Modelling Aquatic Ecosystems Course 701-0426-00, ETH Zürich Spring 2020 Nele Schuwirth & Peter Reichert [email protected], [email protected]ETH Zürich, Department of Environmental Systems Sciences Eawag, Swiss Federal Institute of Aquatic Science and Technology
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Modelling Aquatic EcosystemsCourse 701-0426-00, ETH Zürich
Catalina Chaparro Pedraza ([email protected])PhD in Theoretical Ecology, University of AmsterdamPostdoctoral Scientist at EawagModelling eco-evolutionary dynamics in food webs
Gian Marco Palamara ([email protected])MSc Theoretical Physics, Sapienza University of RomePhD Integrative Ecology, Uni Zurich/Microsoft ResearchResearch Scientist at Eawag, Stochastic ecological models
Lorenz Ammann ([email protected])MSc in Environmental Engineering, ETH ZurichPhD student on herbicide transport modellingEawag/ETHZ.
The students (you!) are able to• build mathematical models of aquatic ecosystems thatconsider the most important biological, biogeochemical,chemical and physical processes.• explain the interactions between these processes and thebehaviour of the system that results from these interactingprocesses.• formulate, implement and apply simple ecological models• consider stochasticity and uncertainty.
Emphasis is on integrating knowledge in the form of models, ontheir use for improving the understanding and management ofaquatic ecosystems and on their limitations.
• Deepen and extend the knowledge gained in the coursethrough implementation, simulation, sensitivity analysis, anddiscussion of the behaviour of a series of ecosystem models ofincreasing complexity introduced in the course.• Learn to implement and use models using the publiclyavailable statistics and graphics software R(http://www.r-project.org) and extensions in the form ofpackages.• Learn to use R (this is also useful for statistical data analysisin future projects).
Emphasis is on improving the understanding of the behaviour ofthe models and the underlying ecosystems through practicalapplication and discussion and not on programming.
Basic knowledge about structure and function of aquatic eco-systems as well as about analysis, differential equations, linearalgebra and probability.
Time for the exercises will be provided during the course. Thiscompresses the lectures to the remaining time and makes themquite intensive. You will need time between the course hours toread the manuscript.
Approximate time budget (3 credit points = 90 hours study time):30 hours: Course including supervised exercise time.30 hours: Reading the manuscript and preparing exercises.30 hours: Preparation of your own model and the oral exam.
There will be an oral exam in the the week after the semester:04./05.06.2020
Course and exercises will take place Wednesday 10:15 - 12:00 inLFW B3.
You will do the exercises on your own notebooks.The current version of R (http://www.r-project.org),the editor R-Studio(http://www.rstudio.org),and the packages deSolve, stoichcalc and ecosim:install.packages(c("ecosim"))have to be installed before the exercise!
During the semester you have to develop and implement your ownmodel (alone or in groups of two), interpret simulation results andperform a simple sensitivity analysis.
We will assign topics on 01.04.2020.
Deadline for code submission 30.04.20You will deliver the R-files and results by 22.05.20.This is mandatory! In the oral exam we will ask you about yourexample (beside other topics).
Use the time in the exercises to ask questions and get help!
I Basic Concepts2 Principles of Modelling Environmental Systems3 Formulation of Mass Balance Equations4 Formulation of Transformation Processes5 Behaviour of Solutions of ODE models
II Formulation of Ecosystem Processes6 Physical Processes7 Chemical Processes8 Biological Processes
III Consideration of Stochasticity and Uncertainty9 Consideration of Stochasticity and Uncertainty
10 Parameter EstimationIV Simple Models of Aquatic Ecosystems
11 Simple Models of Aquatic EcosystemsV Advanced Aquatic Ecosystem Modelling
12 Extensions of Processes and Model Structure13 Research Models of Aquatic Ecosystems
Structure of the Course1. Introduction. Principles of modelling environmental systems.
Mass balance in a mixed reactor. Process table notation.Simple lake phyto- and zooplankton model. ecosim package.
2. Process stoichiometry. stoichcalc package.3. Biological processes in lakes, the metabolic theory of ecology
Chemical equilibriaTwo box lake model for plankton and biogeochemical cycles.
4. Physical processes in lakesMass balance in a system of reactors and in continuous systems.
5. Transport and mixing in rivers.Model of O, N and P household and benthic populations in a river.
6. Additional processes and model extensions.7. Stochasticity and uncertainty. Individual based models.8. Examples and case studies about research models. Preparation of
exam and review.Modelling Aquatic Ecosystems 2020 Lecture 1: Goals and Organizational Issues 13
Exercises
1. Introduction to R and the ecosim package.Demonstration of the implementation of a simple lakephytoplankton model.
2. Phytoplankton-zooplankton model for a mixed lake.
3. Practice of stoichiometric calculations.Introduction to the stoichcalc package.
4. Two box lake model for plankton and biogeochemical cycles.
5. River benthos and water column model with sessile algae andbacteria and O, P and N cycles
6. Consideration of environmental stochasticity and uncertainty.
• Acquire basic knowledge of the formulation of transport andtransformation processes to formulate a simple lake planktonmodel.• Become familiar with the process table notation and rateformulation that will be the basis of the more complex models.
1. Improving understanding of ecosystem function:Test of quantitatively formulated hypotheses about systemmechanisms. Estimation of fluxes and conversion rates.Stimulation of thinking about the function of an ecosystem.
2. Summarizing and communicating knowledge:Ecosystem models are perfect communication tools forexchanging quantitatively formulated knowledge of theprocesses in the ecosystem. A systematic notation facilitatesthe use of models for this purpose significantly.
3. Supporting ecosystem management:Prediction of the consequences of suggested measures.Estimation and consideration of prediction uncertainty isessential for this purpose of ecosystem modelling.
Different models for different purposesThe model is a simplified representation of the real system.Choices have to be made with respect to type and detail ofdescription.
The model to be used depends on the objective of the study!Models for improving the understanding and communicatingknowledge usually have a higher structural resolution of modelcomponents and processes than models for environmentalmanagement.
System: assemblage of interrelated objects comprising a whole.
Environmental system: part of the environment.→ system boundaries.
Model: abstract representation of a system.Internal variables, external influence factors. Different models canrepresent the same system at different levels of resolution.
Adequate model complexity and structure depends on the purposeof modelling and on the data/knowledge available forcalibration/model specification.
There is no prediction without a model - and no modelwithout data!
The spectrum of models used for prediction can range from mentalmodels to simple trend extrapolations to detailed mechanisticsystem descriptions.
As we have an emphasis on the use of models for improving andintegrating our understanding, the focus of this course is on(partially) mechanistic models.
Essential techniques: Empirical relationships and mass-balanceequations.
Typical form of an environmental model:Mechanistic description of mass conservation - use of empiricalexpressions for the formulation of transformation and transferprocesses.
Iterative systems analysis process:• Model formulation (integration of knowledge; understandingthe effect of processes and their interaction)• Parameter estimation• Statistical testing• Uncertainty analysis• Model application
In this course we focus on the first point as this point is specific toaquatic ecosystems. This does not mean that the other pointswould be less important; they build the methodological frameworkof the modelling process in general.
Substance transformation rate in homogeneous environment:
rj =np∑i=1
νij ρi
One of the (non-zero) stoichiometric coefficients, νij , in each row can beselected to be plus or minus unity. This makes the corresponding processrate, ρj , to the (positive or negative) contribution of this process to thetotal transformation rate of the corresponding substance, si.
Process rate with maximum/standard specific growth rate andnon-dimensional modification factors that account for the influenceof temperature, light intensity, nutrients, etc.
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 41
Typical Elements of Process Rates
Temperature dependence factor
Exponential:f exp
temp(T ) = exp(β(T − T0)
)
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 42
Typical Elements of Process Rates
Temperature dependence factor
0 5 10 15 20 25 30
0.0
0.5
1.0
1.5
2.0
2.5
3.0
T
f T
T0=15T0=20T0=25beta=0.046beta=0.08beta=0.1
Exponential:f exp
temp(T ) = exp(β(T − T0)
)Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 43
Typical Elements of Process Rates
Limitation by substance concentrationsMonod:
fMonodlim (C) = C
K + C
Exponential:f exp
lim (C) = 1− exp(−CK
)Blackman:
fBlackmanlim (C) =
C
Kfor C < K
1 for C ≥ KMonod Quadratic:
fMonodquadlim (C) = C2
K2 + C2
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 44
Typical Elements of Process Rates
Limitation by substance concentrations
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
C K
f lim
MonodExponentialBlackmanMonod Quadratic
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 45
Typical Elements of Process Rates
Limitation by substance concentrations
0 2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
C
f Mon
od
K=1K=2K=3K=4K=5
fMonodlim (C) = C
K + C
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 46
Typical Elements of Process Rates
Limitation by multiple substancesProduct:
fN (CHPO4, CNH4, CNO3)
= CHPO4KHPO4 + CHPO4
· CNH4 + CNO3KN + CNH4 + CNO3
Minimum (Liebig’s Law):
fN (CHPO4, CNH4, CNO3)
= min(
CHPO4KHPO4 + CHPO4
,CNH4 + CNO3
KN + CNH4 + CNO3
)
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 47
Typical Elements of Process Rates
Inhibition by substance concentrationsMonod:
fMonodinh (C) = K
K + C
Exponential:f exp
inh (C) = exp(−CK
)Blackman:
fBlackmaninh (C) =
1− C
Kfor C < K
0 for C ≥ K
Monod Quadratic:
fMonodquadinh (C) = K2
K2 + C2
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 48
Typical Elements of Process Rates
Inhibition by substance concentrations
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
C K
f inh
MonodExponentialBlackmanMonod Quadratic
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 49
Typical Elements of Process Rates
Inhibition by substance concentrations
0 2 4 6 8 10 12 14
0.0
0.2
0.4
0.6
0.8
1.0
C
f Mon
od in
h
K=1K=2K=3K=4K=5
fMonodinh (C) = K
K + C
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 50
Typical Elements of Process Rates
Light dependence factorMonod:
fMonodrad (I) = I
KI + I
Smith:fSmith
rad (I) = I√K2
I + I2
Steele:fSteele
rad (I) = I
Ioptexp
(1− I
Iopt
)
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 51
Typical Elements of Process Rates
Light dependence factors
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
I KI, I Iopt
f rad
MonodSmithSteele
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 52
Typical Elements of Process Rates
Light attenuation:I(z) = I0 exp(−λz);
0.0 0.2 0.4 0.6 0.8 1.0
2015
105
0
I I0
z [m
]
lambda = 0.5/mlambda = 0.2/mlambda = 0.1/m
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 53
Typical Elements of Process Rates
Light attenuation
For a model with a mixed reactor, the light dependence factor (andnot the light itself!) has to be averaged across depth.
Average light dependence factor:
f̄rad(I0, λ, h) = 1h
∫ h
0frad
(I0 exp(−λz)
)dz
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 54
Typical Elements of Process Rates
Average light dependence factorsMonod:
f̄Monodrad (I0, λ, h) = 1
λhlog(
KI + I0
KI + I0 exp(−λh)
)Smith:
f̄Smithrad (I0, λ, h) = 1
λhlog
I0
KI+
√1 +
(I0
KI
)2
I0 exp(−λh)KI
+
√1 +
(I0 exp(−λh)
KI
)2
Steele:
f̄Steelerad (I0, λ, h) = e
λh
[exp
(−I0 exp(−λh)
Iopt
)− exp
(− I0
Iopt
)]Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 55
Typical Elements of Process Rates
Preference Among Different Food Sources
Many organisms can grow on different food sources.
As the stoichiometry and kinetics of growth on one food sourcemay be different from that on another, it is best to representgrowth on different food sources by different processes.
The process rates of these processes can still have many terms incommon. But they also need a preference factor that depends onthe concentrations of all food sources.
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 56
Typical Elements of Process Rates
Preference Among Different Food SourcesSimplest conceptually satisfying expression:
f ipref(C1, ..., Cn) = piCi
n∑j=1
pjCj
n: food sources with concentrations C1, ..., Cn,pj : preference coefficient for food source j.
Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 57
Lake Phytoplankton Model
Process TableProcess Substances / Organisms Rate
HPO4 ALG[gP/m3] [gDM/m3]
Growth of algae −αP,ALG 1 ρgro,ALG
Death of algae −1 ρdeath,ALG
Modelling Aquatic Ecosystems 2020 Lecture 1: 11.1 Lake Plankton Model 58
Lake Phytoplankton Model
Process Rates
ρgro,ALG = kgro,ALGCHPO4
KHPO4 + CHPO4CALG
ρdeath,ALG = kdeath,ALG CALG
Modelling Aquatic Ecosystems 2020 Lecture 1: 11.1 Lake Plankton Model 59
Lake Phytoplankton Model
Transformation Rates
rHPO4 = −αP,ALG · kgro,ALGCHPO4
KHPO4 + CHPO4CALG
rALG = kgro,ALGCHPO4
KHPO4 + CHPO4CALG − kdeath,ALG CALG
Modelling Aquatic Ecosystems 2020 Lecture 1: 11.1 Lake Plankton Model 60
Lake Phytoplankton Model
Mass Balance in Well-Mixed Epilimnion
dCdt = Qin
V
(Cin −C
)+ Jint
V+ r
C =(CHPO4CALG
)Cin =
(CHPO4,in
0
)Jint =
(00
)
Modelling Aquatic Ecosystems 2020 Lecture 1: 11.1 Lake Plankton Model 61
Lake Phytoplankton Model
Mass Balance in Well-Mixed Epilimnion
dCdt = Qin
V
(Cin −C
)+ Jint
V+ r
Differential Equations
dCHPO4
dt = Qin
V
(CHPO4,in − CHPO4
)+ rHPO4
dCALG
dt = −Qin
VCALG + rALG
Modelling Aquatic Ecosystems 2020 Lecture 1: 11.1 Lake Plankton Model 62
Lake Phytoplankton Model
Differential Equations
dCHPO4dt = Qin
V
(CHPO4,in − CHPO4
)− αP,ALG · kgro,ALG
CHPO4KHPO4 + CHPO4
CALG
dCALGdt = −Qin
VCALG + kgro,ALG
CHPO4KHPO4 + CHPO4
CALG
− kdeath,ALGCALG
Modelling Aquatic Ecosystems 2020 Lecture 1: 11.1 Lake Plankton Model 63
Lake Phytoplankton Model
Extended Process Rates
Additional influence factors of algae growth rate to account foryearly cycles in temperature and light.
ρgro,ALG = kgro,ALG · exp(βALG(T − T0)
)· 1λh
log(
KI + I0KI + I0 exp(−λh)
)· CHPO4KHPO4 + CHPO4
· CALG
Modelling Aquatic Ecosystems 2020 Lecture 1: 11.1 Lake Plankton Model 64
Lake Phytoplankton Model
Seasonally Varying Environmental Conditions
T (t) = Tmax + Tmin2 + Tmax − Tmin
2 cos(
2π t− tmaxtper
)
I0(t) = I0,max + I0,min2 + I0,max − I0,min
2 cos(
2π t− tmaxtper
)
Modelling Aquatic Ecosystems 2020 Lecture 1: 11.1 Lake Plankton Model 65
Lake Phytoplankton Model
Results for constant environmental conditions
0 50 100 150 200 250 300 350
0.00
00.
002
0.00
40.
006
0.00
8
C.HPO4
t
C.H
PO
4
C.HPO4
0 50 100 150 200 250 300 350
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
C.ALG
t
C.A
LG
C.ALG
Modelling Aquatic Ecosystems 2020 Lecture 1: 11.1 Lake Plankton Model 66
Lake Phytoplankton Model
Results for periodic environmental conditions
0 200 400 600 800 1000 1400
0.00
00.
005
0.01
00.
015
0.02
00.
025
C.HPO4
t
C.H
PO
4
C.HPO4
0 200 400 600 800 1000 1400
01
23
4
C.ALG
t
C.A
LG
C.ALG
Modelling Aquatic Ecosystems 2020 Lecture 1: 11.1 Lake Plankton Model 67
Lecture 1: Goals
• Acquire basic knowledge of the formulation of transport andtransformation processes to formulate a simple lake planktonmodel.• Become familiar with the process table notation and rateformulation that will be the basis of the more complex models.