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Food Webs and Biodiversity: Foundations, Models, Data

ISBN: 978-0-470-97355-4
396 pages
August 2013
Food Webs and Biodiversity: Foundations, Models, Data (0470973552) cover image

Food webs have now been addressed in empirical and theoretical research for more than 50 years. Yet, even elementary foundational issues are still hotly debated. One difficulty is that a multitude of processes need to be taken into account to understand the patterns found empirically in the structure of food webs and communities.

 

Food Webs and Biodiversity develops a fresh, comprehensive perspective on food webs. Mechanistic explanations for several known macroecological patterns are derived from a few fundamental concepts, which are quantitatively linked to field-observables. An argument is developed that food webs will often be the key to understanding patterns of biodiversity at  community level.

 

Key Features:

 

  • Predicts generic characteristics of ecological communities in invasion-extirpation equilibrium.
  • Generalizes the theory of competition to food webs with arbitrary topologies.
  • Presents a new, testable quantitative theory for the mechanisms determining species richness in food webs, and other new results.
  • Written by an internationally respected expert in the field.

 

With global warming and other pressures on ecosystems rising, understanding and protecting biodiversity is a cause of international concern. This highly topical book will be of interest to a wide ranging audience, including not only graduate students and practitioners in community and conservation ecology but also the complex-systems research community as well as mathematicians and physicists interested in the theory of networks.

 

 

“This is a comprehensive work outlining a large array of very novel and potentially game-changing ideas in food web ecology.”

Ken Haste Andersen, Technical University of Denmark

 

 “I believe that this will be a landmark book in community ecology … it presents a well-established and consistent mathematical theory of food-webs. It is testable in many ways and the author finds remarkable agreements between predictions and reality.”

Géza Meszéna, Eötvös University, Budapest

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Acknowledgments xvii

List of Symbols xix

Part I Preliminaries

1 Introduction 3

2 Models and Theories 7

2.1 The usefulness of models 7

2.2 What models should model 8

2.3 The possibility of ecological theory 10

2.4 Theory-driven ecological research 11

3 Some Basic Concepts 13

3.1 Basic concepts of food-web studies 13

3.2 Physical quantities and dimensions 15

Part II Elements of Food-Web Models

4 Energy and Biomass Budgets 19

4.1 Currencies of accounting 19

4.2 Rates and efficiencies 20

4.3 Energy budgets in food webs 21

5 Allometric Scaling Relationships Between Body Size and Physiological Rates 25

5.1 Scales and scaling 25

5.2 Allometric scaling 26

6 Population Dynamics 29

6.1 Basic considerations 29

6.1.1 Exponential population growth 29

6.1.2 Five complications 30

6.1.3 Environmental variability 31

6.2 Structured populations and density-dependence 32

6.2.1 The dilemma between species and stages 32

6.2.2 Explicitly stage-structured population dynamics 32

6.2.3 Communities of structured populations 35

6.3 The Quasi-Neutral Approximation 35

6.3.1 The emergence of food webs 35

6.3.2 Rana catesbeiana and its resources 35

6.3.3 Numerical test of the approximation 38

6.4 Reproductive value 40

6.4.1 The concept of reproductive value 40

6.4.2 The role of reproductive value in the QNA 40

6.4.3 Body mass as a proxy for reproductive value 40

7 From Trophic Interactions to Trophic Link Strengths 45

7.1 Functional and numerical responses 45

7.2 Three models for functional responses 46

7.2.1 Linear response 46

7.2.2 Type II response 46

7.2.3 Type II response with prey switching 47

7.2.4 Strengths and weaknesses of these models 48

7.3 Food webs as networks of trophic link strengths 48

7.3.1 The ontology of trophic link strengths 48

7.3.2 Variability of trophic link strengths 49

8 Tropic Niche Space and Trophic Traits 51

8.1 Topology and dimensionality of trophic niche space 52

8.1.1 Formal setting 52

8.1.2 Definition of trophic niche-space dimensionality 53

8.2 Examples and ecological interpretations 55

8.2.1 A minimal example 55

8.2.2 Is the definition of dimensionality reasonable? 55

8.2.3 Dependencies between vulnerability and foraging traits of a species 56

8.2.4 The range of phenotypes considered affects niche-space dimensionality 56

8.3 Determination of trophic niche-space dimensionality 58

8.3.1 Typical empirical data 58

8.3.2 Direct estimation of dimensionality 59

8.3.3 Iterative estimation of dimensionality 59

8.4 Identification of trophic traits 60

8.4.1 Formal setting 60

8.4.2 Dimensional reduction 62

8.5 The geometry of trophic niche space 65

8.5.1 Abstract trophic traits 65

8.5.2 Indeterminacy in abstract trophic traits 65

8.5.3 The D-dimensional niche space as a pseudo-Euclidean space 66

8.5.4 Linear transformations of abstract trophic traits 67

8.5.5 Non-linear transformations of abstract trophic traits 68

8.5.6 Standardization and interpretation of abstract trophic traits 69

8.5.7 A hypothesis and a convention 72

8.5.8 Getting oriented in trophic niche space 73

8.6 Conclusions 75

9 Community Turnover and Evolution 77

9.1 The spatial scale of interest 77

9.2 How communities evolve 78

9.3 The mutation-for-dispersion trick 79

9.4 Mutation-for-dispersion in a neutral food-web model 80

10 The Population-Dynamical Matching Model 81

Part III Mechanisms and Processes

11 Basic Characterizations of Link-Strength Distributions 87

11.1 Modelling the distribution of logarithmic link strengths 88

11.1.1 General normally distributed trophic traits 88

11.1.2 Isotropically distributed trophic traits 91

11.2 High-dimensional trophic niche spaces 93

11.2.1 Understanding link stengths in high-dimensional trophic niche spaces 93

11.2.2 Log-normal probability distributions 94

11.2.3 The limit of log-normally distributed trophic link strength 95

11.2.4 Correlations between trophic link strengths 96

11.2.5 The distribution of the strengths of observable links 97

11.2.6 The probability of observing links (connectance) 99

11.2.7 Estimation of link-strength spread and Pareto exponent 100

11.2.8 Empirical examples 101

12 Diet Partitioning 103

12.1 The diet partitioning function 103

12.1.1 Relation to the probability distribution of diet proportions 105

12.1.2 Another probabilistic interpretation of the DPF 106

12.1.3 The normalization property of the DPF 106

12.1.4 Empirical determination of the DPF 107

12.2 Modelling the DPF 107

12.2.1 Formal setting 107

12.2.2 Diet ratios 108

12.2.3 The DPF for high-dimensional trophic niche spaces 109

12.2.4 Gini-Simpson dietary diversity 110

12.2.5 Dependence of the DPF on niche-space dimensionality 112

12.3 Comparison with data 113

12.4 Conclusions 114

13 Multivariate Link-Strength Distributions and Phylogenetic Patterns 117

13.1 Modelling phylogenetic structure in trophic traits 118

13.1.1 Phylogenetic correlations among logarithmic link strengths 120

13.1.2 Phylogenetic correlations among link strengths 121

13.1.3 Phylogenetic patterns in binary food webs 122

13.2 The matching model 123

13.2.1 A simple model for phylogenetic structure in food webs 123

13.2.2 Definition of the matching model 124

13.2.3 Sampling steady-state matching model food webs 124

13.2.4 Alternatives to the matching model 126

13.3 Characteristics of phylogenetically structured food webs 126

13.3.1 Graphical representation of food-web topologies 127

13.3.2 Standard parameter values 127

13.3.3 Intervality 128

13.3.4 Intervality and trophic niche-space dimensionality 129

13.3.5 Degree distributions 131

13.3.6 Other phylogenetic patterns 134

13.3.7 Is phylogeny just a nuisance? 135

14 A Framework Theory for Community Assembly 137

14.1 Ecological communities as dynamical systems 137

14.2 Existence, positivity, stability, and permanence 138

14.3 Generic bifurcations in community dynamics and their ecological phenomenology 139

14.3.1 General concepts 139

14.3.2 Saddle-node bifurcations 140

14.3.3 Hopf bifurcations 142

14.3.4 Transcritical bifurcations 142

14.3.5 Bifurcations of complicated attractors 144

14.4 Comparison with observations 144

14.4.1 Extirpations and invasions proceed slowly 145

14.4.2 The logistic equation works quite well 145

14.4.3 IUCN Red-List criteria highlight specific extinction scenarios 147

14.4.4 Conclusion 148

14.5 Invasion fitness and harvesting resistance 148

14.5.1 Invasion fitness 148

14.5.2 Harvesting resistance: definition 149

14.5.3 Harvesting resistance: interpretation 149

14.5.4 Harvesting resistance: computation 151

14.5.5 Interpretation of h → 0 152

14.6 Community assembly and stochastic species packing 152

14.6.1 Community saturation and species packing 152

14.6.2 Invasion probability 154

14.6.3 The steady-state distribution of harvesting resistance 157

14.6.4 The scenario of stochastic species packing 158

14.6.5 A numerical example 160

14.6.6 Biodiversity and ecosystem functioning 162

15 Competition in Food Webs 165

15.1 Basic concepts 166

15.1.1 Modes of competition 166

15.1.2 Interactions in communities 166

15.2 Competition in two-level food webs 167

15.2.1 The Lotka-Volterra two-level food-web model 168

15.2.2 Computation of the equilibrium point 168

15.2.3 Direct competition among producers 169

15.2.4 Resource-mediated competition in two-level food webs 169

15.2.5 Consumer-mediated competition in two-level food webs 170

15.3 Competition in arbitrary food webs 173

15.3.1 The general Lotka-Volterra food-web model 173

15.3.2 The competition matrix for general food webs 174

15.3.3 The L-R-P formalism 176

15.3.4 Ecological interpretations of the matrices L, R, and P 176

15.3.5 Formal computation of the equilibrium point 177

15.3.6 Consumer-mediated competition in general food webs 178

15.3.7 Consumer-mediated competitive exclusion 179

15.3.8 Conclusions 179

16 Mean-Field Theory of Resource-Mediated Competition 181

16.1 Transition to scaled variables 182

16.1.1 The competitive overlap matrix 182

16.1.2 Free abundances 183

16.2 The extended mean-field theory of competitive exclusion 184

16.2.1 Assumptions 184

16.2.2 Separation of means and residuals 186

16.2.3 Mean-field theory for the mean scaled abundance 187

16.2.4 Mean-field theory for the variance of scaled abundance 188

16.2.5 The coefficient of variation of scaled abundance 190

16.2.6 Related theories 191

17 Resource-Mediated Competition and Assembly 193

17.1 Preparation 193

17.1.1 Scaled vs. unscaled variables and parameters 193

17.1.2 Mean-field vs framework theory 195

17.2 Stochastic species packing under asymmetric competition 197

17.2.1 Species richness and distribution of invasion fitness (Part I) 198

17.2.2 Community response to invasion 199

17.2.3 Sensitivity of residents to invaders 200

17.2.4 Species richness and distribution of invasion fitness (Part II) 203

17.2.5 Random walks of abundances driven by invasions 204

17.2.6 Further discussion of the scenario 206

17.3 Stochastic species packing with competition symmetry 207

17.3.1 Community assembly with perfectly symmetric competition 207

17.3.2 Community assembly under nearly perfectly symmetric competition 209

17.3.3 Outline of mechanism limiting competition avoidance 211

17.3.4 The distribution of invasion fitness 212

17.3.5 Competition between residents and invaders 213

17.3.6 Balance of scaled biomass during assembly 214

17.3.7 Competition avoidance 215

17.3.8 Numerical test of the theory 216

18 Random-Matrix Competition Theory 221

18.1 Asymmetric competition 221

18.1.1 Girko’s Law 221

18.1.2 Application to competitive overlap matrices 223

18.1.3 Implications for sensitivity to invaders 223

18.1.4 Relation to mean-field theory 224

18.2 Stability vs feasibility limits to species richness 225

18.2.1 The result of May (1972) 225

18.2.2 Comparison of stability and feasibility criteria 225

18.3 Partially and fully symmetric competition 226

18.4 Sparse overlap matrices 228

18.4.1 Sparse competition 228

18.4.2 Eigenvalue distributions for sparse matrices 228

18.5 Resource overlap matrices 230

18.5.1 Diffuse resource competition 230

18.5.2 Sparse resource competition: the basic problem 232

18.5.3 The effect of trophic niche-space geometry 235

18.5.4 Competition among highly specialized consumers 237

18.5.5 Resource competition for varying ratios of producer to consumer richness 237

18.5.6 Competition for competing resources 239

18.6 Comparison with data 242

18.6.1 Gall-inducing insects on plants 242

18.6.2 Freshwater ecosystems 243

18.6.3 The North Sea 244

18.6.4 Conclusions 244

19 Species Richness, Size and Trophic Level 247

19.1 Predator-prey mass ratios 247

19.2 Modelling the joint distribution of size, trophic level, and species richness 249

19.2.1 Initial considerations 249

19.2.2 Model definition 251

19.2.3 Model simulation and comparison with data 252

20 Consumer-Mediated Competition and Assembly 255

20.1 A two-level food-web assembly model 256

20.2 Analytic characterization of the model steady state 257

20.2.1 Mechanism controlling producer richness 257

20.2.2 Other characteristics of the model steady state 259

20.3 Dependence of invader impacts on dietary diversity 262

20.3.1 Formal setting 262

20.3.2 Invadibility condition 263

20.3.3 Extirpation of resources during invasion 263

20.3.4 Extirpation of resources through consumer-mediated competition 264

20.3.5 Synthesis 264

20.4 Evolution of base attack rates 266

20.4.1 Motivation 266

20.4.2 Model definition 267

20.4.3 Numerical demonstration of attack rate evolution 267

20.4.4 Attack-rate evolution and prudent predation 268

21 Food Chains and Size Spectra 271

21.1 Concepts 271

21.1.1 Community size spectra 271

21.1.2 Species size spectra 273

21.2 Power-law food chains 274

21.2.1 Infinitely long power-law food chains 274

21.2.2 Top-down and bottom-up control 276

21.2.3 Power law-food chains of finite lengths and their stability to pulse

perturbations 278

21.2.4 Food chains as approximations for size spectra 279

21.2.5 Adaptation of attack rates 281

21.3 Food chains with non-linear functional responses 281

21.3.1 Loss of stability with density-independent consumption 282

21.3.2 Linearization of a generalized food chain model 283

21.3.3 Linear responses to press perturbations 284

21.3.4 Linear stability to pulse perturbations 285

21.4 What are the mechanisms controlling the scaling laws? 290

21.4.1 Arguments for biological constraints on transfer efficiency 290

21.4.2 Arguments for stability constraints on transfer efficiency 291

21.4.3 Arguments for ecological constraints on biomass imbalance 291

21.4.4 Arguments for mechanical constraints on PPMR 292

21.4.5 Arguments for dynamical constraints on PPMR 293

21.4.6 Conclusions 293

21.5 Scavengers and detrivores 294

21.5.1 The general argument 294

21.5.2 The microbial loop and other detrital channels 294

22 Structure and Dynamics of PDMM Model Communities 297

22.1 PDMM model definition 298

22.1.1 Model states 298

22.1.2 Species sampling and community assembly 298

22.1.3 Population dynamics 301

22.2 PDMM simulations 303

22.2.1 Trophic niche space and phylogenetic correlations 304

22.2.2 Steady state and invasion fitness 307

22.2.3 Diet partitioning 309

22.2.4 Resource-mediated competition 310

22.2.5 Distribution of species over body sizes and trophic levels 311

22.2.6 The size spectrum and related distributions 312

22.3 The PDMM with evolving attack rates 314

22.3.1 Modelling and tracking evolving attack rates in the PDMM 314

22.3.2 Time series of species richness, aggressivity and dietary diversity 315

22.3.3 Mutual regulation of aggressivity and dietary diversity 316

22.4 Conclusions 318

Part IV Implications

23 Scientific Implications 323

23.1 Main mechanisms identified by the theory 323

23.1.1 Two trades – one currency 323

23.1.2 Resource-mediated competition 324

23.1.3 Randomness and structure in food webs 324

23.1.4 Consumer-mediated competition and attack-rate evolution 325

23.2 Testable assumptions and predictions 325

23.2.1 Link-strength distributions and trophic niche-space geometry 325

23.2.2 Diet-partitioning statistics and sampling curves 325

23.2.3 Prey switching 326

23.2.4 Adapted attack rates 326

23.2.5 Community assembly and turnover 326

23.2.6 Patterns in link-strength matrices 327

23.3 Some unsolved problems 327

23.3.1 Large plants 327

23.3.2 Interactions between modes of competition 327

23.3.3 Absolute species richness: the role of viruses 327

23.3.4 The role of prey switching for community structure 328

23.3.5 The role of phylogenetic correlations for community dynamics 328

23.3.6 Fundamental constraints determining size-spectrum slopes 328

23.3.7 Community assembly with non-trivial attractors 328

23.3.8 Solution of the Riccati Equation for resource competition 328

23.3.9 Eigenvalues of competition matrices 329

23.3.10 Geometry and topology of trophic niche space 329

23.4 The future of community ecology 329

24 Conservation Implications 331

24.1 Assessing biodiversity 331

24.1.1 Quantifying biodiversity 331

24.1.2 Biodiversity supporting biodiversity 331

24.1.3 Assessing community turnover 332

24.2 Modelling ecological communities 333

24.2.1 Unpredictability of long-term community responses 333

24.2.2 Short-term predictions of community responses 334

24.2.3 Coarse-grained and stochastic community models 334

24.3 Managing biodiversity 334

Appendix A 337

A.1 Mathematical concepts, formulae, and jargon 337

A.1.1 Sums 337

A.1.2 Complex numbers 338

A.1.3 Vectors and matrices 339

A.1.4 Sets and functions 343

A.1.5 Differential calculus 343

A.1.6 Integrals 344

A.1.7 Differential equations 345

A.1.8 Random variables and expectation values 346

Bibliography 349

Index 365

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Axel G. Rossberg obtained an M.A. in theoretical physics at the University of Texas at Austin and a Ph.D. in complex-system physics at the University of Bayreuth. Since 2003 he is specializing on food-web theory and community ecology. To foster applications in the management context he recently joined UK’s Centre for Environment, Fisheries & Aquaculture Science (Cefas). He is also Senior Research Fellow at Queen’s University Belfast and Honorary Lecturer at University of East Anglia, and serves on the editorial board of The American Naturalist.

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