A Practical Guide to Ecological Modelling: Using R as a Simulation Platform

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Springer Science & Business Media, Oct 21, 2008 - Science - 372 pages
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Mathematical 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
References
361
Index
367
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