Individual-Based Modeling in Ocean Ecology:
Where Behavior, Physiology and Physics Meet
Hal Batchelder
Oregon State University
Supported by NSF and
NOAA
within the U.S. GLOBEC
Northeast Pacific Program
IBM Outline
• Introduction to i-state distribution and i-state configuration
models
• How they differ
• Why IBM’s
• Advantages and Disadvantages
• Eulerian-Lagrangian Coupled Approaches and Details
• Examples
• Design of Marine Protected Areas for Scallops
• Nearshore retention (copepods in EBC upwelling regions; ADR)
• DVM of dinoflagellates using a cell N quota model
• Connectivity and Retention through Lagrangian Approaches
• Considerations
•Take Home Messages
• Challenges and Opportunities
Individual Based Modeling
Ecosystem Model
N
P
Z
Franks et al., 1986
SOME BIOMASS
Vitals:
380 lbs, 7’1”;
Vitals:
380 lbs, 7’1”;
MORE BIOMASS
Vitals:
~380 lbs,
?
=
Vitals:
380 lbs, 7’1”;
Vitals:
~380 lbs,
?
=
Vitals:
380 lbs, 7’1”; one mouth;
Regularly puts foot in mouth
(figuratively)
Vitals:
~380 lbs, 40 mouths;
Actually able to put foot
in mouth
Euphausia pacifica life stages
N2
Metanauplius
Calyptopi
Adult
Individual Size
•
•
•
•
•
Impacts preferred prey type (abundance/size)
Impacts growth rate
Impacts mortality when size-dependent
Impacts behavior
Impacts internal pools (lipid reserves)
Euphausia pacifica life stages
Stage-specific CW
N2
Metanauplius
~7 μg ind-1
Calyptopi
~3.2 μg ind-1
Adult
~4000 μg ind-1
Euphausia pacifica life stages
Stage-specific CW
N2
Metanauplius
~7 μg ind-1
571 indiv.
Calyptopi
~3.2 μg ind-1
1250 indiv.
Adult
~4000 μg ind-1
1 indiv.
Allometric Relationships are Important
Robin Ross (1982)
Allometric Relationships are Important
(here it is weight specific relation)
Robin Ross (1982)
Euphausia pacifica life stages
Stage-specific CW
N2
Metanauplius
~7 μg ind-1
571 indiv.
ΣR=529.2 ug C d-1
ΣG=519.6 ug C d-1
Calyptopi
~3.2 μg ind-1
1250 indiv.
ΣR=633.6 ug C d-1
ΣG=425 ug C d-1
Adult
ΣR=122.9 ug C d-1
ΣG=26 ug C d-1
~4000 μg ind-1
1 indiv.
Bioenergetics of an Individual Process
R (ug C d-1) = f(Weight, Prey, Temp)
Body Size
Prey
Temperature
A Stage Progression Model
E. pacifica Belehradek function for time to
stage as function of temperature
Basic Form is:
Di = ai (T + b)c
Di is the time (days) from egg to stage i
ai is a stage specific constant
b is a stage-independent shift in temperature
c is assumed to be -2.05 (commonly observed
from experiments; determines the curvature)
What if low food conditions delay development?
Revised Form is:
Di = [ai (T + b)c] / [1 – e-kP]
Data from Ross (1982) and
Feinberg et al. (2006)
Interindividual variation
in lipid weight of C5 stage
of Calanus pacificus
Laboratory reared individuals
(range of hi to low food) varied
by a factor of ca. 2.5; lipid
content in field collected
individuals even more variable
(ca. 2.8)
2.8
2.5
Hakanson (1984, Limnol.Oceanog.)
i-state Distribution Models
•fundamental tools of demographic theory
•produce differential or difference equations
•examples:
•NPZ+ models
•Lotka-Volterra predator-prey models
•McKendrick-von Foerster equations
Suppose:
One population; two important dimensions control
dynamics: individual age and individual size; given the
assumption that all individuals experience the same
environment (global mixing), then all individuals with the
same i-state will have the same dynamics and can be
treated collectively.
Suppose: Only indiv body size and life-stage are important to dynamics…
Then: Could model population using n life-stages, each having mn wt classes.
Within Stage Weight
LS1
LS2
Life Stage
LS3
LS4
…
LSn
W1
W2
W3
W4
…
Wmn
What if: There are many more dimensions important to dynamics?
“It is impossible to predict the response of all but the
very simplest natural systems from knowledge of current
environmental stimuli alone. The problem is that the past
of the system affects its response in the present.”
Caswell and John (1992, p. 37)
System State = f(History,Curr. Envir.)
both are required to describe the systems behavior
(deterministic) or probability distribution of systems
behavior (stochastic)
Individual Size
•
•
•
•
Impacts preferred prey type (abundance/size)
Impacts growth rate
Impacts mortality when size-dependent
Impacts behavior
Some early classic examples…
Intraspecific Effects - Initial Condition Sensitivity
Interspecific Effects – Relative Size
All figures are from Huston, M., D.
DeAngelis, and W. Post. 1988. New
computer models unify ecological
theory. BioScience 38 (10), 682691.
i-state configuration models
(aka Individual Based Models)
Each individual has a vector of characteristics associated with it
Examples are:
•Body size (weight, length)
•Age
•Reproductive Condition
= f (history)
•Nutritional (structural or physiological) Condition
•Behavior
•Location = Defines Present Environment
Conditions in which i-state distribution models are insufficient
and i-state configuration models (IBMs) are necessary:
1) Complicated i-states –
•
Many elements in i-state configuration vector; numerical
solutions as ‘distribution’ difficult
2) Small populations
•
Demographic analysis of endangered species
•
Viability of small populations
3) Local spatial interactions important
•
Spatial heterogeneity of the environment
•
Local interaction of individuals
4) Size- or individual-specific behaviors
ZP
Advantages of i-state configuration (IBMs)
1) Biology is often mechanistically explicit. (not hidden in
differential equations).
2) Biological-Physical-Chemical Interactions are clearly detailed.
3) Individual is the fundamental biological unit, thus it is natural and
intuitive to model at that level, rather than at the population level.
4) Allows explicit inclusion of an individual’s history and behavior.
5) History-Spatial Heterogeneity interactions ‘easily’ handled.
Costs Involved in IBM Approach
1)
Difficult to implement feedback from IBM (Lagrangian) to underlying
Eulerian model, esp. across multiple trophic levels
1)
2)
Consumption (depletion) of prey (E) by predators (L)
•
Assume not important (Batchelder & co. 1989,1995)
•
Conversion to concentrations per grid cell (Carlotti & Wolf
1998)
Requirement for Large Numbers of Particles
•
Difficult to simulate realistic abundances
•
Each particle may represent one (IBM) or a variable number of
identical individuals (Lag. Ens. Method/Superindividuals)
3)
Difficult (Impossible?) to simulate density dependence
4)
Extensive Computation Penalty
•
5)
Biological/biochemical processes for individuals are many and
complex
Increased knowledge about the system (this might be a good thing)
Design of Marine Protected Areas
The NW Atlantic Scallop Example
Scallop Larval Drift from Proposed Closed Regions
Issues: larval repopulation of source regions, as well
as non-closed regions; Long-term effects of
marine protected areas
Transport patterns
Retention effect of
circulation over a single 40day pelagic period within
the fall climatology.
•There is exchange between
closed areas 1 and 2.
•Area 1 is largely selfseeding; Area 2 seeds both
areas.
Source
10 Year Scallop Simulation w/ 1
spawning per year; 40 day larval
drift; individual surviving
scallops plotted (red are oldest
individuals)
No Closed Regions
Closed Regions
Impacts of Dispersal
Single, patchy
Population (open)
Metapopulation
(structured connectance)
High
From C. Grant Law (unpubl.)
Separate
(closed)
Low
Population Connectivity
Modified from Harrison and Taylor (1997)
Transport patterns
From C. Grant Law (unpubl.)
Questions



How connected are different
populations and does
connectivity change with
population structure or
physical forcing?
Are all populations equally
valuable when protected?
Do some regions act
primarily as sources and
others as sinks?



How often is a given area
dependent on recruits from
elsewhere?
Under which conditions is a
given area self-seeding and
how often are those
conditions present?
Are there regions of the coast
that are particularly robust in
terms of self seeding and
which also act frequently as a
source for remote areas?
Modified from C. Grant Law (unpubl.)
Management History
Pre-Closure Distribution



NE side of
Georges Bank
NE side of
Nantucket shoals
Head of Hudson
Canyon
From C. Grant Law (unpubl.)
Management History
Post-Closure Distribution





CLII north & south
CLI SW side of
Georges Bank
NE side of
Nantucket shoals
Head of Hudson
Canyon
Poor recruitment in
NLS and VBC
closed areas
From C. Grant Law (unpubl.)
Zooplankton Population Dynamics in 2D
The Oregon Upwelling System
Processes and Environmental Variables Influencing Organism Growth and Number
Spatially-Explicit Model
Physical Exchange
B, 
Predation T, B, , L
Ingestion T, B, , L, P
Egestion, P
Age, Size, Number
of Organisms
Starvation T, B, P, 
Migration T, B, P
Respiration T, B, 
Bioenergetic Model
T = Temperature; B=Behavior; =Turbulence; P=Prey; L=Light
Modeling Approach
(Eulerian-Lagrangian Coupling)
Physical Forcing
(light,wind,IC's)
2D or 3D
Eulerian
Model
IBM with simulated
Lagrangian Particles
Individual
Zooplankton
Characteristics
(wt,stage,condition,
sex, position)
Eulerian Fields
(velocity,
temperature
light, food, K)
Population Characteristics
(Demography) and
Spatial Distribution
Biological Organisms are not Passive Tracers
Euphausia pacifica
at NH25 (Aug 4, 2000, daytime)
Depth Range of Layer Sampled
0-10m
10-20m
20-50m
Nauplii
Calyptopes
50-100m
Furcilia
All Stages are in upper
Adults
20 m during Night
Juveniles
100-150m
150-200m
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
Density (# m-3 )
Figure courtesy of J. Keister
Magnitude of Diel Vertical Migration by Life Stage
Shelf Stations
Slope Stations
Life History Stage
0
0
25
25
50
50
75
75
100
Depth (m)
Depth (m)
Life History Stage
100
125
The top of the vertical bar
represents the nighttime
Average WMD. The
bottom of the bar
represents the daytime
Average WMD. The height
of the bar therefore
represents the magnitude
of DVM.
150
175
200
Based on 6 day-night paired MOCNESS
From shelf stations and 8 day-night pairs
From slope stations.
Vance et al. (unpublished)
Individual Based Copepod Model (IBM)
• Bioenergetics based model
– dW/dt = Assimilation - Respiration
• Growth is a function of weight, hunger condition, ambient
food
• Reproduction within C6 females with weight specific
allocation between somatic and reproductive growth
• Stage-specific, spatially-constant and weight-based mortality
• Diel Vertical Migration behavior dependent on
–
–
–
–
–
light
size (weight)
hunger condition
food resources
proximity to boundaries
10 m during night
160 m during day
Batchelder et al. (2002, PiO)
Batchelder and Williams (1995) Individual-based modelling of the
population dynamics of Metridia lucens in the North Atlantic. ICES J.
Mar. Sci., 52, 469-482.
Runge, J. A. 1980. Effects of hunger and season on
the feeding behavior of Calanus pacificus. Limnol.
Oceanogr., 25, 134-145.
~2X
starved
Batchelder, H. P. 1986. Phytoplankton
balance in the oceanic subarctic Pacific:
grazing impact of Metridia pacifica.
Mar. Ecol. Prog. Ser., 34, 213-225.
fed
Size (S)
Light
Food (P)
Hunger (H)
Slows upmig
Boundary
(Ns,Nb)
Slows downmig
H
Batchelder et al. (2002, PiO)
Physical Model
• Southward wind-stress
forcing of 0.5 dyne/cm2,
either constant or alternating
on/off with 5 or 10 day
intervals
 200 m 
• 2d (x-z) Vertical slice
• Time-dependent, hydrostatic,
Boussinesq, Navier-Stokes
• Finite difference
• KPP mixing
• Explicit mixing-length
Bottom Boundary Layer
• 500 < dx (m) < 1500
• 1.5 m < dz (m) < 3.7
• Topography for Newport, OR
• Initialized w/ April
climatology
 100 km 
Batchelder et al. (2002, PiO)
2D Upwelling Scenario Simulations
N
P
Z
Batchelder et al. (2002, PiO)
No-DVM Simulation
(PTM forced with Eulerian Concentrations of Prey,
Velocities, and Kv)
Recently layed
clutches in hi food
region
Day 20
Weight loss
below
mixed layer
Day 40
Starvation Mortality
Few Nearshore
Day 80
Size of bubble is proportional to individual weight
Batchelder et al. (2002, PiO)
DVM Simulation
(PTM forced with Eulerian Concentrations of Prey,
Velocities, and Kv)
Middepth aggregation offshore
Large Individuals
Inshore
Day 20
No reproduction &
mortality loss offshore
Nearshore reproduction
and retention
Day 40
Population
nearshore
only
Day 80
Size of bubble is proportional to individual weight
Batchelder et al. (2002, PiO)
Nutrient Quota Based DVM
Of Dinoflagellates
• diverse vertical patterns of populations (subsurface aggregations,
multiple depth aggregations, day-night differences)
• Nitrogen Quota IBM (internal nutrient status impacts VM)
• 1D w/ specified vertical nutrient profiles and vertical diffusivity
• How is the vertical pattern controlled by MLD, internal waves and
light intensity?
• Use average net growth rate as a measure of fitness
• 9 physiological parameters (Qmin, Qmax, α [PvI slope], μmax, Vm, σ
(descent thresh), γ [ascent thresh], λ [resp rate], g0 [dark N
uptake offset]).
Ji and Franks (2007, MEPS)
MLD and Migration Pattern
For both 10m and
20m MLD, cells are
able to balance their
need for light and
nutrients by
occupying the
pycnocline/nutricline.
No DVM.
MLD = 10m
Ji and Franks (2007, MEPS)
Subsurface vs. DVM
10m
Higher light level at 10m yields
higher net growth rates than at
20m for subsurface individuals.
With an imposed photo-/geotaxis
DVM (open bars) ANGR
distribution is shifted to the left
(poorer growth) for 10m MLD, but
shifted to the right (improved
growth) for 20m MLD.
Imposed DVM broadens the
distribution of ANGR in both
cases, reflecting the more diverse
light and nutrient conditions
experienced by individual cells.
20m
Ji and Franks (2007, MEPS)
“AN AVERAGE FISH DIES
WITHIN ITS FIRST WEEK
OF LIFE!” -- Gary Sharp
(in writing)
10m
An average larvae is a dead
larvae… (Gary at a meeting)
The average fish is a dead
fish…
20m
Applies also to individuals at
most LTLs (phytoplankton,
zooplankton) – 60-90% of
copepod eggs do not survive
to hatch
Ji and Franks (2007, MEPS)
Synchronous (tied to light) diel
vertical migrations only occur
for a limited physiological
parameter space (large growth
rate and small difference
between quota thresholds for
ascending and descending).
20m MLD
AVM using quota model
Asynchronous vertical
migrations occur for many
more physiological
combinations. Bimodal depth
distributions day and night.
Ji and Franks (2007, MEPS)
Asynchronous vertical migrations
have higher ANGR than DVM,
esp. when the mixed layer is
deep. Since most grazers on
dinoflagellates are zooplankton,
which generally do not search for
prey using vision, there is no
negative effect of being near the
surface during the day (as there
might be for zooplankton
susceptible to visual fish
predators).
10m
20m
Ji and Franks (2007, MEPS)
Internal Waves
(12 m amplitude)
20m MLD
Case 2a
Case 2b
Ji and Franks (2007, MEPS)
Allocation of
Consumption within the
Adult Female
29 params
Lagrangian Particle and
Individual Based Modeling for
Informing Population
Connectivity and Retention
RCCS ROMS Model
Domain: 41 – 45.5N, -126.7 – 123.5E
166 x 258 x 42 gridpoints (~ 1 km)
Forward run for 2002
Lagrangian Particle Tracking
50,000 initial locations on shelf
(bottom depths < 500m)
(Averages ~ 1-2 indiv/km2)
10-100m depth
3D-advected for 15 days (dt=1 hr)
New simulation begins every 7 days
RCCS ROMS runs provided by Enrique Curchitser (Rutgers)
Untangling spaghetti . . .
RCCS
19 Jun 2002 start
ET = 7 days
Retention Indices and Metrics
• Displacement distance at some elapsed
time
• e-flushing time for a specified control
volume (distance)
Connectivity Indices and
Metrics
• Transition Probability Matrix Plots
• Sources and Destinations (Maps)
Strong Upwelling and Alongshore Flow
From Batchelder (in prep.)
‘Destination maps’ identify potential of a
site to export to other locations.
RCCS
19 Jun 2002 start
ET = 7 days
High potential to
supply other
locations
Strong Upwelling and Alongshore Flow
From Batchelder (in prep.)
‘Source maps’ identify potential of other
sites to supply propagules to this location.
RCCS
19 Jun 2002 start
ET = 7 days
Large number
of sites that
can supply
this location
Strong Upwelling and Alongshore Flow
From Batchelder (in prep.)
‘Destination maps’ identify potential of a
site to export to other locations.
‘Source maps’ identify potential of other
sites to supply propagules to this location.
RCCS
19 Jun 2002 start
ET = 7 days
Large number
of sites that
can supply
this location
High potential to
supply other
locations
Strong Upwelling and Alongshore Flow
From Batchelder (in prep.)
spatial pattern of residence time
Mean
StdDev
Longest residence
time and greatest
variability in inner
Heceta Bank Region
From Batchelder (in prep.)
Considerations
1) Zooplankton and fish behavior has important demographic
consequences—how detailed do we need to model the
processes involved? Small improvements in condition, growth,
or fitness can lead to survival (being in the tail of the
distribution).
2) Zooplankton and larval fish can detect and respond to nonphysical gradients (e.g., food conc.) creating aggregations
(patchiness) due to behavior (rather than physics directly).
3) IBM’s can deal with complex stage, size and history dependent
physiology and behavior at process based level—but at the
expense of generality?
4) Under what scenarios is it critical to model zooplankton with
IBM’s in a Lagrangian framework vs. a stage-structured, agewithin-stage-structured, or physiologically structured Eulerian
framework?
5) Feedbacks across trophic levels and considerations of density
dependence are difficult to model with IBM approaches.
Take Home Messages (1)
• Concentration based (Eulerian) modeling is used in
biogeochemical contexts, with model currency being C, N, or
energy.
– Capable of, but rarely, considers size structure within a population
– Computationally efficient; scales to (number of state variables X
number of grid points)
– Biology is often hidden in non-mechanistic equations
– Difficult (impossible?) to consider behavior and history
It is rare that individual members of populations can be justifiably
aggregated into a single state variable representing abundance
(or total biomass). Consequences of aggregation need to be
considered:
– To lump individuals of various characteristics (as in NPZ+) requires
assumption that individuals are identical, and can be modeled as the
mean individual.
– Ignores nonlinearities in physiology and behavioral complexity.
– Ignores the interesting and evolutionarily significant part
(interindividual variability) of population dynamics.
Take Home Messages (2)
• Individual-based (Lagrangian) models explicitly
consider inter-individual (and potentially interspecies)
variation.
– Biology is mechanistically explicit
– History-behavior-spatial heterogeneity interactions
relatively straightforward
– Downsides
• Can be computationally expensive; scales to the number of
individuals/populations modeled
• Difficult to implement feedback to underlying Eulerian state
variables and density dependence
• Requires more knowledge of the fundamental
biological/ecological system
Take Home Messages (3)
• A simple 3-component NPZ model in an upwelling
circulation reveals
– Physical forcing induces nearshore phytoplankton bloom
– Horizontal offshore extent of the bloom determined
largely by biological parameters
• A Lagrangian zooplankton model within a 2D
upwelling circulation revealed the key role that
DVM plays in facilitating nearshore retention
– Fundamental assumption that individuals reside at times
within the deeper layer onshore flow.
– Physiological and behavioral interaction with high
nearshore phytoplankton fields further enhances
demographic retention resulting from DVM.
Take Home Messages (4)
• As revealed by the dinoflagellate IBM case study
– Physical setting can interact with physiological
demands/constraints to yield diverse outcomes.
• IBM’s are commonly used to evaluate the efficacy
of spatial management options (design of Marine
Protected Areas) for marine fisheries
• Climate change will alter species distributions,
change temperatures (altering PLD), and perhaps
alter current pathways and intensities. Lagrangian
tracking that considers advection-diffusionreaction processes will inform connectivity in
changed ecosystems.
Challenges and Opportunities to Coupling Physical Models,
Lower Trophic Level (NPZ) Models and Higher Trophic
Models (e.g., fish) (1)
Need better winds and heat fluxes in coastal regions;
coastal regions are cloudy, have nearby hills, larger hifreq variability
NPZ+ often run coupled with physics
Higher trophic levels (HTL) are usually run separately
from physics-NPZ+, with the coupling being through
advection and diffusion of the HTL, the prey available to
them and temperature effects
Empirical functional relationships (food-ingestion; foodegg production) are useful for linking species-specific life
history models to NPZ+ models
Challenges and Opportunities to Coupling Physical Models,
Lower Trophic Level (NPZ) Models and Higher Trophic
Models (e.g., fish) (2)
Food type, chemical composition, size distribution and
spatio-temporal distribution of food are important
sources of variability
Simple NPZ models cannot represent the diversity of
prey types
Prey switching and omnivorousness complicate dynamics
Averaging in space, time and trophic complexity (e.g.,
through model resolution) may stabilize models, but
ignores important ecological processes.
Mortality—the great unknown.
Thanks also to the NCAR
ASP Colloquium Organizers.
Conclusions and Lessons Learned (cont’d)
Advective transport alone can be very misleading. Models should include
diffusive effects also. And, in species capable of swimming, even small
active movements can dramatically alter transport pathways.
•Adding vertical diffusion to an advection-only model increases probability of
nearshore retention.
•Adding DVM of only 8-m (cycling between 3-m and 11-m) to an advection or
advection-diffusion model increases probability of nearshore retention.
Initial Locations of Individuals that produced eggs
DVM
Passive
Passive,
reduced
offshore
food
YOY Bloater (a FW fish)
Spatial Arrangement and Local
Interactions
Small differences in individual growth
rates can result in large changes in
size, and this can be strongly
influenced by mortality, esp. if size
based.
From DeAngelis, D. L., and K. A. Rose. 1992. Which individual-based approach
is most appropriate for a given problem? Pp. 67-87 in Individual-Based Models
and Approaches in Ecology, DeAngelis and Gross, Editors. Chapman and Hall
Publishing.
Additional Capabilities of the Oregon Shelf
Forecast Model
Use Lagrangian approach to examine spatio-temporal
connectivity and retention times in shelf environments. Develop
regional and seasonal statistics on connectivity scales and
retention times. Some preliminary results have been completed
for an earlier RCCS simulation using hindcast of 2002.
Adding a Lagrangian tracking component to the coupled model
will allow satellite or in situ observations that define the
presence or intensity of phytoplankton blooms, including HABs,
to be forecast in space/time. Assuming an accurate physical
model, discrepancies between the forecast and the next data
observation are due to production and loss processes not
considered in passive tracking.
Lagrangian back-tracking of observed HAB shore interactions
(toxic shellfish; beach closures) may be able to hindcast
probable trajectories of HABs to identify ocean conditions that
led to HAB blooms.
Ji and Franks (2007, MEPS)
Ji and Franks (2007, MEPS)
Individual-Based Model (IBM) for a Copepod
Bioenergetics based model of
growth and reproduction
• Each individual is represented
by a state-vector
• Mortality is stage specific but
independent of location
• Specific diel vertical migration
(DVM) behaviors, perhaps
dependent on condition, food
resources, etc., hypothesized.
• Growth is a balance of
assimilation and respiration,
and is a function of
•
weight (ugC)
" birthdate (days)
" time of last
reproduction
" time attained present
stage
" position (depth,
distance offshore)
" hunger condition
" most recent food level
"
Individual weight
"hunger condition
"ambient food
"
Most recent
temperature
" preferred daytime
light level
" development stage
" sex
" reproductive weight
" individual ID
"
E. pacifica Juveniles and Adults
• Reached F7 in 60 days
• Reach adult (at 12 mm) within ~ 4 months
• The most fecund adults are ~ 20 mm or
about 12 months of age
• Capable of living up to 2 years
From North et al. (2006, JMS)
Hydrodynamic model output and
particle distributions. (a)
Hydrodynamic model output at day
350. Line contours are salinity and
shaded contours are suspended
sediment concentrations (kg m− 3,
color scale on right). (b) Initial
position of 50,000 particles
randomly distributed throughout
the particle-tracking model domain.
(c) Particle distribution after 6 h
when a random displacement model
was used to simulate sub-grid scale
turbulence in the vertical direction.
(d) Particle distribution after 6 h
when a random walk model was used
to simulate sub-grid scale
turbulence in the vertical direction.
(From North et al. 2006, JMS)
Schematic of Source Region Identification
Assuming Fixed Sensor Location with
Advection Only
Predominant Flow Direction
Integration Period
2 days
4 days
2 days
4 days
Larval Duration
7 days
7 days
14 days
14 days
From Batchelder (2006)
Sensor Volume
Backward-in-TimeTrajectory (BITT)
Simulations