Modelling Aquatic Ecosystems Course 701-0426-00 ETH Z¨urich Spring Semester 2021 Peter Reichert, Johanna Mieleitner and Nele Schuwirth Eawag: Swiss Federal Institute of Aquatic Science and Technology Department of Systems Analysis, Integrated Assessment and Modeling 8600 D¨ubendorf, Switzerland and ETH Z¨urich Department of Environmental Systems Science 8092 Z¨urich, Switzerland February 8, 2021 Contents Preface vii 1 Introduction 1 1.1 Motivation for Modelling Aquatic Ecosystems . 2 1.2 Overview of Processes in Aquatic Ecosystems . 3 1.3 Overview of Biological Model Structures . 7 I Basic Concepts 11 2 Principles of Modelling Environmental Systems 13 2.1 Meaning of Models . 13 2.2 Formulation of Models . 14 2.3 Learning with Models . 14 2.4 Model-Based Decision Support . 15 3 Formulation of Mass Balance Equations 17 3.1 General Concept of Mass Balance Equations . 18 3.2 Mass Balance in a Mixed Reactor . 19 3.3 Mass Balance in a Multi-Reactor System . 21 3.4 Mass Balance in a Continuous System . 24 3.4.1 One-Dimensional Case . 24 3.4.2 Three-Dimensional Case . 26 4 Formulation of Transformation Processes 31 4.1 Process Table Notation . 32 4.2 Typical Elements of Process Rates . 36 4.2.1 Temperature Dependence of Process Rates . 37 4.2.2 Limitation of Process Rates by Substance Concentrations . 37 4.2.3 Inhibition of Process Rates by Substance Concentrations . 39 4.2.4 Light Dependence of Primary Production . 40 4.2.5 Preference Among Different Food Sources . 42 4.2.6 Example of a Process Table . 43 4.3 Derivation of Stoichiometry from Composition . 45 4.3.1 Derivation of Stoichiometry from Chemical Substance Notation (given elemental mass fractions) . 46 4.3.2 Derivation of Stoichiometry from Parameterized Elemental Mass Fractions . 51 i ii CONTENTS 4.3.2.1 Growth and Respiration of Algae: Nutrient Balance . 52 4.3.2.2 Growth, Respiration and Death of Algae and Zooplankton: Consideration of the Elements C, H, O, N and P . 54 4.3.3 General Analysis and Solution of Stoichiometric Equations . 60 4.3.3.1 Overview of General Procedure . 60 4.3.3.2 Constraints on Stoichiometric Conditions . 61 4.3.3.3 Analysing and Calculating Process Stoichiometry . 64 4.3.3.4 Analytical Example: Growth of Algae on Nutrients . 66 4.3.3.5 Numerical Example: Growth of Algae and Zooplankton . 70 5 Behaviour of Solutions of Differential Equation Models 73 5.1 Graphical Interpretation . 74 5.2 Systems of Linear Ordinary Differential Equations . 74 5.3 Systems of Nonlinear Ordinary Differential Equations . 78 5.3.1 Fixed Points and their Stability . 78 5.3.2 Limit Cycles . 79 5.3.3 Chaos . 79 5.4 Numerical Solution of Ordinary Differential Equations . 81 II Formulation of Ecosystem Processes 85 6 Physical Processes 87 6.1 Transport and Mixing . 88 6.1.1 Transport and Mixing in Lakes and Reservoirs . 88 6.1.1.1 Density Stratification of Lakes and Reservoirs . 88 6.1.1.2 Plunging of Inflows . 89 6.1.1.3 Horizontal Spreading of Dissolved Substances . 89 6.1.1.4 Vertical Spreading of Dissolved Substances . 93 6.1.2 Transport and Mixing in Rivers . 95 6.1.2.1 Steady-State River Hydraulics . 95 6.1.2.2 Vertical Mixing . 99 6.1.2.3 Lateral Mixing . 102 6.1.2.4 Transport and Longitudinal Dispersion . 105 6.2 Sedimentation . 108 6.3 Gas Exchange . 109 6.4 Detachment and Resuspension . 111 7 Chemical Processes 113 7.1 Chemical Equilibria . 114 7.2 Sorption . 118 8 Biological Processes 121 8.1 Primary Production . 122 8.2 Respiration . 124 8.3 Death . 125 8.4 Consumption . 127 8.5 Mineralization . 129 8.5.1 Oxic Mineralization . 129 CONTENTS iii 8.5.2 Anoxic Mineralization . 130 8.5.3 Anaerobic Mineralization . 131 8.6 Nitrification . 133 8.6.1 Nitrification as a One Step Process . 133 8.6.2 Nitrification as a Two Step Process . 133 8.7 Hydrolysis . 135 8.8 Bacterial Growth . 137 8.8.1 Growth of Heterotrophic Bacteria . 137 8.8.2 Growth of Nitrifiers . 139 8.9 Colonization . 140 III Stochasticity, Uncertainty and Parameter Estimation 141 9 Consideration of Stochasticity and Uncertainty 143 9.1 Causes of Stochasticity and Uncertainty . 143 9.1.1 Stochasticity of Dynamics . 143 9.1.2 Uncertainty of Model Structure, Input and Parameters . 144 9.2 Probabilistic Framework . 144 9.2.1 Univariate Random Variables and Distributions . 144 9.2.2 Multivariate Random Variables and Distributions . 145 9.2.3 Stochastic Processes . 146 9.3 Stochasticity and Uncertainty in Ecological Models . 148 9.3.1 Genetic or Demographic Stochasticity . 148 9.3.2 Environmental Stochasticity . 149 9.3.3 Uncertainty due to Incomplete Knowledge . 149 9.4 Numerical Approximation by Monte Carlo Simulation . 150 10 Parameter Estimation 153 10.1 Observation Error and Likelihood Function . 153 10.2 Maximum Likelihood Parameter Estimation . 154 10.3 Bayesian Inference . 154 10.4 Markov Chain Monte Carlo for Posterior Sampling . 157 IV Simple Models of Aquatic Ecosystems 165 11 Simple Models of Aquatic Ecosystems 167 11.1 Lake Phytoplankton Model . 168 11.2 Lake Phyto- and Zooplankton Model . 175 11.3 Two Box Oxygen and Phosphorus Lake Model . 178 11.4 Model of Biogeochemical Cycles in a Lake . 182 11.5 Oxygen and Nutrient Household Model of a River . 189 11.6 Benthic Population Model of a River . 192 11.7 Model Predictions with Stochasticity and Uncertainty . 197 iv CONTENTS V Advanced Aquatic Ecosystem Modelling 201 12 Extensions of Processes and Model Structure 203 12.1 Mechanistic Description of Physical Processes . 204 12.2 Important Extensions of Biological Processes . 205 12.2.1 Consideration of Silicon . 205 12.2.2 Variable Phosphorus Stoichiometry . 206 12.2.3 Phosphate Uptake by Organic Particles . 207 12.2.4 Modelling Internal Concentrations . 207 12.2.5 Modelling Functional Groups of Algae, Zooplankton or Invertebrates 209 12.2.6 Modelling Individual Taxa . 209 12.2.7 Consideration of Rapid Evolution . 212 12.3 Important Extensions to the Model Structure . 213 12.3.1 Age-, Size- or Stage-Structured Models . 213 12.3.2 Discrete Individuals Models . 216 12.3.3 Individual-Based Models . 219 13 Research Models of Aquatic Ecosystems 221 13.1 Examples of Models of Aquatic Ecosystems . 222 13.1.1 BELAMO . 222 13.1.2 SALMO . 222 13.1.3 CAEDYM . 223 13.1.4 PROTECH . 224 13.1.5 QUAL2K . 224 13.1.6 RWQM1 . 224 13.1.7 ERIMO . 225 13.1.8 Streambugs . 225 13.2 Case Studies of Aquatic Ecosystem Model Application . 226 13.2.1 Modelling Biogeochemistry and Plankton in Three Lakes of Different Trophic State . 226 13.2.1.1 Model of Lake Zurich . 226 13.2.1.2 Extension to Three Lakes of Different Trophic State . 231 13.2.1.3 Long-Term Simulation . 233 13.2.2 Modelling Benthos Community Dynamics in the River Sihl . 237 13.2.3 Modelling Taxonomic Composition of Steam Benthos Communities . 243 VI Appendix 245 14 Important Univariate Probability Distributions 247 14.1 Uniform Distribution . 247 14.2 Normal Distribution . ..