Modeling is now one of the most efficient methodologies in life sciences. From practice to theory, this book develops this approach illustrated by many examples; general concepts and the current state of the art are also presented and discussed.
An historical and general introduction informs the reader how mathematics and formal tools are used to solve biological problems at all levels of the organization of life. The core of this book explains how this is done, based on practical examples coming, for the most part, from the author’s personal experience. In most cases, data are included so that the reader can follow the reasoning process and even reproduce calculus. The final chapter is devoted to essential concepts and current developments. The main mathematical tools are presented in an appendix to the book and are written in an adapted language readable by scientists, professionals or students, with a basic knowledge of mathematics.

Preface  xi

Introduction  xv

Chapter 1. Methodology of Modeling in Biology and Ecology  1

1.1. Models and modeling  1

1.1.1. Models  2

1.1.2. Modeling  4

1.2. Mathematical modeling  6

1.2.1. Analysis of the biological situation and problem  7

1.2.2. Characterization and analysis of the system  11

1.2.3. Choice or construction of a model  14

1.2.4. Study of the properties of the model  18

1.2.5. Identification  25

1.2.6. Validation  26

1.2.7. Use  31

1.2.8. Conclusion  32

1.3. Supplements  33

1.3.1. Differences between a mathematical object and a mathematical model  33

1.3.2. Different types of objects and formalizations used in mathematical modeling  34

1.3.3. Elements for choosing a mathematical formalism  36

1.3.4. Stochastic and deterministic approaches  37

1.3.5. Discrete and continuous time  39

1.3.6. Biological and physical variables  39

1.3.7. The quantitative – qualitative debate  40

1.4. Models and modeling in life sciences  41

1.4.1. Historical overview  42

1.4.2. Modeling in biological disciplines  46

1.4.3. Modeling in population biology and ecology  47

1.4.4. Actors  48

1.4.5. Modeling and informatics  49

1.4.6. Definition of bioinformatics  49

1.5. A brief history of ecology and the importance of models in this discipline  51

1.6. Systems: a unifying concept  56

Chapter 2. Functional Representations: Construction and Interpretation of Mathematical Models  59

2.1. Introduction  60

2.2. Box and arrow diagrams: compartmental models  62

2.3. Representations based on Forrester diagrams  65

2.4. “Chemical-type” representation and multilinear differential models  66

2.4.1. General overview of the translation algorithm  67

2.4.2. Example of the logistic model  71

2.4.3. Saturation phenomena  73

2.5. Functional representations of models in population dynamics  76

2.5.1. Single population model  76

2.5.2. Models with two interacting populations  79

2.6. General points on functional representations and the interpretation of differential models  84

2.6.1. General hypotheses  84

2.6.2. Interpretation: phenomenological and mechanistic aspects, superficial knowledge and deep knowledge  85

2.6.3. Towards a classification of differential and integro-differential models of population dynamics  86

2.7. Conclusion  87

Chapter 3. Growth Models – Population

Dynamics and Genetics  89

3.1. The biological processes of growth  90

3.2. Experimental data  93

3.2.1. Organism growth data  93

3.2.2. Data relating to population growth  96

3.3. Models  98

3.3.1. Questions and uses of models  99

3.3.2. Some examples of classic growth models  100

3.4. Growth modeling and functional representations  104

3.4.1. Quantitative aspects  106

3.4.2. Qualitative aspects: choice and construction of models  107

3.4.3. Functional representations and growth models  107

3.4.4. Examples of the construction of new models  110

3.4.5. Typology of growth models  115

3.5. Growth of organisms: some examples  117

3.5.1. Individual growth of the European herring gull, Larus argentatus  117

3.5.2. Individual growth of young muskrats, Ondatra zibethica  118

3.5.3. Growth of a tree in a forest: examples of the application of individual growth models  124

3.5.4. Human growth  132

3.6. Models of population dynamics  133

3.6.1. Examples of growth models for bacterial populations: the exponential model, the logistic model, the Monod model and the Contois model  133

3.6.2. Dynamics of biodiversity at a geological level  146

3.7. Discrete time elementary demographic models  153

3.7.1. A discrete time demographic model of microbial populations  153

3.7.2. The Fibonacci model  155

3.7.3. Lindenmayer systems as demographic models  157

3.7.4. Examples of branching processes  164

3.7.5. Evolution of the “Grand Paradis” ibex population  170

3.7.6. Conclusion  172

3.8. Continuous time model of the age structure of a population  173

3.9. Spatialized dynamics: example of fishing populations and the regulation of sea-fishing  174

3.10. Evolution of the structure of an autogamous diploid population  175

3.10.1. The Mendelian system  176

3.10.2. Genetic evolution of an autogamous population  177

Chapter 4. Models of the Interaction Between Populations  183

4.1. The Volterra-Kostitzin model: an example of use

in molecular biology. Dynamics of RNA populations  184

4.1.1. Experimental data  185

4.1.2. Elements of qualitative analysis using the Kostitzin model  187

4.1.3. Initial data  190

4.1.4. Estimation of parameters and analysis of results  190

4.2. Models of competition between populations  193

4.2.1. The differential system  194

4.2.2. Description of competition using functional representations  198

4.2.3. Application to the study of competition between Fusarium populations in soil  203

4.2.4. Theoretical study of competition in an open system  207

4.2.5. Competition in a variable environment  210

4.3. Predator–prey systems  217

4.3.1. The basic model (model I)  218

4.3.2. Model in a limited environment (model II)  222

4.3.3. Model with limited capacities of assimilation of prey by the predator (model III)  227

4.3.4. Model with variable limited capacities for assimilation of prey by the predator  233

4.3.5. Model with limited capacities for assimilation of prey by the predator and spatial heterogeneity  234

4.3.6. Population dynamics of Rhizobium japonicum in soil  237

4.3.7. Predation of Rhizobium japonicum by amoeba in soil  239

4.4. Modeling the process of nitrification by microbial populations in soil: an example of succession  241

4.4.1. Introduction  241

4.4.2. Experimental procedure  243

4.4.3. Construction of the model – identification  244

4.4.4. Results  248

4.4.5. Discussion and conclusion  249

4.5. Conclusion and other details  251

Chapter 5. Compartmental Models
  253

5.1. Diagrammatic representations and associated mathematical models  256

5.1.1. Diagrammatic representations  256

5.1.2. Mathematical models  257

5.2. General autonomous compartmental models  265

5.2.1. Catenary systems  266

5.2.2. Looped systems  267

5.2.3. Mammillary systems  268

5.2.4. Systems representing spatial processes  268

5.2.5. General representation of an autonomous compartmental system  269

5.3. Estimation of model parameters  272

5.3.1. Least squares method (elementary principles)  272

5.3.2. Study of sensitivity functions – optimization of the experimental procedure  274

5.4. Open systems  274

5.4.1. The single compartment  274

5.4.2. The single compartment with input and output  275

5.5. General open compartmental models  278

5.6. Controllabillity, observability and identifiability

of a compartmental system  280

5.6.1. Controllabillity, observability and identifiability  280

5.6.2. Applications of these notions  281

5.7. Other mathematical models  282

5.8. Examples and additional information  283

5.8.1. Model of a single compartment system: application to the definition of optimal posology  283

5.8.2. Reversible two-compartment system  287

5.8.3. Estimation of tracer waiting time in cellular structures  293

5.8.4. Example of construction of the diffusion equation  300

Chapter 6. Complexity, Scales, Chaos, Chance and Other Oddities  305

6.1. Complexity  307

6.1.1. Some aspects of word use for complex and complexity  308

6.1.2. Biodiversity and complexity towards a unifying theory of biodiversity?  325

6.1.3. Random, logical, structural and dynamic complexity  328

6.2. Nonlinearities, temporal and spatial scales, the concept of equilibrium and its avatars  331

6.2.1. Time and spatial scales  335

6.2.2. About the concept of equilibrium  337

6.2.3. Transitions between attractors: are the bifurcations predictable?  342

6.3. The modeling of complexity  344

6.3.1. Complex dynamics: the example of deterministic chaos  344

6.3.2. Dynamics of complex systems and their structure  352

6.3.3. Shapes and morphogenesis – spatial structure dynamics:

Lindenmayer systems, fractals and cellular automata  358

6.3.4. Random behavior  369

6.4. Conclusion  371

6.4.1. Chance and complexity  371

6.4.2. The modeling approach  375

6.4.3. Problems linked to predictions  378

APPENDICES  383

Appendix 1. Differential Equations  385

Appendix 2. Recurrence Equations  465

Appendix 3. Fitting a Model to Experimental Results  489

Appendix 4. Introduction to Stochastic Processes  561

Bibliography  597

Index  617