Ecological modelling: Introductive course

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Ecological modelling Introductive
course Marilaure Grégoire, Liege University,
Belgium
Sesame Second summer school Malta, June 8th-13th.
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Outlines

• What is a model ? Why modelling ?
• The modelling steps How do we make a model ?
• The conceptual model and its components
• Mathematical translation of ecological
  interactions
• Impact of physical conditions
• Parameters calibration
• Fundamental principles
• Examples

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Outlines

• What is a model ? Why modelling ?
• The modelling steps How do we make a model ?
• The conceptual model and its components
• Mathematical translation of ecological
  interactions
• Impact of physical conditions
• Parameters calibration
• Fundamental principles
• Examples

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What is a model ?

• A simplified representation of a complex
  phenomenon
• focus only on the object of interest
• ignoring the (irrelevant) details, making
  suitable abstractions of reality. What
  characteristics are essential? Depends on the
  aims of the model and on the definition of the
  problem. Ideally, the model has to be realistic
  enough in order to provide a good representation
  of the real system but more simple than the real
  system in order to be handled more easily.
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• select temporal and spatial scales of interest
  (not possible to describe everything, from
  molecules and elementary physics up to the
  functioning of the whole earth). Once the
  spectral window is selected processes of smaller
  scale are parameterized while processes at much
  large scale are imposed to the model.

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What is a model

• In this course, mechanistic mathematical models,
  i.e. models where the ecological processes (the
  mechanisms) are described in a mathematical
  sense.
• Express quantitative relationships -gt
  mathematical formulation gt Predictions,
  tested to data
• gt Computers
• Most of the ecological, but also physical,
  chemical models are written as a set of coupled
  differential equations

Example NPZD model
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Why do we use models?

• Basic research
• To complete the observations which have a limited
  coverage in time and space.
• To understand in a quantitative sense how a
  system works test hypotheses. Models as
  Analysing Tools.
• Experiments are more efficient if a model tells
  us what to expect
• Some things cannot be directly measured (or too
  expensive)

The scientific method observations and models
are used for interpretation and hypothesis
generation, and thus explain real-world phenomena.
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Why do we use models

• Interpolation, budgetting
• measurements may not be accurate enough
• Black-box interpolation methods do not tell us
  anything about the functioning of the system.

Black-box methods may give high r2 but they do
not increase our understanding of the system.
Mechanistic models, even if they perform less
well in predicting have the benefit of increased
knowledge of system functioning.
Simple statistical interpolation of sparse data
may be inaccurate
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Why do we use models

• Management tool
• Model predictions may be used to examine the
  consequences of our actions in advance.
• What is the effect of REDUCING the input of
  organic matter to an estuary on the export of
  nitrogen to the sea ?
• MODEL ANSWER it INCREASES the net export.
• O2 improves gt denitrification lower gt removal
  of N in estuary decreases

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Example of the use of models for
quantifying unmeasurable processes and as a
budgeting tool Fate of marine zooplankton in
the Westerschelde estuary (Soeteart and Herman,
1994)

• Question
• Is there net growth of marine zooplankton species
  in the estuary or do they deteriorate?
• What is the net import/export of marine
  zooplankton to the estuary?
• Fact
• Difficult to measure directly (flow in/out
  estuary ?)
• Seasonal time scales, scale of km.
• Tool Simplified physics, simplified biology

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Ex Westerschelde zooplankton
The estuary, from Rupelmonde to Vlissingen, was
divided into 13 large boxes, along the horizontal
axis (1-dimensional).
River inputs
Exchanges due to the action of tides
Fresh water flow
Net growth in the estuary

• 1. Unknown G
• 2. Data monthly transect of zooplankton biomass
  along a transect from the sea to the river.
• Run the model assuming G0 gt Negative/positive
  growth
• Estimate G (calibration)
• 3. Calculating budgets

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Ex Westerschelde zooplankton

• RESULT
• 1. Marine zooplankton dies in the estuary on
  average 5 per day
• 2. Over a year, some 1500 tonnes of zooplankton
  dry weight is imported into the estuary each year
  (4000 dutch cows).

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Outlines

• What is a model ? Why modelling ?
• The modelling steps How do we make a model ?
• The conceptual model and its components
• Mathematical translation of ecological
  interactions
• Impact of physical conditions
• Parameters calibration
• Fundamental principles
• Examples

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Modelling steps
In this course

• Iterative process
• gtImprove if wrong
• gtData

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Outlines

• What is a model ? Why modelling ?
• The modelling steps How do we make a model ?
• The conceptual model and its components
• Mathematical translation of ecological
  interactions
• Impact of physical conditions
• Parameters calibration
• Fundamental principles
• Examples

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Modelling steps
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Conceptual model
"A model should be as simple as possible but not
simpler" (Einstein).

• Selection of model complexity as models are
  simplifications, at this stage we determine the
  model complexity.
• Choice of modelled components and processes
  involves a subtle balance between realism and
  complexity the more we know about a system, the
  more complex the model can be. If knowledge is
  poor, then the model will be unable to contain
  many details. In addition, as the complexity also
  depends on the problem to be solved, this means
  that for one system, there may well exist many
  different models, all resolving a different
  question.
• The structure of the conceptual model depends on
  the problem at hand and also on the modeller.
  This is different from physical models where the
  variables are very similar between different
  models.

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• Black Sea models Different models, different
  aims, different modellers.

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Black Seas models
Lancelot et al., 2002, ESCS
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Black Seas models
Oguz et al. 2001
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Black Seas model NAPZD model (Fasham-like model)
Rivers inputs
Grazing
Phytoplankton
Zooplankton
Nutrients Uptake
Egestion
Mortality
Grazing
NO3-
Mortality
Detritus
Sinking
Nitrification
Ng
NH4
Degradation
Excretion
Gregoire et al., 2004
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Conceptual model

• COMPONENTS
• State variable (biomass, density, concentration)
• Flows or interaction
• Forcing functions (light intensity, Wind, flow
  rates)
• Ordinary variable (Grazing rates, Chlorophyll)
• Parameters (ks, pFaeces)
• Universal constants (e.g. atomic weights)
• TEMPORAL AND SPATIAL SCALE
• MODEL CURRENCY (N, C, DWT, individuals,..)

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The Standard Organism (Functional group
approach)
Vichi et al. 2006
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The chemical currencies flows in the trophic
web Assuming different chemical structures
Vichi et al. 2006
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Elements of a model

• Differential equation
• Time dependentproblem can be expressed by means
  of sources and sinks

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Conceptual model
MODEL CURRENCY N, -gt mmol N m-3 Chlorophyll
PHYTO Chlorophyll/Nitrogen ratio
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Outlines

• What is a model ? Why modelling ?
• The modelling steps How do we make a model ?
• The conceptual model and its components
• Mathematical translation of ecological
  interactions
• Impact of physical conditions
• Parameters calibration
• Fundamental principles
• Examples

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Modelling steps
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Ecological interactions

• Ecological interactions deal with the exchange of
  energy
• INTERACTION MaximalINTERACTION Rate
  limiting_Term(s)
• Compartment that performs the work controls
  maximal strength MaximalINTERACTIONMaximalRate
  Consumer(or compartment performing the work)
• Rate limiting term
• a function of resource (Functional response)
• a function of consumer (Carrying capacity)

PREDATION (mmolC/m3/d) MaximalRate (
/d) Predator (mmolC/m3) f(Prey)
(-)
PREDATION MaximalRate Predator f(Prey)
NUTRIENTUPTAKE MaximalRate Algae
f(Nutrient)
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Ecological interactions

• Biochemical transformation Bacteria perform work
• gt first-order to bacteria
• gt rate limiting term function of source
  compartment

Hydrolysis (mmolC/m3/d)
MaximalHydrolysisRate (
/d) Bacteria
(mmolC/m3) f(SemilabileDOC) (-)
Hydrolysis MaximalHydrolysisRate Bacteria
f(semilabile DOC)
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Ecological interactions

• Rate limiting term functional response
• how a consumption rate is affected by the
  concentration of resource

Low resource linear High resource handling
time
Low resource exponential (learning,switch
behavior) High resource handling time
Monod/Michaelis-Menten
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Ecological interactions

• Carrying capacity model
• Rate limiting term is a function of CONSUMER

K10
Carrying capacity is a proxy for Resource
limitation Predation Space limitation
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Ecological interactions

• More than one limiting resource
• Liebig law of the minimum determined by
  substance least in supply
• Multiplicative effect
• preference factor for multiple food sources

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Ecological interactions

• Relationships between flows
• One flow function of another flow

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Ecological interactions

• CLOSURE TERMS
• Models are simplicifications, not everything is
  explicitly modeled
• gt some processes are Parameterised.

• Closure on mesozooplankton
• gt do NOT model their predators (fishes,
  gelatinous..)
• gt take into account the mortality imposed by
  those predators

Mesozooplankton in mmolC/m3 c1 /day c2
/day/(mmolC/m3) Mortality mmolC/m3/day
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Chemical reactions
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Inhibition
INTERACTION MaximalRate WORK Rate
limiting_TermInhibition_Term

• Exponential
• NO3-uptake of algae inhibited by ammonium

• 1-Monod
• denitrification inhibited by O2

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Coupled reactions
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Coupled reactions
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Coupled reactions

• Coupling via Source-sink (previous examples)
• Stoichiometry cycles of N, C, Si, P are coupled
  through stoichiometric relations. The
  stoichiometry of a compound is defined as the
  proportion of the various quantities.

(CH2O)106(NH3)16(H3PO4) C106H263O110N16P CHON
P ratio of 106263110161.
(CH2O)106(NH3)16(H3PO4) 106 O2
-gt106 CO2 16 NH3
H3PO4 106H2O
Molar ratios OC ratio 1 NC ratio
16/106 PC ratio 1/106
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Outlines

• What is a model ? Why modelling ?
• The modelling steps How do we make a model ?
• The conceptual model and its components
• Mathematical translation of ecological
  interactions
• Impact of physical conditions
• Parameters calibration
• Fundamental principles
• Examples

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Impact of physical conditions

• Currents / turbulence
• pelagic constituents
• benthic animals supply of food / removal of
  wastes
• Hydrodynamical models coupled differential
  equations

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Impact of physical conditions

• Temperature
• Rates (Physical, chemical, physiological,..)
• Solubility of substances -gt exchange across
  air-sea
• Forcing function / hydrodynamical models

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Impact of physical conditions

• Light
• Heats up water and sediment
• PAR photosynthesis
• Forcing function (data/algorithm)

saturation
inhibition
linear
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Model formulation-summary

• Ecological interactions
• first-order to work compartment
• rate limiting terms (functional responses,
  carrying capacity terms)
• inhibition terms
• closure terms - proxy for processes not modeled
• Chemical reactions
• inhibition terms
• Coupled models
• source-sink compartments
• stoichiometry
• Physical conditions
• currents
• temperature
• light
• wind

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Outlines

• What is a model ? Why modelling ?
• The modelling steps How do we make a model ?
• The conceptual model and its components
• Mathematical translation of ecological
  interactions
• Impact of physical conditions
• Parameters calibration
• Fundamental principles
• Examples

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Modelling steps
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Model parameterisation

• Problem
• Values for constants (parameters) in equations?

• Both models the same
• dN / dt a N
• Both models start at same value
• but a1 2 a2

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Sources for parameters

• Direct observation
• Literature
• Calibration

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Direct observation
Example P-I relation Fitting of Eilers-Peeters
eq. -gt pmax, Iopt, beta
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Direct observation beware

• Consider different sources of error !
• experimental error
• spatial variability
• temporal variability
• -gt base estimates on data base geared to time and
  space scales
• of model !

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Literature-derived parameters

• Direct use published values for exact purpose
• Indirect through relationships with other
  (master) variables or parameters

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Literature beware !
Linear scale
Log-log
Parameters from log-log relationships are
excellent when used in a similar context.
Otherwise large uncertainty may arise !
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Calibration
Principle find those parameters for which model
fits data best
minimise
Straightforward for linear models
(Yabx) Non-linear models beware of local (ltgt
global) minima
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Model cost landscape
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Sensitivity analysis

• Vary parameters in the model within reasonable
  range
• Look at change in model outcome as a consequence
• Large model change for small parameter change -gt
  sensitive parameter !
• Concentrate efforts on sensitive parameters

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Outlines

• What is a model ? Why modelling ?
• The modelling steps How do we make a model ?
• The conceptual model and its components
• Mathematical translation of ecological
  interactions
• Impact of physical conditions
• Parameters calibration
• Fundamental principles
• Examples

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Modelling steps
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2 fundamental principles

• More robust model applications
• Dimensional homogeneity and consistency of units
• Conservation of energy and mass

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1. Consistency of units

• All quantities have a unit attached
• S.I. units m (length)
  kg (mass) s (time)
  K (temperature)
  mol (amount of substance)
• Derived units C K -
  273.15 (C-1K-1) N kg m
  s-2 (force) J kg m2 s-2
  N m (energy) W kg m2
  s-3 J s-1 (power)
• An equation is dimensionally homogeneous and has
  consistent units if the units and quantities at
  two sides of an equation balance

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Consistency of units

• To a certain extent units can be manipulated like
  numbers
• J kg-1 (kg m2 s-2)/kg m2 s-2 (units
  mass-specific energy)
• Relative density of females in a population
• (number of females m-2) / (total individuals m-2)
  (-)
• it is not allowed to add mass to length, length
  to area, ..
• It is not allowed to add grams to kilograms
• Before calculating with the numbers, the units
  must be written to base S.I. Units

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Consistency of units

• Units on both sides of the sign must match
• gt can be used to check the consistency of a
  model
• ex the rate of change of detrital nitrogen in a
  water column

rPHYmort phytoplankton mortality rate
(d-1) PHYC phytoplankton concentration (mmol C
m-3) NDET detrital Nitrogen (mmol N m-3)
mmol N m-3 d-1 ( d-1)
(mmol C m-3) NOT CONSISTENT !
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Consistency of units
mmol N m-3 d-1 mmol N m-3 3
d-1 CONSISTENT !
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2. Conservation of mass and energy

• neither total mass nor energy can be created or
  destroyed
• Sum of all rate of changes and external sources /
  sinks constant

If no external sources/sinks CLOSED
SYSTEM total load must be constant.
PhytoZooFish Detritus NH3Bottom detritus
Ct
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2. Conservation of mass and energy

• neither total mass nor energy can be created or
  destroyed
• Sum of all rate of changes and external sources /
  sinks constant

3 state variables FOOD, DAPHNIA, EGGS
(mmolC/m3) 2 external sinks OPEN
SYSTEM dFood/dt -Ingestion dDaphnia/dtIngestion
-Faeces production -Respiration(basal and growth)
Reproduction dEggs/dtReproduction dDaphnia/dtd
Eggs/dtdFood/dt -(BasalrespGrowthRespFaecesP
rod)
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Outlines

• What is a model ? Why modelling ?
• The modelling steps How do we make a model ?
• The conceptual model and its components
• Mathematical translation of ecological
  interactions
• Impact of physical conditions
• Parameters calibration
• Fundamental principles
• Examples

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NPZD model

• 4 state variables - mmol N/m3 - rates per day
• 1 ordinary variable chlorophyll (calculated
  based on PHYTO)
• 1 Forcing function Light

More details tomorrow, Marco Zavatarelli lecture
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AQUAPHY

• Physiological model of unbalanced algal growth
• Algae have variable stoichiometry due to
  uncoupling of
• photosynthesis (C-assimilation)
• protein synthesis (N-assimilation)

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AQUAPHY

• 4 state variables Reserve, LMW, proteins (mmol
  C/m3), DIN (mmolN/m3)

-ProteinSynthesis
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Modelling steps
Momme Butensch?n, Wednesday lecture
Marcello Vichi, Thursday lecture
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REFERENCE
Materials for this course are basically derived
from the MARELAC course given by K. Soetaert and
P. Herman, NIOO CEME, Yerseke, the Netherlands
http//194.171.24.200/ppages/ksoetaert/