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Simple coupled physical-biogeochemical models of marine ecosystems

Formulating quantitative mathematical models of conceptual ecosystems

MS320: John Wilkin

mathematic

al ^

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Why use mathematical models?

• Conceptual models often characterize an ecosystem as a set of “boxes” linked by processes

• Processes e.g. photosynthesis, growth, grazing, and mortality link elements of the …

• State (“the boxes”) e.g. nutrient concentration, phytoplankton abundance, biomass, dissolved gases, of an ecosystem

• In the lab, field, or mesocosm, we can observe some of the complexity of an ecosystem and quantify these processes

• With quantitative rules for linking the boxes, we can attempt to simulate the changes over time of the ecosystem state

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What can we learn?

• Suppose a model can simulate the spring bloom chlorophyll concentration observed by satellite using: observed light, a climatology of winter nutrients, ocean temperature and mixed layer depth …

• Then the model rates of uptake of nutrients during the bloom and loss of particulates below the euphotic zone give us quantitative information on net primary production and carbon export – quantities we cannot easily observe directly

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Reality Model

• Individual plants and animals • Many influences from nutrients and trace elements• Continuous functions of space and time• Varying behavior, choice, chance• Unknown or incompletely understood interactions

• Lump similar individuals into groups– express in terms of biomass and C:N ratio

• Small number of state variables (one or two limiting nutrients)

• Discrete spatial points and time intervals• Average behavior based on ad hoc assumptions• Must parameterize unknowns

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The steps in constructing a model

1) Identify the scientific problem(e.g. seasonal cycle of nutrients and plankton in mid-latitudes; short-term blooms associated with coastal upwelling events; human-induced eutrophication and water quality; global climate change)

2) Determine relevant variables and processes that need to be considered

3) Develop mathematical formulation

4) Numerical implementation, provide forcing, parameters, etc.

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State variables and Processes

“NPZD”: model named for and characterized by its state variables

State variables are concentrations (in a common “currency”) that depend on space and time

Processes link the state variable boxes

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Processes

• Biological:– Growth– Death– Photosynthesis– Grazing– Bacterial regeneration of nutrients

• Physical:– Mixing– Transport (by currents from tides, winds …)– Light– Air-sea interaction (winds, heat fluxes, precipitation)

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State variables and Processes

Can use Redfield ratio to give e.g. carbon biomass from nitrogen equivalent

Carbon-chlorophyll ratio

Where is the physics?

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Examples of conceptual ecosystems that have been modeled

• A model of a food web might be relatively complex– Several nutrients– Different size/species classes of phytoplankton– Different size/species classes of zooplankton– Detritus (multiple size classes)– Predation (predators and their behavior)

• Multiple trophic levels– Pigments and bio-optical properties

• Photo-adaptation, self-shading– 3 spatial dimensions in the physical environment, diurnal cycle of

atmospheric forcing, tides

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Silicic acid – im

portant

limitin

g nutrient in

N. Pacific

gelatinous zooplankton, euphausids, krill

copepods

ciliates

particulate silic

on

Fig. 1 – Schematic view of the NEMURO lower trophic level ecosystem model. Solid black arrows indicate nitrogen flows and dashed blue arrows indicate silicon. Dotted black arrows represent the exchange or sinking of the materials between the modeled box below the mixed layer depth.

Kishi, M., M. Kashiwai, and others, (2007), NEMURO - a lower trophic level model for the North Pacific marine ecosystem, Ecological Modelling, 202(1-2), 12-25.

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Soetaert K, Middelburg JJ, Herman PMJ, Buis K. 2000. On the coupling of benthic and pelagic bio-geochemical models. Earth-Sci. Rev. 51:173-201

Schematic of ROMS “Fennel” ecosystem model

Phytoplankton concentration absorbs light Att(x,z) = AttSW + AttChl*Chlorophyll(x,z,t)

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Examples of conceptual ecosystems that have been modeled

• In simpler models, elements of the state and processes can be combined if time and space scales justify this– e.g. bacterial regeneration can be treated as a flux from

zooplankton mortality directly to nutrients

• A very simple model might be just:

N – P – Z – Nutrients– Phytoplankton– Zooplankton

… all expressed in terms of equivalent nitrogen concentration

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ROMS fennel.h(carbon off, oxygen off, chl not shown)

http://clover.ocean.washington.edu/~neil/NPZvisualizer

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Mathematical formulation

• Mass conservation– Mass M (kilograms) of

e.g. carbon or nitrogen in the system

• Concentration Cn (kg m-3) of state variable n is mass per unit volume V

• Source for one state variable will be a sink for another

sinkssourcesMdt

d

n

nVCM

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e.g. inputs of nutrients from rivers or sediments

e.g. burial in sediments

e.g. nutrient uptake by phytoplankton

The key to model building is finding appropriate formulations for transfers, and not omitting important state variables

Mathematical formulation

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Baron Gottfried Wilhelm von Leibniz 1646-1716

Slope of a continuous function of x is dx

dffslope

Some calculus

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For example:

State variables: Nutrient and PhytoplanktonProcess: Photosynthetic production of organic matter

max ( )dP v f N P

dt

Nk

NNf

N )(

Large N

Small N

Michaelis and Menten (1913)

vmax is maximum growth rate (units are time-1) kn is “half-saturation” concentration; at N=kn f(kn)=0.5

max

max

( ) 1

( ) /

/n

n

f N

dP dt v P

f N N k

dP dt v N k P

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State variables: Nutrient and Phytoplankton

Process: Photosynthetic production of organic matter

max

max

( )

( )

dP v f N P

dtdN v f N P

dt

The nitrogen consumed by the phytoplankton for growth must be lost from the Nutrients state variable

The total inventory of nitrogen is conserved

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• Suppose there are ample nutrients so N is not limiting: then f(N) = 1

• Growth of P will be exponential

max

max

v t

dPv P

dt

P Ae

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• Suppose the plankton concentration held constant, and nutrients again are not limiting: f(N) = 1

• N will decrease linearly with time as it is consumed to grow P

max

dNv P

dt

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• Suppose the plankton concentration held constant, but nutrients become limiting: then f(N) = N/kn

• N will exponentially decay to zero until it is exhausted

max

max

n

n

v Pt

k

v PdNN

dt k

N Ae

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Can the right-hand-side of the P equation be negative?

Can the right-hand-side of the N equation be positive?

… So we need other processes to complete our model.

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Coupling to physical processes

Advection-diffusion-equation:

)()( ClossCgainCDCvCt

C is the concentration of any biological state variable

advectionturbulent mixing

Biological dynamics

physics

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winter spring summer fallI0

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Simple 1-dimensional vertical model of mixed layer and N-P ecosystem

• Windows program and inputs files are at: http://marine.rutgers.edu/dmcs/ms320/Phyto1d/– Run the program

called Phyto_1d.exe using the default inputfiles

• Sharples, J., Investigating theseasonal vertical structure of phytoplankton in shelf seas, Marine Models Online, vol 1, 1999, 3-38.

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Fig. 1. Schematic of the model grid, and the physical processes. Velocities and scalars are associated with the centres of a grid cell, and vertical turbulent fluxes with the lower boundary of a grid cell.

Fig. 2. Schematic diagram of the biological scalars and processes at each grid cell.

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vertical turbulent

mixing of phytoplankton

phytoplankton growth

grazing mortality

vertical sinking at velocity ws

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Change settings ….

Physicsc.dat: stronger PAR attenuation eliminates mid-depth chl-max

Phyto1d.dat: greater respiration rate delaysbloom until photosynthesis rate is greater

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winter spring summer fallI0

bloom

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winter spring summer fallI0

bloom secondary bloom

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winter spring summer fallI0

bloom secondary bloom