Parameter Identification in Partial Integro-Differential ... · 1/53 Temporal mismatch Population...
Transcript of Parameter Identification in Partial Integro-Differential ... · 1/53 Temporal mismatch Population...
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1/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Parameter Identificationin Partial Integro-Differential Equations
for Physiologically Structured Populations
Sylvia Moenickes, Otto Richter, and Klaus SchmalstiegInstitut fur Geookologie, Braunschweig
Presented at the COMSOL Conference 2008 Hannover
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2/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Temporal mismatch
Population dynamics of G. pulex
Identifiability of parameters
Future Applications
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3/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Temporal mismatch
Population dynamics of G. pulex
Identifiability of parameters
Future Applications
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4/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Climate change
I arboreal population dynamics
I microbial population dynamics
I freshwater amphipod population dynamics
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5/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Climate change
I arboreal population dynamics
I microbial population dynamics
I freshwater amphipod population dynamics
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6/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Interdependency in ecology
I birthI growth
I food availabilityI temperature conditionsI . . .
I reproduction
I food availabilityI weightI . . .
I death
I food availabilityI temperature conditionsI ageI . . .
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7/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Interdependency in ecology
I birthI growth
I food availabilityI temperature conditionsI . . .
I reproduction
I food availabilityI weightI . . .
I death
I food availabilityI temperature conditionsI ageI . . .
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8/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Interdependency in ecology
I birthI growth
I food availabilityI temperature conditionsI . . .
I reproductionI food availabilityI weightI . . .
I death
I food availabilityI temperature conditionsI ageI . . .
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9/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Interdependency in ecology
I birthI growth
I food availabilityI temperature conditionsI . . .
I reproductionI food availabilityI weightI . . .
I deathI food availabilityI temperature conditionsI ageI . . .
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10/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Measuring interdependency in ecology
I in-situ, highly dynamic
I in lab, highly specific
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11/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Measuring I
Suhling et al., 2008
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12/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Measuring I
Suhling et al., 2008
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13/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Measuring I
Suhling et al., 2008
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14/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Measuring I
Suhling et al., 2008
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15/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Measuring II
Schneider et al., 2008
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16/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Measuring II
Schneider et al., 2008
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17/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Temporal mismatch
Population dynamics of G. pulex
Identifiability of parameters
Future Applications
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18/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Individual growth g(F , T , w)
dwdt = g(F , T , w)
dwdt = γw
23 − ρw
dwdt = γ F
F+Fhw
23 − ρw
dwdt = Φ(T )(γ F
F+Fhw
23 − ρw)
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19/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Individual growth g(F , T , w)
dwdt = g(F , T , w)
dwdt = γw
23 − ρw
dwdt = γ F
F+Fhw
23 − ρw
dwdt = Φ(T )(γ F
F+Fhw
23 − ρw)
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20/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Individual growth g(F , T , w)
dwdt = g(F , T , w)
dwdt = γw
23 − ρw
dwdt = γ F
F+Fhw
23 − ρw
dwdt = Φ(T )(γ F
F+Fhw
23 − ρw)
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21/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Individual growth g(F , T , w)
dwdt = g(F , T , w)
dwdt = γw
23 − ρw
dwdt = γ F
F+Fhw
23 − ρw
dwdt = Φ(T )(γ F
F+Fhw
23 − ρw)
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22/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Weight structured population n(w , t)
Population density n varying with time t and weight w :
∂n(w ,t)∂t = −∂g(F ,T ,w)n(w ,t)
∂w
+ λ∂2n(w ,t)∂w2
− p−(F ,T ,w , . . .)
+ p+(F ,T ,w , . . .)
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23/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Weight structured population n(w , t)
Population density n varying with time t and weight w :
∂n(w ,t)∂t = −∂g(F ,T ,w)n(w ,t)
∂w + λ∂2n(w ,t)∂w2
− p−(F ,T ,w , . . .)
+ p+(F ,T ,w , . . .)
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24/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Weight structured population n(w , t)
Population density n varying with time t and weight w :
∂n(w ,t)∂t = −∂g(F ,T ,w)n(w ,t)
∂w + λ∂2n(w ,t)∂w2
− p−(F ,T ,w , . . .)
+ p+(F ,T ,w , . . .)
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25/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Weight structured population n(w , t)
Population density n varying with time t and weight w :
∂n(w ,t)∂t = −∂g(F ,T ,w)n(w ,t)
∂w + λ∂2n(w ,t)∂w2
− p−(F ,T ,w , . . .)
+ p+(F ,T ,w , . . .)
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26/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Mortality and Fertility
Mortality
p−(F ,T ,w , . . .) = µ(F ,T ,w , . . .)n(w , t)
1− µ(F ,T ,w , . . .) = (1− µ0)(1− µT (1− Φ(T )))(1− µFFµ
Fµ+F )
Fertility
p+(F ,T ,w , . . .) = B(F ,T ,w , . . .)Π(w)
B(F , t) =∫ wmax
wminrmax
FF+Fh
w23 n(w , t)dw
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27/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Mortality and Fertility
Mortality
p−(F ,T ,w , . . .) = µ(F ,T ,w , . . .)n(w , t)
1− µ(F ,T ,w , . . .) = (1− µ0)
(1− µT (1− Φ(T )))(1− µFFµ
Fµ+F )
Fertility
p+(F ,T ,w , . . .) = B(F ,T ,w , . . .)Π(w)
B(F , t) =∫ wmax
wminrmax
FF+Fh
w23 n(w , t)dw
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28/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Mortality and Fertility
Mortality
p−(F ,T ,w , . . .) = µ(F ,T ,w , . . .)n(w , t)
1− µ(F ,T ,w , . . .) = (1− µ0)(1− µT (1− Φ(T )))
(1− µFFµ
Fµ+F )
Fertility
p+(F ,T ,w , . . .) = B(F ,T ,w , . . .)Π(w)
B(F , t) =∫ wmax
wminrmax
FF+Fh
w23 n(w , t)dw
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29/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Mortality and Fertility
Mortality
p−(F ,T ,w , . . .) = µ(F ,T ,w , . . .)n(w , t)
1− µ(F ,T ,w , . . .) = (1− µ0)(1− µT (1− Φ(T )))(1− µFFµ
Fµ+F )
Fertility
p+(F ,T ,w , . . .) = B(F ,T ,w , . . .)Π(w)
B(F , t) =∫ wmax
wminrmax
FF+Fh
w23 n(w , t)dw
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30/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Mortality and Fertility
Mortality
p−(F ,T ,w , . . .) = µ(F ,T ,w , . . .)n(w , t)
1− µ(F ,T ,w , . . .) = (1− µ0)(1− µT (1− Φ(T )))(1− µFFµ
Fµ+F )
Fertility
p+(F ,T ,w , . . .) = B(F ,T ,w , . . .)Π(w)
B(F , t) =∫ wmax
wminrmax
FF+Fh
w23
n(w , t)dw
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31/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Mortality and Fertility
Mortality
p−(F ,T ,w , . . .) = µ(F ,T ,w , . . .)n(w , t)
1− µ(F ,T ,w , . . .) = (1− µ0)(1− µT (1− Φ(T )))(1− µFFµ
Fµ+F )
Fertility
p+(F ,T ,w , . . .) = B(F ,T ,w , . . .)Π(w)
B(F , t) =∫ wmax
wminrmax
FF+Fh
w23
n(w , t)dw
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32/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Mortality and Fertility
Mortality
p−(F ,T ,w , . . .) = µ(F ,T ,w , . . .)n(w , t)
1− µ(F ,T ,w , . . .) = (1− µ0)(1− µT (1− Φ(T )))(1− µFFµ
Fµ+F )
Fertility
p+(F ,T ,w , . . .) = B(F ,T ,w , . . .)Π(w)
B(F , t) =∫ wmax
wminrmax
FF+Fh
w23 n(w , t)dw
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33/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Temporal mismatch
Population dynamics of G. pulex
Identifiability of parameters
Future Applications
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34/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Parameter identification
I Minimization of
L(θ) =∑
j
∑i
(ni (tj)− ni (tj , θ))2
I throughI direct simulation with Comsol in MatlabI optimization toolbox of Matlab
I with model efficiency ME
ME = 1−∑
j
∑i (ni (tj)− ni (tj))
2∑j
∑i (ni (tj)− n(tj))2
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35/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Process analysis
I Statistical method for comparison of nested models:Likelihood ratio test
I The test statistic is −2 log(Lcomplex(θ)Lsimple(θ)
)n2 .
I It is χ2 distributed with the difference in numbers ofparameters as degrees of freedom.
I A significance level of 0.05 is applied.
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36/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Process analysis
I Statistical method for comparison of nested models:Likelihood ratio test
I The test statistic is −2 log(Lcomplex(θ)Lsimple(θ)
)n2 .
I It is χ2 distributed with the difference in numbers ofparameters as degrees of freedom.
I A significance level of 0.05 is applied.
![Page 37: Parameter Identification in Partial Integro-Differential ... · 1/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications Parameter](https://reader036.fdocuments.in/reader036/viewer/2022081404/5f0437ed7e708231d40ce5ab/html5/thumbnails/37.jpg)
37/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Process analysis
I Statistical method for comparison of nested models:Likelihood ratio test
I The test statistic is −2 log(Lcomplex(θ)Lsimple(θ)
)n2 .
I It is χ2 distributed with the difference in numbers ofparameters as degrees of freedom.
I A significance level of 0.05 is applied.
![Page 38: Parameter Identification in Partial Integro-Differential ... · 1/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications Parameter](https://reader036.fdocuments.in/reader036/viewer/2022081404/5f0437ed7e708231d40ce5ab/html5/thumbnails/38.jpg)
38/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Accounting for continuity
I Experiments on temperature response are based onhomogeneity assumption over intervals.
I Adequateness depends on response itself.
I Our idea: dynamic parameter identification under controlledtemperature regimes.
O’Neill temperature response function:
Φ(T ) = k( Tmax−TTmax−Topt
)pep
T−ToptTmax−Topt
with
p = 1400W 2(1 +
√1 + 40
W )2
W = (q10 − 1)(Tmax − Topt)
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39/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Accounting for continuity
I Experiments on temperature response are based onhomogeneity assumption over intervals.
I Adequateness depends on response itself.
I Our idea: dynamic parameter identification under controlledtemperature regimes.
O’Neill temperature response function:
Φ(T ) = k( Tmax−TTmax−Topt
)pep
T−ToptTmax−Topt
with
p = 1400W 2(1 +
√1 + 40
W )2
W = (q10 − 1)(Tmax − Topt)
![Page 40: Parameter Identification in Partial Integro-Differential ... · 1/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications Parameter](https://reader036.fdocuments.in/reader036/viewer/2022081404/5f0437ed7e708231d40ce5ab/html5/thumbnails/40.jpg)
40/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Accounting for continuity
I Experiments on temperature response are based onhomogeneity assumption over intervals.
I Adequateness depends on response itself.
I Our idea: dynamic parameter identification under controlledtemperature regimes.
O’Neill temperature response function:
Φ(T ) = k( Tmax−TTmax−Topt
)pep
T−ToptTmax−Topt
with
p = 1400W 2(1 +
√1 + 40
W )2
W = (q10 − 1)(Tmax − Topt)
![Page 41: Parameter Identification in Partial Integro-Differential ... · 1/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications Parameter](https://reader036.fdocuments.in/reader036/viewer/2022081404/5f0437ed7e708231d40ce5ab/html5/thumbnails/41.jpg)
41/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Accounting for continuity
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42/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Accounting for continuity
γ ρ Topt q10
A 0.32 0.06 18.3 2.58 97.9B 0.30 0.07 17.7 2.04 98.5C 0.23 0.05 17.6 2.11 99.1D 0.15 0.03 17.7 1.75 99.2
0.16 0.04 17.5 1.7
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43/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Accounting for unmanageability
I Experiments on food response would require food control.
I Food control is unjustifiably expensive.
I Our idea: dynamic parameter identification including fooddynamic.
Litter dynamics
dF
dt= L(t,T )− ImaxΦ(T )
F
F + Fh
∫ wmax
0w
23 n(w , t)dw
Possible approximation
F (t) = F0e−(t/t1)p1 (1− e−(t/t2)p2 )
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44/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Accounting for unmanageability
I Experiments on food response would require food control.
I Food control is unjustifiably expensive.
I Our idea: dynamic parameter identification including fooddynamic.
Litter dynamics
dF
dt= L(t,T )− ImaxΦ(T )
F
F + Fh
∫ wmax
0w
23 n(w , t)dw
Possible approximation
F (t) = F0e−(t/t1)p1 (1− e−(t/t2)p2 )
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45/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Accounting for unmanageability
I Experiments on food response would require food control.
I Food control is unjustifiably expensive.
I Our idea: dynamic parameter identification including fooddynamic.
Litter dynamics
dF
dt= L(t,T )− ImaxΦ(T )
F
F + Fh
∫ wmax
0w
23 n(w , t)dw
Possible approximation
F (t) = F0e−(t/t1)p1 (1− e−(t/t2)p2 )
![Page 46: Parameter Identification in Partial Integro-Differential ... · 1/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications Parameter](https://reader036.fdocuments.in/reader036/viewer/2022081404/5f0437ed7e708231d40ce5ab/html5/thumbnails/46.jpg)
46/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Accounting for unmanageability
Fh p1 p2 t1 t24.27 18.4 25.1 84.0 97.9
4.7 20 25.0 82 98
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47/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Identifying processes
I Mortality depends onmultiple processes.
I Effect of individual processoften unknown.
I Adequate simplificationscan be tested.
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48/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Identifying processes
µ0 µT µF RMSE ME (%)
• 268 99.0• 364 98.7
• 285 99.0
• • 263 99.0• • 264 99.0
• • 266 99.0
• • • 260 99.0
0.002 0.002 0.002
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49/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Identifying processes
µ0 µT µF RMSE Λ
0.004 268 •0.01 364
0.008 285 ◦0.003 0.004 263 •0.003 0.002 264 •
0.005 0.005 266 •0.002 0.003 0.004 260
0.002 0.002 0.002
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50/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Completing the circle
µ0 rmax ME (%)
0.006 0.007 94.1
0.01 0.008
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51/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
Temporal mismatch
Population dynamics of G. pulex
Identifiability of parameters
Future Applications
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52/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
What we learned
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53/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications
What we learned