modelling aquatic ecosystems - course 701-0426-00, eth zürich · 2020-02-18 · modelling aquatic...

Modelling Aquatic EcosystemsCourse 701-0426-00, ETH Zürich

Spring 2020Nele Schuwirth & Peter Reichertnele.schuwirth@eawag.ch, reichert@eawag.ch

ETH Zürich, Department of Environmental Systems SciencesEawag, Swiss Federal Institute of Aquatic Science and Technology

Lecturer

Nele Schuwirth (nele.schuwirth@eawag.ch)PhD in GeologyGroup leader Ecological Modelling at Eawag,Privatdozentin ETHZ

Peter Reichert (peter.reichert@eawag.ch)PhD in Theoretical PhysicsDepartment head of Systems Analysis and Modelling atEawag, Adjunct professor at ETHZ.

Modelling Aquatic Ecosystems 2020 Lecture 1: Goals and Organizational Issues 1

Course Assistants

Catalina Chaparro Pedraza (catalina.chaparro@eawag.ch)PhD in Theoretical Ecology, University of AmsterdamPostdoctoral Scientist at EawagModelling eco-evolutionary dynamics in food webs

Gian Marco Palamara (gianmarco.palamara@eawag.ch)MSc Theoretical Physics, Sapienza University of RomePhD Integrative Ecology, Uni Zurich/Microsoft ResearchResearch Scientist at Eawag, Stochastic ecological models

Lorenz Ammann (lorenz.ammann@eawag.ch)MSc in Environmental Engineering, ETH ZurichPhD student on herbicide transport modellingEawag/ETHZ.

Goals of the Course

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.

Goals of the Exercises

• 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.

Course Elements

• Concepts of Model FormulationMass balance: One box, multiple box, continuous.Transformation: Process table, kinetics, stoichiometry.

• Formulation of Ecosystem ProcessesPhysical: Transport and mixing, gas exchange,

sedimentation, detachment and resuspension.Chemical: Chemical equilibria, sorption.Biological: Production, respiration, death, hydrolysis, miner-

alization, consumption, nitrification, bact. growth.• Ecosystem Models

Simple Models: Stepwise build-up of lake and river ecosystemmodels; basis for lectures and exercises.

Advanced Mod.: Additional processes, model structures; examples.• Technical Issues

R and packages ecosim, stoichcalc, (deSolve)

Prerequisites

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.

Administrative Aspects

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!

Administrative Aspects

Program, manuscript, exercises etc. can be downloaded from:

http://www.eawag.ch/forschung/siam/lehre/modaqecosys

If you want a printed version of the manuscript let me know in thebreak.

Develop your own model

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!

Have Fun!

Questions?

Structure of the Manuscript1 Introduction

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.

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.

0 200 400 600 800 1000 1400

C.HPO4

0 200 400 600 800 1000 1400

Modelling Aquatic Ecosystems 2020 Lecture 1: Goals 15

Motivation for ecosystem modelling

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.

Modelling Aquatic Ecosystems 2020 Lecture 1: 1 Introduction 16

Model Choice

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.

Zonation of aquatic ecosystems

Pelagic zone:Water body not close to sediment and shore or bank.

Litoral zone:Water body close to the shore or bank and the adjacentperiodically inundated area.

Benthic zone:Water body above the sediment and the top sediment layers.

Interstitial zone:Pore space in the sediment below the benthic zone.

Pelagic Food Web

dissolved oxygen

herbivorous zooplankton

omnivorous zooplankton

planktivorous fish

carnivorous fish

consumers

phytoplankton

primary producers

particulate org. mat.

dissolved org. mat.

organic matter

nutrients

dissolved oxygen

microorganisms

bacteria

Benthic Food Web

dissolved oxygen

grazers

collectors-filterers

collectors-gatherers

predators

consumers

periphyton

primary producers

susp. part. org. mat.

dissolved org. mat.

organic matter

nutrients

dissolved oxygen

sed. part. org. mat.

microorganisms

bacteria

Transport Processes in a Lake

exchange of radiation, heat, momentum and gases

r colu

mol. diffusionsedimentation

horizontal transport/mixing

inflow

outflow

Transport Processes in a River

sedimentation

exchange of radiation, heat, momentum and gases

resuspension, detachment, drift

transport/mixinginflow

outflow

Possible model structures

individual-based,stochastic model

trait-structured, stochasticdiscrete individual model

trait-structured, stochasticorganism density model

trait-structured, determini-stic organism density model

stochastic, discrete individual model

stochastic organismdensity model

deterministic organismdensity model

all individuals equalexcept specific trait(s)

consider differencesbetween individuals

limit of large numberof individuals

consider discrete nature of individuals

average overstochasticity

consider environmental/demo-graphic/genetic stochasticity

average overstochasticity

individuals equal

distinguish trait(s)

individuals equal

limit of large numberof individuals

Principles of Modelling

Meaning of models

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.

Modelling Aquatic Ecosystems 2020 Lecture 1: 2 Principles of Modelling 24

Meaning of models

system

system boundary

external influence factors

abstraction

extrapolation

of behaviour

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.

Formulation of ecosystem 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.

Learning with models

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.

Learning with models

generation and revision ofhypotheses (in model form)

new theories

comparisonof model predictionswith available data(test of the model)

planning and carrying out ofexperiments or measurements

that are expected to be sensitive to the hypotheses

newdataagreement

disagree-ment

Model-based decision support:

9. Analysis of results,search for better alternatives

8. Ranking of alternativesbased on expected degree

of achievement of objectives

4. Quantification of preferences

1. Problem definition

2. Stakeholder analysis

5. Identification of deficits

6. Construction of alternatives

7. Prediction of consequences

3. Formulation and structuringof objectives

General Mass Balance

Modelling Aquatic Ecosystems 2020 Lecture 1: 3 Mass Balance Equations 31

m "masses" [m], J (net) inputs [m/T], R (net) production [m/T]

Integral form: calculate "mass" at tend from "mass" at tini byadding net inputs and net production:

m(tend) = m(tini) +∫ tend

tiniJ(t) dt+

∫ tend

tiniR(t) dt

Differential form: substitute tend with t and differentiate:

dmdt (t) = J(t) + R(t)

Mass Balance in a Mixed Reactor

Qin, Cin

JintQout, C

C concentration [M/L3]D surface density [M/L2]Qin inflow, Qout outflow [L3/T]Jint flux across the interface [M/T]

VV C1V C2

...V Cnv

AD1AD2

...ADna

Qin − QoutQinCin,1 − QoutC1 + Jint,1QinCin,2 − QoutC2 + Jint,2

...QinCin,ns − QoutCns + Jint,nv

00...0

0V rC1

V rC2...

V rCnvArD1

ArD2...

C1C2...

, Jint =

Jint,1Jint,2...

Jint,nv

, rC =

rC2...

D1D2...

, rD =

rD2...

dVdt = Qin −Qout

ddt(VC

)= QinCin −QoutC + Jint + V rC

ddt(AD

)= ArD

dVdt = Qin −Qout

dCdt = Qin

(Cin −C

)+ Jint

dDdt = rD

Process Table Notation

Process Substances Rates1 s2 s3 · · · sns

p1 ν11 ν12 ν13 · · · ν1ns ρ1p2 ν21 ν22 ν23 · · · ν2ns ρ2...

......

... . . . ......

pnp νnp1 νnp2 νnp3 · · · νnpns ρnp

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.

Modelling Aquatic Ecosystems 2020 Lecture 1: 4.1 Process Table Notation 39

Lake Phytoplankton Model

Modelling Aquatic Ecosystems 2020 Lecture 1: 4.1 Process Table Notation 40

Typical Elements of Process Rates

Process rate with maximum/standard specific growth rate andnon-dimensional modification factors that account for the influenceof temperature, light intensity, nutrients, etc.

ρgro,ALG = kgro,ALG,T0 · ftemp(T ) · frad(I)· flim(CHPO2−

4, CNH+

4, CNO−

3) · CALG

ρminer,anox,POM = kminer,anox,POM,T0 · ftemp(T )· finh(CO2) · flim(CNO−

3) · CPOM

Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 41

Temperature dependence factor

Exponential:f exp

temp(T ) = exp(β(T − T0)

Temperature dependence factor

0 5 10 15 20 25 30

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

Limitation by substance concentrationsMonod:

fMonodlim (C) = C

Exponential:f exp

lim (C) = 1− exp(−CK

)Blackman:

fBlackmanlim (C) =

Kfor C < K

1 for C ≥ KMonod Quadratic:

fMonodquadlim (C) = C2

K2 + C2

Limitation by substance concentrations

0 1 2 3 4 5

MonodExponentialBlackmanMonod Quadratic

Limitation by substance concentrations

0 2 4 6 8 10 12 14

K=1K=2K=3K=4K=5

fMonodlim (C) = C

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

Inhibition by substance concentrationsMonod:

fMonodinh (C) = K

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

Inhibition by substance concentrations

0 1 2 3 4 5

MonodExponentialBlackmanMonod Quadratic

Inhibition by substance concentrations

0 2 4 6 8 10 12 14

K=1K=2K=3K=4K=5

fMonodinh (C) = K

Light dependence factorMonod:

fMonodrad (I) = I

KI + I

Smith:fSmith

rad (I) = I√K2

I + I2

Steele:fSteele

rad (I) = I

Ioptexp

(1− I

Light dependence factors

0 1 2 3 4 5

I KI, I Iopt

MonodSmithSteele

Light attenuation:I(z) = I0 exp(−λz);

0.0 0.2 0.4 0.6 0.8 1.0

lambda = 0.5/mlambda = 0.2/mlambda = 0.1/m

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

(I0 exp(−λz)

Average light dependence factorsMonod:

f̄Monodrad (I0, λ, h) = 1

λhlog(

KI + I0

KI + I0 exp(−λh)

)Smith:

f̄Smithrad (I0, λ, h) = 1

λhlog

√1 +

I0 exp(−λh)KI

√1 +

(I0 exp(−λh)

Steele:

f̄Steelerad (I0, λ, h) = e

(−I0 exp(−λh)

)− exp

(− I0

)]Modelling Aquatic Ecosystems 2020 Lecture 1: 4.2 Elements of Process Rates 55

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.

Preference Among Different Food SourcesSimplest conceptually satisfying expression:

f ipref(C1, ..., Cn) = piCi

n∑j=1

n: food sources with concentrations C1, ..., Cn,pj : preference coefficient for food source j.

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

Process Rates

ρgro,ALG = kgro,ALGCHPO4

KHPO4 + CHPO4CALG

ρdeath,ALG = kdeath,ALG CALG

Transformation Rates

rHPO4 = −αP,ALG · kgro,ALGCHPO4

KHPO4 + CHPO4CALG

rALG = kgro,ALGCHPO4

KHPO4 + CHPO4CALG − kdeath,ALG CALG

Mass Balance in Well-Mixed Epilimnion

dCdt = Qin

(Cin −C

)+ Jint

C =(CHPO4CALG

)Cin =

(CHPO4,in

)Jint =

Mass Balance in Well-Mixed Epilimnion

dCdt = Qin

(Cin −C

)+ Jint

Differential Equations

dCHPO4

dt = Qin

(CHPO4,in − CHPO4

)+ rHPO4

dt = −Qin

VCALG + rALG

Differential Equations

dCHPO4dt = Qin

(CHPO4,in − CHPO4

)− αP,ALG · kgro,ALG

CHPO4KHPO4 + CHPO4

dCALGdt = −Qin

VCALG + kgro,ALG

CHPO4KHPO4 + CHPO4

− kdeath,ALGCALG

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

KI + I0KI + I0 exp(−λh)

)· CHPO4KHPO4 + CHPO4

· CALG

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

Results for constant environmental conditions

0 50 100 150 200 250 300 350

C.HPO4

0 50 100 150 200 250 300 350

Results for periodic environmental conditions

0 200 400 600 800 1000 1400

C.HPO4

0 200 400 600 800 1000 1400

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.

0 200 400 600 800 1000 1400

C.HPO4

0 200 400 600 800 1000 1400

Modelling Aquatic Ecosystems 2020 Lecture 1: Goals 68

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.

Modelling Aquatic Ecosystems 2020 Lecture 1: Next Exercises 69

Preparation Exercise 1 - Homework

1. Install a current version of R and R-Studio and the ecosim-packageon your notebook -> see Programm 2020

2. Read chapter 11.1 about the first didactical model

3. Read chapters 16.1 and 16.2 about the ecosim-package.

4. Think about your open questions.

Modelling Aquatic Ecosystems 2020 Lecture 1: Next Exercises 70

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