title bo deng unl. b. blaslus, et al nature 1999 b. blaslus, et al nature 1999
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Bo DengUNL
B. Blaslus, et al Nature1999
B. Blaslus, et al Nature1999
Mark O’Donoghue, et al Ecology 1998
Mark O’Donoghue, et al Ecology 1998
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.
ljkiyt jjiji ,...,2,1,,...,2,1),,(
.0)1( jiji tt
)0(
),(
0xx
pxFdt
dx
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 .
)),,,(( 0 ppxtxf ijj
).,( ijij yt
l
j
k
iijijjijfF
j
yppxtxfwxpE1 1
2
02
0),( )),,,((),(
),(),(min),( 00),(),( 0
xpExpEfF fFxp
),( 0 xp
),(),( fGfF
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.
),( 0xp
))(,( 0 sxp
),())0(),0((
)),,,(()),,,((2
),(),(
0,000
1 10),(0
2
020
0
xpxp
ppxtxfDyppxtxfw
xpEs
xp
l
j
k
iijjxpijijjij
j
))(,(lim),( 00 sxpxps
Dimensional Analysis by the Buckingham Theorem:
Old Dimension = m + nOld Dimension = m + n
New Dimension = (m – n – 1) + n + l + 1 = n + m – ( n – l ) 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 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
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.
,
with 1
)1('
1
)1(' Kh
r
Ka
x
xxxx
xh
xa
K
xrxx
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
XC
1/hhaX
aX
T
X C
1
XaXhTX CC ) (
Prey captured during T period of time
Type I Form, h = 0
Type II Form, h > 0
Dimensional ModelDimensional Model
Dimensionless ModelDimensionless Model
By Method of Line Search for local extrema
By Method of Line Search for local extrema
iv 1iv
kvv ii
1 chiralityright , 0
chiralityleft , 0
Left Chirality and Right Chirality :
By Taylor,s expansion:
Best-Fit Sensitivity : ,
By Taylor,s expansion:
Best-Fit Sensitivity : ,
20
2
)(
),(
2
1
ii
p pp
xpES
i 20,0,
02
)(
),(
2
10,
ii
x xx
xpES
i
...)(
),(
2
1
)(
),(
2
1),(),(
2
0,
0,0,2
0,0,
022
20
2
00
i
ii
iii
ii
ii x
xx
xx
xpE
p
pp
pp
xpExpExpE
Best-Fit Sensitivity : ,
Best-Fit Sensitivity : ,
20
2
)(
),(
2
1
ii
p pp
xpES
i2
0,0,
02
)(
),(
2
10,
ii
x xx
xpES
i
All models are constructed to fail against the test of time. All models are constructed to fail against the test of time.
S. Ellner & P. TurchinAmer. Nat.1995
Is Hare-Lynx Dynamics Chaotic? Rate of Expansion along Time Series ~ exp()Lyapunov Exponent > 0 Chaos
N.C. StensethScience1995
1844 -- 1935
Alternative Title:
Holling made trappers to drive hares to eat lynx
Dimension: n + m Dimension: n + m
Dimension: n + m - n - 1 + l +1 = n + m - n + l Dimension: n + m - n - 1 + l +1 = n + m - n + l
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