 1959) For One Predator: Prey captured during T period of time where T = given time a = encounter probability rate h = handling time per prey Solve it for the per-predator Predation Rate: XC Type I Form, h = 0 1/h Type II Form, h > 0 X

  10. Dimensional Model Dimensionless Model

  11. By Method of Line Search for local extrema

  12. Left Chirality and Right Chirality :

  13. By Taylor,s expansion: • Best-Fit Sensitivity : • ,

  14. Best-Fit Sensitivity : • ,

  15. All models are constructed to fail against the test of time.

  16. Is Hare-Lynx Dynamics Chaotic? Rate of Expansion along Time Series ~ exp(l) l = Lyapunov Exponent > 0  Chaos Field Data S. Ellner & P. Turchin Amer. Nat. 1995

  17. Canadian Snowshoe Hare -Lynx Dynamics 1844 -- 1935 N.C. Stenseth Science 1995

  18. Alternative Title: Holling made trappers to drive hares to eat lynx

  19. Dimension: n + m Dimension: n + m - n - 1 + l+1 = n + m - n + l

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