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

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

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

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

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

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 . . .

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 . . .

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 . . .

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 . . .

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

11/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications

Measuring I

Suhling et al., 2008

12/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications

Measuring I

Suhling et al., 2008

13/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications

Measuring I

Suhling et al., 2008

14/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications

Measuring I

Suhling et al., 2008

15/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications

Measuring II

Schneider et al., 2008

16/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications

Measuring II

Schneider et al., 2008

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

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)

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)

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)

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)

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 , . . .)

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 , . . .)

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 , . . .)

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 , . . .)

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

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

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

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

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

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

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

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

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

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.

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.

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.

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)

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)

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)

41/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications

Accounting for continuity

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

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 )

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 )

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 )

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

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.

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

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

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

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

52/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications

What we learned

53/53 Temporal mismatch Population dynamics of G. pulex Identifiability of parameters Future Applications

What we learned