Simple coupled physical-biogeochemical
models of
marine ecosystems
Formulating quantitative
mathematical models of
conceptual ecosystems
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MS320: John Wilkin
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
• 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
Model
• 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 midlatitudes; 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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gelatinous
zooplankton,
euphausids,
krill
copepods
ciliates
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 biogeochemical models. Earth-Sci. Rev.
51:173-201
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Schematic of ROMS “Fennel”
ecosystem model
Phytoplankton concentration absorbs light
Att(x,z) = AttSW + AttChl*Chlorophyll(x,z,t)
dI
= Att (z) * I(z)
dz
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
d
M  sources  sinks
dt
M   CnV
n
• Source for one state
variable will be a sink for
another
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Mathematical formulation
d
V Cn = sourcesn - sinksn + å transfern, j
dt
j
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
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Some calculus
Slope of a
continuous
function of x is
df
slope  f  
dx
Baron Gottfried
Wilhelm von Leibniz
1646-1716
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For example:
State variables: Nutrient and Phytoplankton
Process: Photosynthetic production of organic matter
d
P  vmax f ( N ) P
dt
N
f (N ) 
kN  N
Large N
Small N
f (N )  1
dP dt  vmax P
f ( N )  N / kn
dP dt   vmax N / kn  P
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
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State variables: Nutrient and Phytoplankton
Process: Photosynthetic production of organic matter
d
P  vmax f ( N ) P
dt
d
N  vmax f ( N ) P
dt
d
(P + N ) = 0
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
dP
 vmax P
dt
• Growth of P will be exponential
P  Ae
vmax t
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• Suppose the plankton concentration held
constant, and nutrients again are not limiting:
f(N) = 1
dN
 vmax P
dt
• N will decrease linearly with time as it is
consumed to grow P
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• Suppose the plankton concentration held
constant, but nutrients become limiting: then
f(N) = N/kn
vmax P
dN

N
dt
kn
• N will exponentially decay to zero until it is
exhausted
N  Ae
vmax P

t
kn
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d
P = vmax f (N )P - ...
dt
d
N = -vmax f (N )P + ...
dt
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:

C    v C  DC  gain (C )  loss (C )
t
turbulent
Biological dynamics
advection
mixing
C is the concentration of any biological state variable
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I0
winter
spring
summer
fall
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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 input
files
• Sharples, J., Investigating the
seasonal 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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Change settings ….
Physicsc.dat: stronger PAR attenuation
eliminates mid-depth chl-max
Phyto1d.dat: greater respiration rate delays
bloom until photosynthesis rate is greater
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I0
winter
spring
bloom
summer
fall
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I0
winter
spring
bloom
summer
fall
secondary bloom
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I0
winter
spring
bloom
summer
fall
secondary bloom
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