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:
(ti j , yi j ), i  1,2,..., k j , j  1,2,..., l
with
t( i 1) j  ti j  0.
 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:
 dx
  F ( x, p )
 dt
 x(0)  x0
with
t
having the same time dimension as
variables, and
p
the parameters.
t ij , x
the state
Inverse Problem :
 Let
f j ( x(tij , x0 , p ), p ) be the predicted states by the model to the
observed states, (tij , yij ). Then the inverse problem is to fit the model
to the data with the least dimensionless error between the predicted
and the observed:
E( F , f ) ( p, x0 ) 

l
kj
 w
j 1 i 1
2
ij
f j ( x(tij , x0 , p), p)  yij
2
The least error of the model for the process is
 ( F , f )  min E( F , f ) ( p, x0 )  E ( p , x0 )
( p , x0 )


with the minimizer ( p , x0 )
being the best fit of the model to
the data.

The best model for the process F satisfies
for all proposed models G .
 ( F , f )   (G, f )

Gradient Search Method for Local Minimizers:
In the parameter and initial state space
( p, x0 ) , a search path
( p, x0 )( s) satisfies the gradient search equation:
  ( p, x0 )
2


E
( p, x0 )
 s

l kj

 2 w2 ij  f j ( x(tij , x0 , p), p)  yij D( p , x0 ) f j ( x(tij , x0 , p), p)

j 1 i 1


 ( p(0), x (0))  ( p , x )
0
0
0, 0

A local minimizer is found as
( p , x0 )  lim ( p, x0 )( s)
s 
 My belief: The fewer the local minima,
the better the model.

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.
Example: Logistic equation with Holling Type II harvesting
x
ax
x
aK
x'  rx (1  ) 
 x '  x(1  x) 
with  
,  hK
K 1 h x
1  x
r
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.
Holling’s Type II Form (Can. Ent. 1959)
For One Predator:
Prey captured during T period of time
X C  (T  h X C ) a X
where
T = given time
a = encounter probability rate
h = handling time per prey
Solve it for the per-predator Predation Rate:
XC
aX

T
1  haX
XC
Type I Form, h = 0
1/h
Type II Form, h > 0
X
Dimensional Model
Dimensionless Model
By Method of Line Search
for local extrema
Left Chirality and Right Chirality :

vi 1

vi

 
vi  vi1  k

 0,
right chirality
 0,
left chirality
,
 By Taylor s expansion:
2

 2 E ( p  , x0 )  xi , 0  xi ,0 
1  2 E ( p  , x0 )  pi  pi  1



  
E ( p, x0 )  E ( p , x0 )  
 ...
 2 

 2 


2
( pi pi )  pi  2 ( xi ,0 xi ,0 )  xi ,0 
2
 Best-Fit Sensitivity :
1  2 E ( p  , x0 )
1  2 E ( p  , x0 )
S xi , 0 
S pi 
 2 ,
2  ( xi , 0 xi, 0 ) 2
2  ( pi pi )
 Best-Fit Sensitivity :
S pi 


0
 2
i
1  E( p , x )
2  ( pi p )
2
, Sx
i,0
1  2 E ( p  , x0 )

2  ( xi , 0 xi, 0 ) 2
 All models are constructed to fail against the test of time.
 Is Hare-Lynx Dynamics Chaotic?
Rate of Expansion along Time Series ~ exp(l)
l  Lyapunov Exponent > 0  Chaos
S. Ellner & P. Turchin
Amer. Nat.
1995
1844 -- 1935
N.C. Stenseth
Science
1995
Alternative Title:
Holling made trappers to drive hares to eat lynx
Dimension: n + m
Dimension: n + m - n - 1 + l +1 = n + m - n + l