## A Practical Guide to Ecological Modelling: Using R as a Simulation PlatformMathematical modelling is an essential tool in present-day ecological research. Yet for many ecologists it is still problematic to apply modelling in their research. In our experience, the major problem is at the conceptual level: proper understanding of what a model is, how ecological relations can be translated consistently into mathematical equations, how models are solved, steady states calculated and interpreted. Many textbooks jump over these conceptual hurdles to dive into detailed formulations or the mathematics of solution. This book attempts to fill that gap. It introduces essential concepts for mathematical modelling, explains the mathematics behind the methods, and helps readers to implement models and obtain hands-on experience. Throughout the book, emphasis is laid on how to translate ecological questions into interpretable models in a practical way. The book aims to be an introductory textbook at the undergraduate-graduate level, but will also be useful to seduce experienced ecologists into the world of modelling. The range of ecological models treated is wide, from Lotka-Volterra type of principle-seeking models to environmental or ecosystem models, and including matrix models, lattice models and sequential decision models. All chapters contain a concise introduction into the theory, worked-out examples and exercises. All examples are implemented in the open-source package R, thus taking away problems of software availability for use of the book. All code used in the book is available on a dedicated website. |

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### Contents

Introduction | 1 |

Zooplankton Energy Balance | 3 |

12 Why Do We Need Models? | 5 |

122 Models as Interpolation Extrapolation and Budgeting Tools | 7 |

123 Models to Quantify Immeasurable Processes | 9 |

124 Model Prediction as a Management Tool | 10 |

14 The Modellers Toolkit | 13 |

Model Formulation | 15 |

642 Approximating Dispersion | 177 |

643 Approximating Advection | 178 |

644 The Boundaries with the External World | 179 |

65 Numerical Dispersion | 180 |

Sediment Model | 182 |

66 Case Studies in R | 183 |

661 Implementing the Enzymatic Reaction Model | 184 |

662 Growth of a Daphnia Individual | 188 |

211 The Balance Equation of a State Variable | 17 |

Conceptual Model of a Lake Ecosystem | 19 |

213 Conservation of Mass and Energy as a Consistency Check | 21 |

214 Dimensional Homogeneity and Consistency of Units | 23 |

22 Mathematical Formulations | 24 |

23 Formulation of Chemical Reactions | 25 |

A Simple Chemical Reaction | 26 |

24 Enzymatic Reactions | 27 |

25 Basic Formulation of Ecological Interactions | 28 |

252 Maximal Interaction Strength Rate Limitation and Inhibition | 31 |

253 One RateLimiting Resource 3 Types of Functional Responses | 35 |

254 More than One Limiting Resource | 37 |

255 Inhibition Terms | 38 |

26 Coupled Model Equations | 40 |

261 Flows Modelled as Fractions of Other Flows | 41 |

262 Coupled Dynamics of Source and Sink Compartments | 42 |

263 Stoichiometry and Coupling of Element Cycles | 43 |

27 Model Simplifications | 44 |

271 Carrying Capacity Formulation | 45 |

272 Closure Terms at the Highest Trophic Level | 48 |

28 Impact of Physical Conditions | 49 |

282 Light | 50 |

283 Other Physical Impacts | 53 |

29 Examples | 54 |

292 AQUAPHY a Physiological Model of Unbalanced Algal Growth | 58 |

210 Case Studies in R | 63 |

2102 One Formula Several Parameter Values | 64 |

211 Projects | 65 |

NutrientLimited Batch Culture | 66 |

Detritus Degradation | 67 |

An Autocatalytic Reaction | 69 |

Spatial Components and Transport | 70 |

31 Microscopic and Macroscopic Models | 72 |

32 Representing Space in Models | 74 |

323 Continuous Spatial Models | 76 |

33 Transport in a ZeroDimensional Model | 77 |

34 Transport in a OneDimensional Model | 79 |

341 Flux Divergence | 80 |

Advection and Dispersion | 82 |

343 The General 1D AdvectionDispersionReaction Equation | 84 |

344 The 1D AdvectionDispersionReaction Equation in Estuaries Rivers and Lakes | 85 |

345 The 1D AdvectionDispersionReaction Equation in Shapes with Different Symmetries | 86 |

346 Onedimensional Diffusion in Porous Media Sediments | 89 |

347 The 3D AdvectionDispersionReaction Equation | 92 |

351 Boundary Conditions in Discrete Models | 94 |

352 Boundary Conditions in Continuous Models | 95 |

353 Boundary Conditions in Multilayered Models | 98 |

36 Case Studies in R | 102 |

362 A 1D Microscopic and Macroscopic Model of Diffusion | 103 |

363 Cellular Automaton Model of Diffusion | 107 |

364 Competition in a Lattice Grid | 110 |

Silicate Diagenesis | 114 |

Parameterization | 117 |

42 LiteratureDerived Parameters | 118 |

43 Calibration | 119 |

431 Linear Regression | 120 |

432 Nonlinear Fitting | 122 |

44 Case Studies in R | 123 |

Sediment Bioturbation | 125 |

443 PseudoRandom Search a RandomBased Minimization Routine | 128 |

444 Calibration of a Simple Model | 132 |

Model Solution Analytical Methods | 139 |

52 Finding an Analytical Solution | 140 |

53 Examples | 141 |

532 The Logistic Equation | 143 |

Carbon Dynamics in Sediments | 144 |

534 Coupled BOD and Oxygen Equations | 146 |

535 Multilayer Differential Equations | 147 |

54 Case Studies in R | 150 |

542 Transient DiffusionReaction on a 2Dimensional Surface | 151 |

543 SteadyState Oxygen Budget in Small Organisms Living in Suboxic Conditions | 152 |

544 Analytical Solution of the NonLocal Exchange Sediment Model | 158 |

55 Projects | 161 |

552 Oxygen Dynamics in the Sediment | 162 |

553 Carbon Dynamics in the Sediment | 164 |

Model SolutionNumerical Methods | 165 |

62 Numerical Approximation and Numerical Errors | 167 |

63 Numerical Integration in Time Basics | 169 |

631 Euler Integration | 170 |

632 Criteria for Numerical Integration | 171 |

633 Interpolation Methods 4th Order RungeKutta | 173 |

634 Flexible Time Step Methods 5th Order RungeKutta | 174 |

636 Which Integrator to Choose? | 175 |

64 Approximating Spatial Derivatives | 176 |

663 ZeroDimensional Estuarine Zooplankton Model | 195 |

Numerical Solution of a DispersionReaction Model | 197 |

665 Fate of Marine Zooplankton in an Estuary | 200 |

67 Projects | 206 |

672 Numerical Solution of a NutrientAlgae Chemostat Model Euler Integration | 207 |

Numerical Solution of the AdvectionReaction Model | 209 |

Stability and SteadyState | 210 |

72 Stability of One FirstOrder Differential Equation | 213 |

722 Multiple Steady States | 215 |

723 Bifurcation | 216 |

73 Stability of Two Differential Equations PhasePlane Analysis | 218 |

731 Example The LotkaVolterra PredatorPrey Equation | 220 |

74 Multiple Equations | 224 |

Analytical Solution | 225 |

76 Formal Analysis of Stability | 226 |

77 Limit Cycles | 231 |

78 Case Studies in R | 232 |

The LotkaVolterra Competition Equations | 237 |

783 The Lorenz Equations Chaos | 242 |

784 SteadyState Solution of the Silicate Diagenetic Model | 243 |

785 Fate of Marine Zooplankton in an Estuary Equilibrium Condition | 248 |

79 Projects | 250 |

792 A Fisheries Model with Allee Effect | 252 |

793 EcologicalEconomical Fisheries Model | 253 |

794 PredatorPrey System with TypeII Functional Response | 254 |

Multiple Time Scales and Equilibrium Processes | 257 |

Ammonia and Ammonium | 258 |

82 Chemical Equilibrium Combined with a Slow Reaction Process | 259 |

83 General Approach to Equilibrium Reformulation | 261 |

832 Equilibrium Adsorption in Porous Media | 264 |

84 Examples in R | 267 |

842 A Model of pH Changes Due to Algal Growth | 269 |

Discrete Time Models | 273 |

91 Difference Equations | 274 |

92 Discrete Logistic Models | 275 |

93 HostParasitoid Interactions | 276 |

94 Dynamic Matrix Models | 278 |

942 Matrix Notation | 280 |

943 Stable Age Distribution and Rate of Increase | 281 |

944 The Reproductive Value | 282 |

95 Case Studies in R | 283 |

953 Attractors in the HostParasitoid Model | 285 |

954 Population Dynamics of Teasel | 287 |

96 Projects | 292 |

Dynamic Programming | 295 |

101 Sequential Decisions | 296 |

103 A Simple Example | 297 |

The PatchSelection Model | 301 |

Testing and Validating the Model | 308 |

112 Testing the Correctness of the Model Solution | 310 |

113 Testing the Internal Logic of the Model | 312 |

114 Model Verification and Validity | 313 |

115 Model Sensitivity | 314 |

116 Case Studies in R | 315 |

1162 R for Validation and Verification | 319 |

1164 Bivariate Local Sensitivity Analysis | 325 |

Further Reading and References | 329 |

122 Spatial Pattern | 330 |

123 Parameterization | 331 |

125 Stability and Equilibrium Analysis | 332 |

About R | 335 |

A2 A Very Short Introduction | 336 |

A22 Getting Help | 337 |

A24 More Complex Data Structures | 339 |

A25 Userdefined Functions and Programming | 340 |

A26 R Packages | 341 |

A28 Minor Things to Remember | 342 |

The Competition in a Lattice Grid Model Revisited | 343 |

Derivatives and Differential Equations | 347 |

B2 Taxonomy of Differential Equations | 348 |

B3 General Solutions of Often Used Differential Equations | 349 |

B32 SteadyState Transport Reaction in 1D Constant Surface | 350 |

B34 SteadyState Transport Reaction in 1D Spherical Coordinates | 351 |

B5 Derivatives and Integrals | 352 |

Matrix Algebra | 353 |

C2 Linear Equations | 354 |

C3 Eigenvalues and Eigenvectors Determinants | 355 |

C5 The Jacobian Matrix | 357 |

Statistical Distributions | 359 |

D3 The Poisson Distribution | 360 |

361 | |

367 | |