1. title Bo Deng UNL

3. B. Blaslus, et al Nature 1999 B. Blaslus, et al Nature 1999

4. Mark O’Donoghue, et al Ecology 1998 Mark O’Donoghue, et al Ecology 1998

5.  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 t having the same time dimension as t i j, x the state variables, and p the parameters.

6. 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.

7.  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.

10.  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.

11. 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. 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.

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

17. Dimensional Model Dimensionless Model

19. By Method of Line Search for local extrema By Method of Line Search for local extrema

20. Left Chirality and Right Chirality :

29.  By Taylor, s expansion:  Best-Fit Sensitivity :,  By Taylor, s expansion:  Best-Fit Sensitivity :,

30.  Best-Fit Sensitivity :,  Best-Fit Sensitivity :,

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

32. S. Ellner & P. Turchin Amer. Nat. 1995  Is Hare-Lynx Dynamics Chaotic? Rate of Expansion along Time Series ~ exp( )  Lyapunov Exponent > 0  Chaos

35. N.C. Stenseth Science 1995 1844 -- 1935

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

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