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 vi1 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
0
