A practical guide to ecological modelling : using R as a simulation platform / Karline Soetaert and Peter M.J. Herman.

Format:

Book

Author/Creator:

Soetaert, Karline.

Contributor:

Herman, P. M. J. (Peter M. J.)

Rudolph G. Schmieder Fund.

Language:

English

Subjects (All):

Ecology--Mathematical models.

Ecology.

R (Computer program language).

Ecology--methods.

Medical Subjects:

Ecology--methods.

Physical Description:

xv, 372 pages : illustrations, map ; 25 cm

Place of Publication:

[Dordrecht] : Springer, [2009]

Summary:

Many texts on ecological models jump to describing either particular relations or computational results, without treating in detail the conceptual and mathematical basis of many steps in modelling: why set up models, what are basic conceptual models, how do conservation laws come in, how are models solved, what are steady states. This book is intended to bridge this gap. It is intended as an introductory text for graduate and post-graduate students, but also as a help for experienced ecologists who want to make more of their data by modelling. It contains many examples, all worked out in the open-source package R, providing the reader the opportunity to practise all methods and get hands-on experience.

Contents:

1.1 What is a Model? 1

1.1.1 A Simple Example: Zooplankton Energy Balance 3

1.2 Why Do We Need Models? 5

1.2.1 Models as Analysing Tools 5

1.2.2 Models as Interpolation, Extrapolation, and Budgeting Tools 7

1.2.3 Models to Quantify Immeasurable Processes 9

1.2.4 Model Prediction as a Management Tool 10

1.3 Modelling Steps and Ingredients 10

1.4 The Modeller's Toolkit 13

2 Model Formulation 15

2.1 Conceptual Model 15

2.1.1 The Balance Equation of a State Variable 17

2.1.2 Example: Conceptual Model of a Lake Ecosystem 19

2.1.3 Conservation of Mass and Energy as a Cosistency Check 21

2.1.4 Dimensional Homogeneity and Consistency of Units 23

2.2 Mathematical Formulations 24

2.3 Formulation of Chemical Reactions 25

2.3.1 The Law of Mass Action 25

2.3.2 Example: A Simple Chemical Reaction 26

2.4 Enzymatic Reactions 27

2.5 Basic Formulation of Ecological Interactions 28

2.5.1 Example: Flows to and from Phytoplankton in the Lake Ecosystem 28

2.5.2 Maximal Interaction Strength, Rate Limitation and Inhibition 31

2.5.3 One Rate-Limiting Resource, 3 Types of Functional Responses 35

2.5.4 More than One Limiting Resource 37

2.5.5 Inhibitino Terms 38

2.6 Coupled Model Equations 40

2.6.1 Flows Modelled as Fractions of Other Flows 41

2.6.2 Coupled Dynamics of Source and Sink Compartments 42

2.6.3 Stoichiometry and Coupling of Element Cycles 43

2.7 Model Simplifications 44

2.7.1 Carrying Capacity Formulation 45

2.7.2 Closure Terms at the Highest Trophic Level 48

2.7.3 Simplification by Deletion of Intermediate Levels 48

2.8 Impact of Physical Conditions 49

2.8.1 Temperature 49

2.8.2 Light 50

2.8.3 Other Physical Impacts 53

2.9.1 NPZD, a Simple Ecosystem Model for Aquatic Environments 54

2.9.2 AQUAPHY, a Physiological Model of Unbalanced Algal Growth (**) 58

2.10 Case Studies in R 63

2.10.1 Making Sense Out of Mathematical Formulations 63

2.10.2 One Formula, Several Parameter Values 64

2.11.1 Conceptual Model: Lake Eutrophication 65

2.11.2 Model Formulation: Nutrient-Limited Batch Culture 66

2.11.3 Model Formulation: Detritus Degradation 67

2.11.4 Model Formulation: An Autocatalytic Reaction 69

3 Spatial Components and Transport 71

3.1 Microscopic and Macroscopic Models 72

3.2 Representing Space in Models 74

3.2.1 Spatial Dimensions 74

3.2.2 Discrete Spatial Models 74

3.2.3 Continuous Spatial Models 76

3.3 Transport in a Zero-Dimensional Model 77

3.4 Transport in a One-Dimensional Model 79

3.4.1 Flux Divergence 80

3.4.2 Macroscopic Formulation of Fluxes: Advection and Dispersion 82

3.4.3 The General 1-D Advection-Dispersion-Reaction Equation 84

3.4.4 The 1-D Advection-Dispersion-Reaction Equation in Estuaries, Rivers and Lakes 85

3.4.5 The 1-D Advection-Dispersion-Reaction Equation in Shapes with Different Symmetries 86

3.4.6 One-dimensional Diffusion in Porous Media (Sediments) (**) 89

3.4.7 The 3-D Advection-Dispersion-Reaction Equation (*) 92

3.5 Boundary Conditions in Spatially Explicit Models 92

3.5.1 Boundary Conditions in Discrete Models 94

3.5.2 Boundary Conditions in Continuous Models 95

3.5.3 Boundary Conditions in Multi-layered Models (**) 98

3.6 Case Studies in R 102

3.6.1 An Autocatalytic Reaction in a Flow-Through Stirred Tank 102

3.6.2 A 1-D Microscopic and Macroscopic Model of Diffusion 103

3.6.3 Cellular Automaton Model of Diffusion (**) 107

3.6.4 Competition in a Lattice Grid 110

3.6.5 Transport and Reaction in Porous Media: Silicate Diagenesis 114

4 Parameterization 117

4.1 In Situ Measurement 117

4.2 Literature-Derived Parameters 118

4.3 Calibration 119

4.3.1 Linear Regression 120

4.3.2 Nonlinear Fitting 122

4.4 Case Studies in R 123

4.4.1 Nonlinear Parameter Estimation: P-I Curve 123

4.4.2 Linear Versus Non-Linear Parameter Estimation: Sediment Bioturbation 125

4.4.3 Pseudo-Random Search, a Random-Based Minimization Routine 128

4.4.4 Calibration of a Simple Model 132

5 Model Solution-Analytical Methods 139

5.1 An Everyday Life Example 139

5.2 Finding an Analytical Solution 140

5.3.1 A Very Simple First-Order Differential Equation 141

5.3.2 The Logistic Equation 143

5.3.3 A Second-Order Differential Equation: Carbon Dynamics in Sediments (*) 144

5.3.4 Coupled BOD and Oxygen Equations (*) 146

5.3.5 Multilayer Differential Equations (**) 147

5.4 Case Studies in R 150

5.4.1 Transient Dispersion-Reaction in One Dimension 150

5.4.2 Transient Diffusion-Reaction on a 2-Dimensional Surface 151

5.4.3 Steady-State Oxygen Budget in Small Organisms Living in Suboxic Conditions 152

5.4.4 Analytical Solution of the Non-Local Exchange Sediment Model (***) 158

5.5.1 Organic Matter Sinking Through a Water Column 161

5.5.2 Oxygen Dynamics in the Sediment 162

5.5.3 Carbon Dynamics in the Sediment 164

6 Model Solution - Numerical Methods 165

6.1 Taylor Expansion 165

6.2 Numerical Approximation and Numerical Errors 167

6.3 Numerical Integration in Time-Basics 169

6.3.1 Euler Integration 170

6.3.2 Criteria for Numerical Integration 171

6.3.3 Interpolation Methods-4th Order Runge-Kutta (**) 173

6.3.4 Flexible Time Step Methods-5th Order Runge-Kutta (**) 174

6.3.5 Implicit and Semi-Implicit Integration Routines (**) 174

6.3.6 Which Integrator to Choose? 175

6.4 Approximating Spatial Derivatives (*) 176

6.4.1 Approximating the Flux Divergence Equation 176

6.4.2 Approximating Dispersion 177

6.4.3 Approximating Advection 178

6.4.4 The Boundaries with the External World 179

6.5 Numerical Dispersion (** *) 180

6.5.1 Example: Sediment Model 182

6.6 Case Studies in R 183

6.6.1 Implementing the Enzymatic Reaction Model 184

6.6.2 Growth of a Daphnia Individual 188

6.6.3 Zero-Dimensional Estuarine Zooplankton Model 195

6.6.4 Aphids on a Row of Plants: Numerical Solution of a Dispersion-Reaction Model 197

6.6.5 Fate of Marine Zooplankton in an Estuary (** *) 200

6.7.1 Numerical Solution of the Autocatalytic Reaction in a Flow-Through Stirred Tank 206

6.7.2 Numerical Solution of a Nutrient-Algae Chemostat Model-Euler Integration 207

6.7.3 Rain of Organic Matter in the Ocean: Numerical Solution of the Advection-Reaction Model 209

6.7.4 AQUAPHY Model Implementation 209

7 Stability and Steady-State 211

7.2 Stability of One First-Order Differential Equation 213

7.2.1 Equilibrium Points, Stability, Domain of Attraction 213

7.2.2 Multiple Steady States 215

7.2.3 Bifurcation 216

7.3 Stability of Two Differential Equations-Phase-Plane Analysis 218

7.3.1 Example.

The Lotka-Volterra Predator-Prey Equation 220

7.4 Multiple Equations 224

7.5 Steady-State Solution of Differential Equations (*) 224

7.5.1 Direct Root Finding: Analytical Solution 225

7.5.2 Iterative Root Finding 225

7.5.3 From Partial to Ordinary Differential Equations 225

7.6 Formal Analysis of Stability (**) 226

7.7 Limit Cycles (***) 231

7.8 Case Studies in R 232

7.8.1 Multiple Stable States: the Spruce Budworm Model 232

7.8.2 Phase-Plane Analysis: The Lotka-Volterra Competition Equations 237

7.8.3 The Lorenz Equations-Chaos 242

7.8.4 Steady-State Solution of the Silicate Diagenetic Model (**) 243

7.8.5 Fate of Marine Zooplankton in an Estuary-Equilibrium Condition (**) 248

7.9.1 The Schaefer Model of Fisheries 250

7.9.2 A Fisheries Model with Allee Effect 252

7.9.3 Ecological-Economical Fisheries Model 253

7.9.4 Predator-Prey System with Type-II Functional Response 254

7.9.5 Succession of Nutrients, Phytoplankton and Zooplankton in a River 254

8 Multiple Time Scales and Equilibrium Processes 257

8.1 Simple Chemical Equilibrium Calculation: Ammonia and Ammonium 258

8.2 Chemical Equilibrium Combined with a Slow Reaction Process 259

8.3 General Approach to Equilibrium Reformulation 261

8.3.1 Enzymatic Equilibrium in a Slow Reaction Process: The Michaelis-Menten Equation (**) 261

8.3.2 Equilibrium Adsorption in Porous Media (**) 264

8.4 Examples in R 267

8.4.1 Solving pH in Aquatic Systems 267

8.4.2 A Model of pH Changes Due to Algal Growth (**) 269

9 Discrete Time Models 273

9.1 Difference Equations 274

9.2 Discrete Logistic Models 275

9.3 Host-Parasitoid Interactions 276

9.4 Dynamic Matrix Models 278

9.4.1 Example: Age Structured Population Model 278

9.4.2 Matrix Notation 280

9.4.3 Stable Age Distribution and Rate of Increase 281

9.4.4 The Reproductive Value 282

9.5 Case Studies in R 283

9.5.1 Bifurcations in the Discrete Logistic Model 283

9.5.2 Bifurcations in the Host-Parasitoid Model 283

9.5.3 Attractors in the Host-Parasitoid Model 285

9.5.4 Population Dynamics of Teasel 287

9.6.1 Bifurcation of the Logistic Map 292

9.6.2 Equilibrium Dynamics of a Simple Age-Class Matrix Model 292

9.6.3 Equilibrium Dynamics of a US Population 292

10 Dynamic Programming 295

10.1 Sequential Decisions 296

10.2 Finding the Optimal Solution 296

10.4 Case Study in R: The Patch-Selection Model 301

11 Testing and Validating the Model 309

11.1 Coupled BOD-O2 Model Revisited 309

11.2 Testing the Correctness of the Model Solution 310

11.3 Testing the Internal Logic of the Model 312

11.4 Model Verification and Validity 313

11.5 Model Sensitivity 314

11.6 Case Studies in R 315

11.6.1 Time-Varying Oxygen Consumption in a Small Cylindrical Organism 315

11.6.2 R for Validation and Verification 319

11.6.3 Univariate Local Sensitivity Analysis 319

11.6.4 Bivariate Local Sensitivity Analysis 325

12 Further Reading and References 329

12.1 Model Formulations 329

12.2 Spatial Pattern 330

12.3 Parameterization 331

12.4 Model Solution 331

12.5 Stability and Equilibrium Analysis 332

12.6 Discrete Time and Dynamic Programming Models 332

Appendix A About R 335

A.1 Installing R 335

A.2 A Very Short Introduction 336

A.2.1 Console Versus Scripts 336

A.2.2 Getting Help 337

A.2.3 Vectors and Matrices 337

A.2.4 More Complex Data Structures 339

A.2.5 User-defined Functions and Programming 340

A.2.6 R Packages 341

A.2.7 Graphics 341

A.2.8 Minor Things to Remember 342

A.3 Interfacing R with Low-Level Languages: The Competition in a Lattice Grid Model Revisited 343

Appendix B Derivatives and Differential Equations 347

B.1 Derivatives 347

B.2 Taxonomy of Differential Equations 348

B.3 General Solutions of Often Used Differential Equations 349

B.3.1 Simple Time-Dependent Equations 349

B.3.2 Steady-State Transport-Reaction in 1-D, Constant Surface 350

B.3.3 Steady-State Transport-Reaction in 1-D, Cylindrical Coordinates 350

B.3.4 Steady-State Transport-Reaction in 1-D, Spherical Coordinates 351

B.4 Particular Solutions of Dynamic Diffusion-Reaction Equation 351

B.5 Derivatives and Integrals 352

Appendix C Matrix Algebra 353

C.1 Matrices 353

C.2 Linear Equations 354

C.3 Eigenvalues and Eigenvectors, Determinants 355

C.4 R-Examples 355

C.5 The Jacobian Matrix 357

Appendix D Statistical Distributions 359

D.1 Probability Distribution 359

D.2 Normal Distribution 359

D.3 The Poisson Distribution 360.

Notes:

Includes bibliographical references (pages 361-365) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Rudolph G. Schmieder Fund.

ISBN:

9781402086236

1402086237

OCLC:

234146516