individual-based modeling in ocean ecology: where behavior, physiology and physics meet hal...

Individual-Based Modeling in Ocean Ecology:

Where Behavior, Physiology and Physics Meet

Hal BatchelderOregon State University

Supported by NSF and NOAA

within the U.S. GLOBECNortheast 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

Franks et al., 1986

N

ZP

Vitals:380 lbs, 7’1”;

SOME BIOMASS

Vitals:380 lbs, 7’1”;

Vitals:~380 lbs,

MORE BIOMASS

=?

Vitals:380 lbs, 7’1”;

Vitals:~380 lbs,

=?

Vitals:380 lbs, 7’1”; one mouth;

Vitals:~380 lbs, 40 mouths;

Regularly puts foot in mouth (figuratively)

Actually able to put foot in mouth

Euphausia pacifica life stages

N2 Metanauplius

Adult

Calyptopi

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

N2 Metanauplius

Adult

Calyptopi~3.2 μg ind-1

~7 μg ind-1

~4000 μg ind-1

Stage-specific CW

Euphausia pacifica life stages

N2 Metanauplius

Adult

Calyptopi~3.2 μg ind-1

~7 μg ind-1

~4000 μg ind-1

Stage-specific CW

1250 indiv.

571 indiv.

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

N2 Metanauplius

Adult

Calyptopi~3.2 μg ind-1

~7 μg ind-1

~4000 μg ind-1

Stage-specific CW

1250 indiv.

571 indiv.

1 indiv.

ΣR=633.6 ug C d-1

ΣG=425 ug C d-1

ΣR=529.2 ug C d-1

ΣG=519.6 ug C d-1

ΣR=122.9 ug C d-1

ΣG=26 ug C d-1

( ) 10

max 10,

kref

k

mT TC

ingestion f mC F k

W P vF a I Q

W P v P

Body Size

Prey

Temperature

R (ug C d-1) = f(Weight, Prey, Temp)

Bioenergetics of an Individual Process

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)

Data from Ross (1982) and Feinberg et al. (2006)

What if low food conditions delay development?

Revised Form is: Di = [ai (T + b)c] / [1 – e-kP]

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)

2.5

2.8

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.

LS1 LS2 LS3 LS4 … LSn

W1

W2

W3

W4

…

Wmn

What if: There are many more dimensions important to dynamics?

Wit

hin

Sta

ge W

eigh

t

Life Stage

“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…

All figures are from Huston, M., D. DeAngelis, and W. Post. 1988. New computer models unify ecological theory. BioScience 38 (10), 682-691.

Intraspecific Effects - Initial Condition Sensitivity

Interspecific Effects – Relative Size

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

•Nutritional (structural or physiological) Condition

•Behavior

•Location

= f (history)

= 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 Approach1) Difficult to implement feedback from IBM (Lagrangian) to

underlying Eulerian model, esp. across multiple trophic levels

1) Consumption (depletion) of prey (E) by predators (L)

• Assume not important (Batchelder & co. 1989,1995)

• Conversion to concentrations per grid cell (Carlotti & Wolf 1998)

2) 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

• Biological/biochemical processes for individuals are many and complex

5) Increased knowledge about the system (this might be a good thing)

Design of Marine Protected AreasThe 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 patternsTransport patterns

Retention effect of circulation over a single 40-day pelagic period within the fall climatology.

•There is exchange between closed areas 1 and 2.

•Area 1 is largely self-seeding; Area 2 seeds both areas.

Source

No Closed Regions

Closed Regions

10 Year Scallop Simulation w/ 1 spawning per year; 40 day larval drift; individual surviving scallops plotted (red are oldest individuals)

Impacts of DispersalImpacts of Dispersal

High Low

Population ConnectivityModified from Harrison and Taylor (1997)

Single, patchyPopulation (open)

Metapopulation(structured connectance)

Separate(closed)

From C. Grant Law (unpubl.)

Transport patternsTransport patterns

From C. Grant Law (unpubl.)

QuestionsQuestions

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 HistoryManagement History

NE side of Georges Bank

NE side of Nantucket shoals

Head of Hudson Canyon

Pre-Closure Distribution

From C. Grant Law (unpubl.)

Management HistoryManagement History

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

Post-Closure Distribution

From C. Grant Law (unpubl.)

Zooplankton Population Dynamics in 2D

The Oregon Upwelling System

Age, Size, Number of Organisms

Egestion, P

Predation T, B, , L

Starvation T, B, P,

Respiration T, B,

Physical Exchange B,

Ingestion T, B, , L, P

Migration T, B, P

Processes and Environmental Variables Influencing Organism Growth and Number

T = Temperature; B=Behavior; =Turbulence; P=Prey; L=Light

Bioenergetic Model

Spatially-Explicit Model

Physical Forcing(light,wind,IC's)

2D or 3D EulerianModel

Eulerian Fields(velocity,

temperaturelight, food, K)

IBM with simulatedLagrangian Particles

Individual Zooplankton

Characteristics (wt,stage,condition,

sex, position)

Population Characteristics(Demography) and Spatial Distribution

Modeling Approach

(Eulerian-Lagrangian Coupling)

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0

Density (# m-3)

150-200m

100-150m

50-100m

20-50m

10-20m

0-10m

Dep

th R

ang

e o

f L

ayer

Sam

ple

dEuphausia pacifica

at NH25 (Aug 4, 2000, daytime)

Nauplii

Calyptopes

Furcilia

Juveniles

Adults

Biological Organisms are not Passive Tracers

Figure courtesy of J. Keister

All Stages are in upper 20 m during

Night

Magnitude of Diel Vertical Migration by Life Stage

Shelf Stations

0

25

50

75

100

Egg Calypt

opis

F1 F2 F3 F4 F6 F7 Juve

nile

Adult

Life History Stage

Dep

th (

m)

Slope Stations

0

25

50

75

100

125

150

175

200

Egg Calypt

opis

F1 F2 F3 F4 F6 F7 Juve

nile

Adult

Life History Stage

Dep

th (

m)

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.

Based on 6 day-night paired MOCNESSFrom shelf stations and 8 day-night pairsFrom 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 night160 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.

Batchelder, H. P. 1986. Phytoplankton balance in the oceanic subarctic Pacific: grazing impact of Metridia pacifica. Mar. Ecol. Prog. Ser., 34, 213-225.

~2X

starved

fed

Hunger (H)

Light

H

Size (S)

Food (P)

Boundary(Ns,Nb)

Slows downmigSlows upmig

Batchelder et al. (2002, PiO)

Physical Model

• 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

• Southward wind-stress forcing of 0.5 dyne/cm2, either constant or alternating on/off with 5 or 10 day intervals

100 km

200

m

Batchelder et al. (2002, PiO)

2D Upwelling Scenario Simulations

N

ZPBatchelder et al. (2002, PiO)

Day 20

Day 40

Day 80

Size of bubble is proportional to individual weight

Recently layed clutches in hi food

region

Weight lossbelow

mixed layer

Starvation Mortality

Few Nearshore

No-DVM Simulation(PTM forced with Eulerian Concentrations of Prey,

Velocities, and Kv)

Batchelder et al. (2002, PiO)

DVM Simulation(PTM forced with Eulerian Concentrations of Prey,

Velocities, and Kv)

Day 20

Day 40

Day 80

Size of bubble is proportional to individual weight

Middepth aggregation offshore Large Individuals Inshore

Nearshore reproductionand retentionNo reproduction &

mortality loss offshore

Populationnearshore

only

Batchelder et al. (2002, PiO)

Nutrient Quota Based DVMOf Dinoflagellates

Ji and Franks (2007, MEPS)

• 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

MLD = 10m

For both 10m and 20m MLD, cells are able to balance their need for light and nutrients by occupying the pycnocline/nutricline. No DVM.

Ji and Franks (2007, MEPS)

Subsurface vs. DVM

Higher light level at 10m yields higher net growth rates than at 20m for subsurface individuals.

10m

20m

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.

“AN AVERAGE FISH DIES WITHIN ITS FIRST WEEK OF LIFE!” -- Gary Sharp (in writing)

An average larvae is a dead larvae… (Gary at a meeting)

The average fish is a dead fish…

Ji and Franks (2007, MEPS)

10m

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)

AVM using quota model

Asynchronous vertical migrations occur for many more physiological combinations. Bimodal depth distributions day and night.

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

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

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)

RCCS19 Jun 2002 start

ET = 7 days

Strong Upwelling and Alongshore Flow

Untangling spaghetti . . .

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)

From Batchelder (in prep.)

RCCS19 Jun 2002 start

ET = 7 days

Strong Upwelling and Alongshore Flow

‘Destination maps’ identify potential of a site to export to other locations.

High potential to supply other locations

From Batchelder (in prep.)

RCCS19 Jun 2002 start

ET = 7 days

Strong Upwelling and Alongshore Flow

‘Source maps’ identify potential of other sites to supply propagules to this location.

Large number of sites that can supply this location

From Batchelder (in prep.)

RCCS19 Jun 2002 start

ET = 7 days

Strong Upwelling and Alongshore Flow

‘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.

High potential to supply other locations

Large number of sites that can supply this location

From Batchelder (in prep.)

spatial pattern of residence time

Longest residence time and greatest variability in inner Heceta Bank Region

StdDevMean

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 non-physical 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, age-within-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

• 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 (3)

• 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-diffusion-reaction processes will inform connectivity in changed ecosystems.

Take Home Messages (4)

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 hi-freq 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; food-egg 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

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.

Spatial Arrangement and Local Interactions

YOY Bloater (a FW fish)

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.

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

Most recent temperature

preferred daytime light level

development stagesexreproductive weightindividual ID

weight (ugC)birthdate (days)time of last reproduction

time attained present stage

position (depth, distance offshore)

hunger conditionmost recent food level

Individual weighthunger conditionambient food

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)

Sensor Volume

Integration Period Larval Duration2 days 7 days4 days 7 days2 days 14 days4 days 14 days

Schematic of Source Region IdentificationAssuming Fixed Sensor Location with

Advection Only

Predominant Flow Direction

Backward-in-Time-Trajectory (BITT)

Simulations

From Batchelder (2006)