optimality-based modelling: from theory to …...2015/03/04 · 3. current distribution of trait...

Marine Ecosystem Modelling Symposium, AORI Mar. 4, 2015S. Lan Smith

S. Lan SmithMarine Ecosystem Dynamics Research Group, RCGC, JAMSTEC, Yokohama, Japan

CREST, Japan Science and Technology Agency, Tokyo, Japan

Optimality-based Modelling: from theory to implementation

It’s all about combining Traits & Trade-offs (Smith et al. L&O Review 2011, Smith et al. JPR Horizons 2014)

Sunlight

Nutrient Supply

αVmax

μmax

chl

α

Vmax

μmax

chl

Marine Ecosystem Modelling Symposium, AORI Mar. 4, 2015S. Lan Smith p. 2

マラソン選手は体重が少ないので力は強くないが速く走る事ができる

力士は体重が重いので力は強いが速く走ることはできない

体重

走る速さ

マラソン選手力士一般的な体型の人

Trade-offs: apply to humans as well as plankton

Different body types perform Optimally for different goals.


Theory: Optimality-based models of plankton

Assumptions

Natural Selection produces optimally adapted organisms

Fitness = Growth Rate Goal: Maximize Growth Rate

Considerable success over the past decade

Recent Advances in: ‘optimal foraging’, Photoacclimation & Primary Production

Major Challenge

Dynamic models consistent with Evolutionarily Stable Strategy (ESS) ESS formulated in terms of steady state (John Maynard Smith)

i.e., how to describe short-term dynamics, e.g., acclimation, consistent with evolutionary adaptation of the very ability to acclimate.

from the review by Smith et al. (Limnology & Oceanography 56, 2011)


Trait x should change in proportion to its effect on fitness, F:

dx = dx∂F(x,E)

dt ∂x dx: flexibility ~ diversity (trait distribution) E: Environment (nutrients, light, temperature, etc.)

For plankton F = Growth; dx/dt depends on E (Smith et al. L&O, 2011)

Acclimation Rates depend on: 1. possible range of trait values (adaptive capacity), 2. environmental variability 3. current distribution of trait values

Remaining Challenge: modelled diversity tends to collapse immigration required to maintain diversity (Bruggeman & Kooijman L&O 2007)

‘Adaptive Dynamics’: evolutionary changes McGill and Brown (An. Rev. Ecol. Evol. Syst. 2007), Litchman et al. (PNAS 2009)

‘adaptive dynamics’: species succession, communities Wirtz & Eckhardt (Ecol. Modell. 1996), Wirtz (J. Biotech. 2002), Abrams (J. Evol. Biol. 2005)

‘adaptive dynamics’: modelling changing trait values

Different trait values are optimal under different environmental conditions.


What can optimality be used to model?

Acclimation short-term changes e.g., seasonal change of a dog’s coat of hair changes in chl content of phytoplankton

Ecological Dynamics

species succession (changes in community composition)

Adaptation

long(er)-term changes evolution (genetic changes in a species)

The same approaches can be used to model all three. e.g., Wirtz and Eckhardt (Ecol. Mod., 1996), Merico et al. (Ecol. Mod., 2009)

What I do

Others also model


What to Optimize?

Nutrient uptake, e.g., Optimal Uptake

as in Smith et al. (2009)

N used for other enzymes

(max. rate)

N used for uptake

sites(affinity)

nutrient uptake

Autotrophic growth, e.g., iron-light-nitrogen

colimitation as in Armstrong (1999)

phytoplankton

iron used for light

harvesting

iron used for N

assimilation

N uptakelight

phytoplankton

nutrients zooplankton

defenseuptake

Community dynamics, e.g., grazers and phytoplankton

as in Merico et al. (2009)

Maximal net growth rate

Maximal specific growth rate

Maximal uptake rate

Quantity being

optimized

trade-off

(Smith et al. L&O 2011, fig. 2)


Trade-offs as ‘Hyper-Parameterizations’

‘Hyper-Parameters’ in hierarchical Bayesian modeling specify prior distributions of model parameters (e.g., Gelman et al. Bayesian Data Analysis, 2nd edition, 2004)

Trade-offs specify how the shapes of functional relationships may change, rather than fixing their shapes. Optimal Uptake kinetics

log NO3 (in seawater)

log

K NO

3

-2.5 -1.0 0.0

-3

-2

-1

0

n = 61 data pts.

0 200 400 600 800 1000

0.0

0.2

0.4

0.6

0.8

1.0

0 200 400 600 800 1000 0 200 400 600 800 1000

Upt

ake

Rat

e

NO3 in incubation expts.

Adaptive Response

Smith et al. (MEPS 2009)

Trade-offV m

ax

α


Shape-shifting of the Growth vs. Irradiance curve

RubiscoCHLa CarotPigments

Q Q0

?

f R

DIN

loss

f θ

1-f θ-f R

lossCO2

C-uptake

N-uptake

PONPOC

Gro

wth

Rat

e

Max

. Gro

wth

Rat

eLight

Flexible Response

Wirtz & Pahlow (MEPS 2010)

Optimal Resource Allocation subject to Trade-offs

Optimality-based Photoacclimation models have advantages compared to empirically-based functions (Smith & Yamanaka. Ecol. Modell. 2007)


Selected Examples of Trade-offs for Plankton

Affinity for nutrient vs. Maximum Uptake Rate Pahlow (MEPS 2005), Smith et al. (MEPS 2009, 2014)

Light Harvesting vs. Nutrient Uptake Pahlow (MEPS 2005, 2009, 2013), Wirtz and Pahlow (MEPS 2010) and effectively all “photo-acclimation” models *although many (e.g. Geider type) do not explicity employ trade-offs

Ingestion of Prey vs. Cost of Foraging Pahlow and Prowe (MEPS 2010)

a few examples from Smith et al. (L&O, 2011)

Trade-offs allow simple models to Represent Flexible Response

Costs vs. benefits in allocating limited resources (energy, nutrients)


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V� �=� μQ

�A� �s�i�n�g�l�e�,� �e�x�p�l�i�c�i�t� �e�q�u�a�t�i�o�n�:� μ� �=� �f�(�N�,� �I�)

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Vmax

α

�O�p�t�i�m�a�l� �U�p�t�a�k�e� �k�i�n�e�t�i�c�s� �(�P�a�h�l�o�w�.� �M�E�P�S�,� �2�0�0�5�;� �S�m�i�t�h� �e�t� �a�l�.� �M�E�P�S�,� �2�0�0�9�)�a�f�f�i�n�i�t�y� �(α�)� � �v�s�.� �m�a�x�.� �u�p�t�a�k�e� �r�a�t�e� �(Vmax�)

� � ∝ fA� � � � � �v�s�.� � � � � � � � � � � � � � � � � � � � ∝ �(1 − fA�)� � � � � � � � � � �.

1 fA 0

�O�p�t�i�m�a�l� �G�r�o�w�t�h� �m�o�d�e�l� �(�P�a�h�l�o�w� �a�n�d� �O�s�c�h�l�i�e�s�.� �M�E�P�S�,� �2�0�1�3�)

� � � � �N� �u�p�t�a�k�e� �(V�^ �)� � �v�s�.� � � � � �C� �f�i�x�a�t�i�o�n� �(μ�^ �I�)� � � � � � � � � � � � � ∝� fV ∝� �(�1� −� �

Qs� − fV�)� � QN

0.5 fV 0

New Simple, Flexible Phytoplankton model: FlexPFT

Just 1 diff. eqn. for dynamics of C biomass.

+ simple calc`ns for chl:C & N:C

submitted with revisions to JPR (March, 2015)


0-D (box) model of the mixed layer at stns. K2 & S1

−200

−150

−100

−50

0

MLD

(m)

Time

39.0 39.5 40.0 40.5 41.0 41.5 42.00.00

0.02

0.04

0.06

0.08

0.10

0.12

Day of Year

PAR

(μE/

sq. c

m/s

)

�P�A�R� �F�o�r�c�i�n�g�:� � �C�r�u�i�s�e� �O�b�s�.� �w�h�e�n� �a�v�a�i�l�a�b�l�e� �+� �i�n�t�e�r�p�o�l�a�t�i�o�n� �b�y� �I�d�e�a�l� �A�s�t�r�o�n�o�m�i�c�a�l� �E�q�n�.�

Surface

MLD

Light...

attenuation with depth

Variable Mixed Layer Depth (Forcing)

entrainment ofNO3 below ML

mixing acrossbase of ML

Inflexible PFT vs. FlexPFTa single PFT for each (NPD model), respectively

embedded in the same physical modelFitted to Obs. Data for NO3, chl and PP


Tim

e (d

ays)

Tim

e (d

ays)

fA fV

0

200

400

600

800

1000

1.0 0.8 0.6 0.4 0.2 0

0

200

400

600

800

1000

0.5 0.4 0.3 0.2 0.1 0

Optimal Growth model (Pahlow and Oschlies 2013)

N uptake (V^ ) vs. C fixation (μ̂I) ∝ fV ∝ (1 − Qs − fV) QN

0.5 fV 0

Vmax

α

Optimal Uptake kinetics (Pahlow 2005; Smith et al. 2009)

affinity (α) vs. max. uptake rate (Vmax) ∝ fA vs. ∝ (1 − fA)

.

1 fA 0

2010

20

11

2012

2010

20

11

2012

Intracellular Resource Allocation

Inflexible Control:Turn off Optimization:

constant fA constant fV

FlexPFT model Optimal Allocation

optimize fA | N

optimize fV | N, I

stn. K2

stn. S1

more allocation to light gathering at stn. K2

=> lower fA & fV (lower in winter)

more allocation to nutrient uptake at stn. K2 higher fA & fV <=(higher in summer)

Applied in a 0-D (box) model of the mixed layer for two time-series stns.


Inflexible PFT vs. FlexPFT applied to stns. K2 & S1

Less Difference at N-poor stn. S1 only

At N-rich stn. K2 Seasonal cycle of chl differs greatly

Each model was fitted to data (red dots) using the Adaptive Metropolis algorithm (Smith JGR 2011)

05

101520253035

0

5

10

15

20

25

30

0.5

1.0

1.5

0.30.40.50.60.70.80.9

0.5

1.0

1.5

2.0

0.12

0.14

0.16

0.18

0.20

Time (days) Time (days)

N (mmol L-1)

chl (mg m-3)

chl:C (g mol-1)

Phy Biomass (mmol C m-3) Q (mol N (mol C)-1)

0 200 400 600 800 10000 200 400 600 800 1000

P. Production(mg C m-3 d-1)

FlexPFT in colorsInflexible Control (grey)

2010 2011 2012 2010 2011 2012

02468

1012

0.0

0.5

1.0

1.5

2.0

0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

0.20.40.60.81.01.2

0.040.060.080.100.120.140.16

Time (days) Time (days)0 200 400 600 800 10000 200 400 600 800 1000

2010 2011 2012 2010 2011 2012

N (mmol L-1)

chl (mg m-3) chl:C (g mol-1)

Phy Biomass (mmol C m-3) Q (mol N (mol C)-1)

P. Production(mg C m-3 d-1)


FlexPFT gives different vertical dist. for chl vs. N biomass, PP 1-D GOTM model, 3 year simulation of subtropical stn. S1.

stn. S1

Importance of ‘Photo-acclimation’ is well known for subtropics (Ayata et al. JMS 2013)

most recent results 2015.01

0.0 0.2 0.4 0.6 0.8 1.0−200

−150

−100

−50

0

0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 30

obs. avg.FlexPFT

obs. avg.FlexPFTchl:N = 1.59

obs. avg.FlexPFT

Dep

th (m

)

(mg m-3) (mmol N m-3) (mg C m-3 d-1)

mean vertical chl mean vertical pon mean vertical P. Prod.

An Inflexible Control Model, chl:N:C ratios

Here the two models perform similarly.

In both, chl decreases too steeply with depth.

Need to add DON to the model to stop ‘nutrient trapping’.

0.0 0.2 0.4 0.6 0.8 1.0−200

−150

−100

−50

0

0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30

obs. avg.Inflexible PFT



Dep

th (m

)


(mg m-3) (mg C m-3 d-1) (mmol N m-3)


FlexPFT gives different vertical dist. for chl vs. N biomass, PP 1-D GOTM model, 3 year simulation of subarctic stn. K2

stn. K2


An Inflexible Control Model, chl:N:C ratios

The inflexible model cannot consistently reproduce both chl & PON.

mean vertical pon

(mmol N m-3)0.0 0.5 1.0 1.5 2.0

mean vertical P. Prod.

(mg C m-3 d-1)0 10 20 30

mean vertical chl

0.0 0.5 1.0 1.5−100

−80

−60

−40

−20

0

(mg m-3)

obs. avg.FlexPFT

obs. avg.FlexPFTchl:N = 1.59

obs. avg.FlexPFT

Dep

th (m

)

(mg m-3) (mmol N m-3) (mg C m-3 d-1)


0.0 0.5 1.0 1.5−100

−80

−60

−40

−20

0

0.0 0.5 1.0 1.5 2.0 0 10 20 30




Dep

th (m

)


(mg m-3) (mg C m-3 d-1) (mmol N m-3)

Flexible chl:N:C is important at subarctic stn. K2 & likely over much of the ocean -- NOT only at low latitudes.


FlexPFT gives dynamic vertical distr. for chl:N and Q1-D GOTM model, 3 year simulations of stns. S1 & K2.


The Importance of ‘Photo-acclimation’ is well known for sub-tropics (Ayata et al. JMS 2013)

stn. S1

stn. S1


New Flexible ZooEFT model developed in FY2014

0 90 180 270 3600

0.1

0.2

0.3

0.4

DIN (nutrient) (mm

ol m

-3)

0 90 180 270 360Grazing Rate

KTWNo KTW

0

0.2

0.4

0.6

0.8

1.0

(d-1

)

Time (d)

Emergent Trade-off KTW: High Phy Size Diversity, more even Growth No KTW: Faster Peak Growth, but more variable, Low Phy size diversity

ZooEFT

Grazing PreferenceRel.

Abu

ndan

ce

0 90 180 270 3600

0.2

0.4

0.6

0.8 Phy size

log

ESD

(μm

)

Time (d)

0 90 180 270 3600

0.2

0.4

0.6

0.8

1.0

Phy Growth Rate

(d-1

)

0 90 180 270 3600

0.2

0.4

0.6

0.8

1.0Phy Size Diversity

log σ

(μm

)

Size as ESD (μm) 0.2 0.5 2.0 5.0 20.0 100.0

0

1

2

3

4

5 N = 10 N = 1 N = 0.1

Gro

wth

Rat

e (d

-1)Simple Trade-off

assumed for Phy:Smaller is better at low NutrientBigger is better at high Nutrient

Periodic Nutrient supply was imposed.

Based on the ‘Kill-the-Winner’ model (Vallina et al. Nat. Comm. 2014; Vallina et al. Prog. Oceanogr. 2014)

Results here are from a simple size-based model using Monod Growth kinetics. We plan to publish this soon.

The ZooEFT model has also been coupled with our PhyEFT model in the 0-D setup for stns. K2 & S1. Next step: try it in the 1-D setup.


Non-Adaptive vs. Flexible models

How do changing env. conditions impact:

Overal ecosystem response to, e.g., climate change or human nutrient inputs?

Biodiversity?

B o t h t h e b i o m a s s & r e s p o n s e o f e a c h

F l e x P F T c a n c h a n g e .

Bio

mas

s EnvironmentalChange

・Each FlexPFT adapts to changing environment (e.g., light, nutrients).

Different AdaPFTs, each having different flexible abilities

(b) New Flexible FlexPFT model

O n l y t h e d i s t r i b u t i o n o f b i o m a s s c a n c h a n g e .

Plankton Functional Types (PFT) (a) Existing PFT models

Bio

mas

s

EnvironmentalChange

C o m p e t i t i v e a b i l i t i e s r e m a i n

f i x e d f o re a c h P F T

N uptakeability

C fix’nability

Different PFTs, each having different abilities


New size-based PhyEFT model Size scaling of Traits (input parameters)

as in Wirtz (Mar. Biol. 2013)Dark Light

Adaptive Response to env. (light & N)

Big is better at high N

Small is better at low N

(esp. at high light)

0.2 1.0 5.0 20.0 100.01

2

3

4

5

6

7

8

0.2 1.0 5.0 20.0 100.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.2 1.0 5.0 20.0 100.0

1.0

1.5

2.0

2.5

0.2 1.0 5.0 20.0 100.0

0.02

0.03

0.04

0.05

Pot. max. growth rate

μm

ax (d

-1)

Pot. max. affinity Subsistence cell quota

ESD (micrometers) ESD (micrometers)

Pot. max. uptake rate

A0

(m3

(mm

ol C

)-1 d

-1)

V0

(mol

N (m

ol C

)-1 d

-1)

Q0

(mol

N (m

ol C

)-1)

0

1

2

3

4

0.2 0.5 2 5 20 50

N = 10 mmol m-3

N = 1 mmol m-3

N = 0.1 mmol m-3

Effe

ctiv

e m

ax. (

Mon

od) g

row

th ra

te

μm

ax (d

-1)

Strong Light Limitation, S(I) = 0.2

0

1

2

3

4

0.2 0.5 2 5 20 50

Effe

ctiv

e m

ax. (

Mon

od) g

row

th ra

te μ

max

(d-1

)

Light Replete, S(I) = 1.0

empirical allometriesLitchman et al. (Ecol. Lett. 2007) Edwards et al. (L&O 2007) Marañon et al. (Ecol. Lett. 2013)


V� �=� μQ



Vmax

α�O�p�t�i�m�a�l� �U�p�t�a�k�e� �k�i�n�e�t�i�c�s�

�(�P�a�h�l�o�w�.� �M�E�P�S�,� �2�0�0�5�;� �S�m�i�t�h� �e�t� �a�l�.� �M�E�P�S�,� �2�0�0�9�)�a�f�f�i�n�i�t�y� �(α�)� � �v�s�.� �m�a�x�.� �u�p�t�a�k�e� �r�a�t�e� �(Vmax�)

� � ∝ fA� � � � � �v�s�.� � � � � � � � � � � � � � � � � � � � ∝ �(1 − fA�)� � � � � � � � � � �.

1 fA 0



Qs� − fV�)� � QN

0.5 fV 0

Cell Size (ESD, mm) Cell Size (ESD, mm)


Old NPZD approach vs. New Ecologically Flexible Types

‘adaptive dynamics’ can represent the dynamics of the trait distribution (α,σ) using only two differential equations per EFT.

Environmental Change

D i s c r e t e t r a i t v a l u e s

Because traits are fixed, model predictions may become invalid

Flexible trait distributions change to suit environ-mental conditions.

C o n t i n u o u s d i s t r i b u t i o n o f t r a i t v a l u e s

Plankton Functional Types (PFT) Ecologically Flexible Types

Biom

ass

Biom

ass

Biom

ass

Biom

ass

Trait value Trait value

Trait value (here, affinity)Environmental Change

(a) Existing NPZD models (b) EFT-model to be developed

・separate differential eqs. for each PFT・traits are fixed (constant) for each PFT

・・・

・・・

Ks Vmax

Gro

wth

Rate

Upta

ke R

ate

nutrient conc. (S)

Affinity-based kinetics

affinity

Trade off*e.g., affinity, Vmax, size

mean trait value:

std. dev. of trait value:

PhyPhyZoo

Zoo


Size-Scaling of Traits => Size-Scaling of Growth

0.2 1.0 5.0 20.0 100.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.2 1.0 5.0 20.0 100.0

1.0

1.5

2.0

2.5

0.2 1.0 5.0 20.0 100.0

0.02

0.03

0.04

0.05

Pot. max. affinity

Subsistence cell quota

ESD (μm)


A^ 0 (

m3

(mm

ol C

)-1 d

-1)

V^ 0 (

mol

N (m

ol C

)-1 d

-1)

Q0

(mol

N (m

ol C

)-1)

Empirical Size-scalings

Response depends on both cell size and EnvironmentNew modelling

framework to relate lab. measurements to the dynamic response of phytoplankton communities.

0

5

10

15

0.0

0.2

0.4

0.6

0.8 αuptake (m

3 (mm

ol C) -1 d -1)

αgr

owth

(m3

(mm

ol N

)-1 d

-1)

0.2 0.5 2.0 5.0 20 50

Light Replete

Effective values of Affinityfor Growth & Uptake, respectively

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5Effective max. (Monod) growth rate

0.2 0.5 2.0 5.0 20 50

μm

ax (d

-1)

eff

0

2

4

6

8

10

w

k

0.2 0.5 2.0 5.0 20 50

KV (dashed)

ESD (μm)

(μm

ol L

-1)

Effective half-saturation ‘constants’for Growth & Uptake, respectively

Kμ (solid)

Very Dark

0.0

0.1

0.2

0.3

0.4

0.5

Effective max. (Monod) growth rate

0.2 0.5 2.0 5.0 20 50

μm

ax (d

-1)

eff

0

2

4

6

8

10

0.0

0.1

0.2

0.3

0.4

0.5

αuptake (m

3 (mm

ol N) -1 d -1)

αgr

owth

(m3

(mm

ol C

)-1 d

-1)

0.2 0.5 2.0 5.0 20 50

Effective values of Affinityfor Growth & Uptake, respectively

ESD (μm)

0

2

4

6

8

10

w

k

0.2 0.5 2.0 5.0 20 50

(μm

ol L

-1)

Effective half-saturation ‘constants’for Growth & Uptake, respectively

KV (dashed)

Kμ (solid)

N = 0.1 μMN = 1.0 μMN = 10 μM

Data for KV (open circles)

(Edwards et al. 2012)

Km < KV (growth) (uptake)

Morel (J. Phycol. 1987)

Here size-scaling for Km depends on light and nutrient environment.


V� �=� μQ



Vmax

α�O�p�t�i�m�a�l� �U�p�t�a�k�e� �k�i�n�e�t�i�c�s�

�(�P�a�h�l�o�w�.� �M�E�P�S�,� �2�0�0�5�;� �S�m�i�t�h� �e�t� �a�l�.� �M�E�P�S�,� �2�0�0�9�)�a�f�f�i�n�i�t�y� �(α�)� � �v�s�.� �m�a�x�.� �u�p�t�a�k�e� �r�a�t�e� �(Vmax�)

� � ∝ fA� � � � � �v�s�.� � � � � � � � � � � � � � � � � � � � ∝ �(1 − fA�)� � � � � � � � � � �.

1 fA 0



Qs� − fV�)� � QN

0.5 fV 0


Phy+ZooEFT in 0-D model of stns. K2 & S1 (9 yr. sim.)

stn. S1, subtropicalstn. K2, subarctic

Decreasing diversity

Decreasing diversity

KTW sustains Phy diversity, & its seasonality with modest effects on chl, PP

Next: Examine how diversity relates to function Compare to Obs. size distr.

Kill-the-Winner Passive Prey Switching Kill-the-Winner Passive Prey Switching


Summary & Conclusions

Optimality-based modelling has well established theory.and is recently being applied to model plankton in the ocean.

Flexible Phytoplankton Functional Type: FlexPFT developed Optimizes specific growth rate w.r.t. to 2 trade-offs: 1. Optimal Uptake (OU) kinetics 2. C (energy) vs. N aquisition Only 1 diff. eq. per PFT Trait-based => can Directly use reported size-scalings Framework for modelling Size Diversity of phy.

Flexible models respond quite differently vs. Inflexible PFTs e.g., Variable chl & N content => more dynamic Some models of optimal Grazing developed, Much more work needed ...


May impact dynamic functioning & resilience of ecosystems e.g., predator-prey feedbacks (Tirok et al. PLoS One 6, 2011)

Major Challenge: How to model realistic biodiversity in dynamic, heterogenous environments?

1. Add more species competitive displacement of many species e.g., Moisan et al. (Ecol. Modell., 2002)

2. Resolve traits better differentiate strategies more precisely e.g., Litchman and Klausmeier (Am. Naturalist 157, 2001)

3. ‘adaptive dynamics’ acclimation within species competitive displacement between species & trait evolution (Wirtz. J. Biotechnol. 97, 2002; Smith et al. L&O. 56, 2011; Tirok et al. PLoS One 6, 2011)

Biodiversity: beyond coarse large-scale patterns

Easiest

MoreDifficult

Simplistic

MoreRealistic


Effective Monod (growth) params. vs. MM params.Burmaster (Am. Nat. 1979) MM kinetics (uptake) + Droop model (Growth) => Monod kinetics (growth)Similarly, we combine OU kinetics (uptake) + Optimal Growth (OG) model => Effective Monod Params.

where

or, equivalently

Now, observed size-scalings for MM & Droop parameters can be used to predict size-scalings for Monod parameters.


Shannon Diversity Index Differential entropy for N discrete Obs. or PFTs for log-normal distribution (model)

H = -∑i

N pi ln(pi ) h = 12 + ln(s√2p) + m

where pi is the probability of where m is mean (in log space) belonging to the i-th class s is the std. deviation depends on ‘binning’ choice depends on the assumed distr’n. and on N ,which is arbitrary log-normal is reasonable for plankton

Instead, for obs. & PFT models (Quintana et al. 2008, Schartau et al. 2010)

Diversity Index: Discrete vs. Continuous

Qintana et al. (L&O. Methods 2008) who denote diff. entropy as m

applied to obs. by Schartau et al. (JPR 2010)

Non-parametric estimate of h from discrete data (yi)

h

Advantages of h Consistently quantifies diversity despite different # of obs. or different # of PFTs (in models)

Dis-advantages of h not bounded: can be < 0 not as intuitive for intrepretation


Comparing h vs. H

0 200 400 600 800 1000012345

initial N = 200

initial N = 50

initial N = 20

0 200 400 600 800 1000−1.0

−0.5

0.0

0.5

1.0

overlayed forN = 20, 50, 200

H

h

Time (days)

FlexPFT (red)Non-adaptive PFT (grey)

2010 2011 2012

Model Set-up for different number, N of either PFTs, or AdaPFTs

in a simple NO3 - PxN - D model (no Zoo)

mortality for mi = m0PavgPi which maintains biodiversity (Record et al. ICES J. Mar. Sci. 2013)

0.5 50cell size (ESD, mm)

PTot

PTot/20PTot/50

PTot/20

= initial total biomass, divided evenly among N

N = 20


N = 50


N = 200

h is independent of N (# of PFTs), whereas H increases with NOnly slight difference between FlexPFT vs. control

h is better for obs.-model & model-model comparsions

with Size scaling of

Traits

0.2 1.0 5.0 20.0 100.01

2

3

4

5

6

7

8

0.2 1.0 5.0 20.0 100.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.2 1.0 5.0 20.0 100.0

1.0

1.5

2.0

2.5

0.2 1.0 5.0 20.0 100.0

0.02

0.03

0.04

0.05

Pot. max. growth rate

μm

ax (d

-1)

Pot. max. affinity Subsistence cell quota

ESD (micrometers) ESD (micrometers)


A0

(m3

(mm

ol C

)-1 d

-1)

V0

(mol

N (m

ol C

)-1 d

-1)

Q0

(mol

N (m

ol C

)-1)

3 yr. sim-ulation for

stn. K2


Comparing h: FlexPFT vs. Inflexible Control200 PFTs in a simple 0-D model: NO3 - Px - D model (no Zoo)

simplified Kill-the-Winner mortality, mPi = m0PavgPi , maintains biodiversity

(Record et al. ICES J. Mar. Sci. 2013)

0.2

0.51.02.0

5.010.0

0 200 400 600 800 10001.77

1.78

1.79

1.80

1.81

1.82

0.2

0.51.02.0

5.010.0

0 200 400 600 800 10001.0

1.2

1.4

1.6

med

ian

size

(μm

)si

ze d

iver

sity

, h


Diversity maintained by inter-dependent mortality ratesstn. K2 stn. S1

FlexPFTcontrol

2010 2011 2012 2010 2011 2012


Comparing h: FlexPFT vs. Inflexible Control200 PFTs in a simple 0-D model: NO3 - Px - D model (no Zoo)

independent quadratic mortality, mPi = m0Pi

2, gives Competitive Exclusion (Record et al. ICES J. Mar. Sci. 2013)

0.2

0.51.02.0

5.01020

0 200 400 600 800 1000

0.0

0.5

1.0

1.5

0.2

0.51.02.0

5.01020

0 200 400 600 800 1000

−0.5

0.0

0.5

1.0

1.5


med

ian

size

(μm

)si

ze d

iver

sity

, h

stn. K2 stn. S1Competitive exclusion (independent mortality rates)

2010 2011 2012 2010 2011 2012

optimality-based modelling: from theory to …...2015/03/04 · 3. current distribution of trait...

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