Alain Pavé is Emeritus Professor of University, Member of the French National Academy of Technologies and Associate Member of the French Academy of Agriculture, Lyon, France.

• Preface xiIntroduction xvChapter 1. Methodology of Modeling in Biology and Ecology 11.1. Models and modeling 11.1.1. Models 21.1.2. Modeling 41.2. Mathematical modeling 61.2.1. Analysis of the biological situation and problem 71.2.2. Characterization and analysis of the system 111.2.3. Choice or construction of a model 141.2.4. Study of the properties of the model 181.2.5. Identification 251.2.6. Validation 261.2.7. Use 311.2.8. Conclusion 321.3. Supplements 331.3.1. Differences between a mathematical object and a mathematical model 331.3.2. Different types of objects and formalizations used in mathematical modeling 341.3.3. Elements for choosing a mathematical formalism 361.3.4. Stochastic and deterministic approaches 371.3.5. Discrete and continuous time 391.3.6. Biological and physical variables 391.3.7. The quantitative – qualitative debate 401.4. Models and modeling in life sciences 411.4.1. Historical overview 421.4.2. Modeling in biological disciplines 461.4.3. Modeling in population biology and ecology 471.4.4. Actors 481.4.5. Modeling and informatics 491.4.6. Definition of bioinformatics 491.5. A brief history of ecology and the importance of models in this discipline 511.6. Systems: a unifying concept 56Chapter 2. Functional Representations: Construction and Interpretation of Mathematical Models 592.1. Introduction 602.2. Box and arrow diagrams: compartmental models 622.3. Representations based on Forrester diagrams 652.4. “Chemical-type” representation and multilinear differential models 662.4.1. General overview of the translation algorithm 672.4.2. Example of the logistic model 712.4.3. Saturation phenomena 732.5. Functional representations of models in population dynamics 762.5.1. Single population model 762.5.2. Models with two interacting populations 792.6. General points on functional representations and the interpretation of differential models 842.6.1. General hypotheses 842.6.2. Interpretation: phenomenological and mechanistic aspects, superficial knowledge and deep knowledge 852.6.3. Towards a classification of differential and integro-differential models of population dynamics  862.7. Conclusion 87Chapter 3. Growth Models – Population Dynamics and Genetics 893.1. The biological processes of growth 903.2. Experimental data 933.2.1. Organism growth data 933.2.2. Data relating to population growth 963.3. Models 983.3.1. Questions and uses of models 993.3.2. Some examples of classic growth models 1003.4. Growth modeling and functional representations 1043.4.1. Quantitative aspects 1063.4.2. Qualitative aspects: choice and construction of models 1073.4.3. Functional representations and growth models 1073.4.4. Examples of the construction of new models 1103.4.5. Typology of growth models 1153.5. Growth of organisms: some examples 1173.5.1. Individual growth of the European herring gull, Larus argentatus 1173.5.2. Individual growth of young muskrats, Ondatra zibethica 1183.5.3. Growth of a tree in a forest: examples of the application of individual growth models 1243.5.4. Human growth 1323.6. Models of population dynamics 1333.6.1. Examples of growth models for bacterial populations: the exponential model, the logistic model, the Monod model and the Contois model 1333.6.2. Dynamics of biodiversity at a geological level 1463.7. Discrete time elementary demographic models 1533.7.1. A discrete time demographic model of microbial populations 1533.7.2. The Fibonacci model 1553.7.3. Lindenmayer systems as demographic models 1573.7.4. Examples of branching processes 1643.7.5. Evolution of the “Grand Paradis” ibex population 1703.7.6. Conclusion 1723.8. Continuous time model of the age structure of a population 1733.9. Spatialized dynamics: example of fishing populations and the regulation of sea-fishing 1743.10. Evolution of the structure of an autogamous diploid population 1753.10.1. The Mendelian system 1763.10.2. Genetic evolution of an autogamous population 177Chapter 4. Models of the Interaction Between Populations 1834.1. The Volterra-Kostitzin model: an example of use in molecular biology. Dynamics of RNA populations 1844.1.1. Experimental data 1854.1.2. Elements of qualitative analysis using the Kostitzin model 1874.1.3. Initial data 1904.1.4. Estimation of parameters and analysis of results 1904.2. Models of competition between populations 1934.2.1. The differential system 1944.2.2. Description of competition using functional representations 1984.2.3. Application to the study of competition between Fusarium populations in soil 2034.2.4. Theoretical study of competition in an open system 2074.2.5. Competition in a variable environment 2104.3. Predator–prey systems 2174.3.1. The basic model (model I) 2184.3.2. Model in a limited environment (model II) 2224.3.3. Model with limited capacities of assimilation of prey by the predator (model III) 2274.3.4. Model with variable limited capacities for assimilation of prey by the predator 2334.3.5. Model with limited capacities for assimilation of prey by the predator and spatial heterogeneity  2344.3.6. Population dynamics of Rhizobium japonicum in soil 2374.3.7. Predation of Rhizobium japonicum by amoeba in soil 2394.4. Modeling the process of nitrification by microbial populations in soil: an example of succession  2414.4.1. Introduction 2414.4.2. Experimental procedure 2434.4.3. Construction of the model – identification 2444.4.4. Results 2484.4.5. Discussion and conclusion 2494.5. Conclusion and other details 251Chapter 5. Compartmental Models
2535.1. Diagrammatic representations and associated mathematical models 2565.1.1. Diagrammatic representations 2565.1.2. Mathematical models 2575.2. General autonomous compartmental models 2655.2.1. Catenary systems 2665.2.2. Looped systems 2675.2.3. Mammillary systems 2685.2.4. Systems representing spatial processes 2685.2.5. General representation of an autonomous compartmental system 2695.3. Estimation of model parameters 2725.3.1. Least squares method (elementary principles) 2725.3.2. Study of sensitivity functions – optimization of the experimental procedure 2745.4. Open systems 2745.4.1. The single compartment 2745.4.2. The single compartment with input and output 2755.5. General open compartmental models 2785.6. Controllabillity, observability and identifiability of a compartmental system 2805.6.1. Controllabillity, observability and identifiability 2805.6.2. Applications of these notions 2815.7. Other mathematical models 2825.8. Examples and additional information 2835.8.1. Model of a single compartment system: application to the definition of optimal posology 2835.8.2. Reversible two-compartment system 2875.8.3. Estimation of tracer waiting time in cellular structures 2935.8.4. Example of construction of the diffusion equation 300Chapter 6. Complexity, Scales, Chaos, Chance and Other Oddities 3056.1. Complexity 3076.1.1. Some aspects of word use for complex and complexity 3086.1.2. Biodiversity and complexity towards a unifying theory of biodiversity? 3256.1.3. Random, logical, structural and dynamic complexity 3286.2. Nonlinearities, temporal and spatial scales, the concept of equilibrium and its avatars 3316.2.1. Time and spatial scales 3356.2.2. About the concept of equilibrium 3376.2.3. Transitions between attractors: are the bifurcations predictable? 3426.3. The modeling of complexity 3446.3.1. Complex dynamics: the example of deterministic chaos 3446.3.2. Dynamics of complex systems and their structure 3526.3.3. Shapes and morphogenesis – spatial structure dynamics: Lindenmayer systems, fractals and cellular automata 3586.3.4. Random behavior 3696.4. Conclusion 3716.4.1. Chance and complexity 3716.4.2. The modeling approach 3756.4.3. Problems linked to predictions 378APPENDICES 383Appendix 1. Differential Equations 385Appendix 2. Recurrence Equations 465Appendix 3. Fitting a Model to Experimental Results 489Appendix 4. Introduction to Stochastic Processes 561Bibliography 597Index 617

Alain Pavé is Emeritus Professor of University, Member of the French National Academy of Technologies and Associate Member of the French Academy of Agriculture, Lyon, France.

• Preface xiIntroduction xvChapter 1. Methodology of Modeling in Biology and Ecology 11.1. Models and modeling 11.1.1. Models 21.1.2. Modeling 41.2. Mathematical modeling 61.2.1. Analysis of the biological situation and problem 71.2.2. Characterization and analysis of the system 111.2.3. Choice or construction of a model 141.2.4. Study of the properties of the model 181.2.5. Identification 251.2.6. Validation 261.2.7. Use 311.2.8. Conclusion 321.3. Supplements 331.3.1. Differences between a mathematical object and a mathematical model 331.3.2. Different types of objects and formalizations used in mathematical modeling 341.3.3. Elements for choosing a mathematical formalism 361.3.4. Stochastic and deterministic approaches 371.3.5. Discrete and continuous time 391.3.6. Biological and physical variables 391.3.7. The quantitative – qualitative debate 401.4. Models and modeling in life sciences 411.4.1. Historical overview 421.4.2. Modeling in biological disciplines 461.4.3. Modeling in population biology and ecology 471.4.4. Actors 481.4.5. Modeling and informatics 491.4.6. Definition of bioinformatics 491.5. A brief history of ecology and the importance of models in this discipline 511.6. Systems: a unifying concept 56Chapter 2. Functional Representations: Construction and Interpretation of Mathematical Models 592.1. Introduction 602.2. Box and arrow diagrams: compartmental models 622.3. Representations based on Forrester diagrams 652.4. “Chemical-type” representation and multilinear differential models 662.4.1. General overview of the translation algorithm 672.4.2. Example of the logistic model 712.4.3. Saturation phenomena 732.5. Functional representations of models in population dynamics 762.5.1. Single population model 762.5.2. Models with two interacting populations 792.6. General points on functional representations and the interpretation of differential models 842.6.1. General hypotheses 842.6.2. Interpretation: phenomenological and mechanistic aspects, superficial knowledge and deep knowledge 852.6.3. Towards a classification of differential and integro-differential models of population dynamics  862.7. Conclusion 87Chapter 3. Growth Models – Population Dynamics and Genetics 893.1. The biological processes of growth 903.2. Experimental data 933.2.1. Organism growth data 933.2.2. Data relating to population growth 963.3. Models 983.3.1. Questions and uses of models 993.3.2. Some examples of classic growth models 1003.4. Growth modeling and functional representations 1043.4.1. Quantitative aspects 1063.4.2. Qualitative aspects: choice and construction of models 1073.4.3. Functional representations and growth models 1073.4.4. Examples of the construction of new models 1103.4.5. Typology of growth models 1153.5. Growth of organisms: some examples 1173.5.1. Individual growth of the European herring gull, Larus argentatus 1173.5.2. Individual growth of young muskrats, Ondatra zibethica 1183.5.3. Growth of a tree in a forest: examples of the application of individual growth models 1243.5.4. Human growth 1323.6. Models of population dynamics 1333.6.1. Examples of growth models for bacterial populations: the exponential model, the logistic model, the Monod model and the Contois model 1333.6.2. Dynamics of biodiversity at a geological level 1463.7. Discrete time elementary demographic models 1533.7.1. A discrete time demographic model of microbial populations 1533.7.2. The Fibonacci model 1553.7.3. Lindenmayer systems as demographic models 1573.7.4. Examples of branching processes 1643.7.5. Evolution of the “Grand Paradis” ibex population 1703.7.6. Conclusion 1723.8. Continuous time model of the age structure of a population 1733.9. Spatialized dynamics: example of fishing populations and the regulation of sea-fishing 1743.10. Evolution of the structure of an autogamous diploid population 1753.10.1. The Mendelian system 1763.10.2. Genetic evolution of an autogamous population 177Chapter 4. Models of the Interaction Between Populations 1834.1. The Volterra-Kostitzin model: an example of use in molecular biology. Dynamics of RNA populations 1844.1.1. Experimental data 1854.1.2. Elements of qualitative analysis using the Kostitzin model 1874.1.3. Initial data 1904.1.4. Estimation of parameters and analysis of results 1904.2. Models of competition between populations 1934.2.1. The differential system 1944.2.2. Description of competition using functional representations 1984.2.3. Application to the study of competition between Fusarium populations in soil 2034.2.4. Theoretical study of competition in an open system 2074.2.5. Competition in a variable environment 2104.3. Predator–prey systems 2174.3.1. The basic model (model I) 2184.3.2. Model in a limited environment (model II) 2224.3.3. Model with limited capacities of assimilation of prey by the predator (model III) 2274.3.4. Model with variable limited capacities for assimilation of prey by the predator 2334.3.5. Model with limited capacities for assimilation of prey by the predator and spatial heterogeneity  2344.3.6. Population dynamics of Rhizobium japonicum in soil 2374.3.7. Predation of Rhizobium japonicum by amoeba in soil 2394.4. Modeling the process of nitrification by microbial populations in soil: an example of succession  2414.4.1. Introduction 2414.4.2. Experimental procedure 2434.4.3. Construction of the model – identification 2444.4.4. Results 2484.4.5. Discussion and conclusion 2494.5. Conclusion and other details 251Chapter 5. Compartmental Models
2535.1. Diagrammatic representations and associated mathematical models 2565.1.1. Diagrammatic representations 2565.1.2. Mathematical models 2575.2. General autonomous compartmental models 2655.2.1. Catenary systems 2665.2.2. Looped systems 2675.2.3. Mammillary systems 2685.2.4. Systems representing spatial processes 2685.2.5. General representation of an autonomous compartmental system 2695.3. Estimation of model parameters 2725.3.1. Least squares method (elementary principles) 2725.3.2. Study of sensitivity functions – optimization of the experimental procedure 2745.4. Open systems 2745.4.1. The single compartment 2745.4.2. The single compartment with input and output 2755.5. General open compartmental models 2785.6. Controllabillity, observability and identifiability of a compartmental system 2805.6.1. Controllabillity, observability and identifiability 2805.6.2. Applications of these notions 2815.7. Other mathematical models 2825.8. Examples and additional information 2835.8.1. Model of a single compartment system: application to the definition of optimal posology 2835.8.2. Reversible two-compartment system 2875.8.3. Estimation of tracer waiting time in cellular structures 2935.8.4. Example of construction of the diffusion equation 300Chapter 6. Complexity, Scales, Chaos, Chance and Other Oddities 3056.1. Complexity 3076.1.1. Some aspects of word use for complex and complexity 3086.1.2. Biodiversity and complexity towards a unifying theory of biodiversity? 3256.1.3. Random, logical, structural and dynamic complexity 3286.2. Nonlinearities, temporal and spatial scales, the concept of equilibrium and its avatars 3316.2.1. Time and spatial scales 3356.2.2. About the concept of equilibrium 3376.2.3. Transitions between attractors: are the bifurcations predictable? 3426.3. The modeling of complexity 3446.3.1. Complex dynamics: the example of deterministic chaos 3446.3.2. Dynamics of complex systems and their structure 3526.3.3. Shapes and morphogenesis – spatial structure dynamics: Lindenmayer systems, fractals and cellular automata 3586.3.4. Random behavior 3696.4. Conclusion 3716.4.1. Chance and complexity 3716.4.2. The modeling approach 3756.4.3. Problems linked to predictions 378APPENDICES 383Appendix 1. Differential Equations 385Appendix 2. Recurrence Equations 465Appendix 3. Fitting a Model to Experimental Results 489Appendix 4. Introduction to Stochastic Processes 561Bibliography 597Index 617

Alain Pavé is Emeritus Professor of University, Member of the French National Academy of Technologies and Associate Member of the French Academy of Agriculture, Lyon, France.

• Preface xiIntroduction xvChapter 1. Methodology of Modeling in Biology and Ecology 11.1. Models and modeling 11.1.1. Models 21.1.2. Modeling 41.2. Mathematical modeling 61.2.1. Analysis of the biological situation and problem 71.2.2. Characterization and analysis of the system 111.2.3. Choice or construction of a model 141.2.4. Study of the properties of the model 181.2.5. Identification 251.2.6. Validation 261.2.7. Use 311.2.8. Conclusion 321.3. Supplements 331.3.1. Differences between a mathematical object and a mathematical model 331.3.2. Different types of objects and formalizations used in mathematical modeling 341.3.3. Elements for choosing a mathematical formalism 361.3.4. Stochastic and deterministic approaches 371.3.5. Discrete and continuous time 391.3.6. Biological and physical variables 391.3.7. The quantitative – qualitative debate 401.4. Models and modeling in life sciences 411.4.1. Historical overview 421.4.2. Modeling in biological disciplines 461.4.3. Modeling in population biology and ecology 471.4.4. Actors 481.4.5. Modeling and informatics 491.4.6. Definition of bioinformatics 491.5. A brief history of ecology and the importance of models in this discipline 511.6. Systems: a unifying concept 56Chapter 2. Functional Representations: Construction and Interpretation of Mathematical Models 592.1. Introduction 602.2. Box and arrow diagrams: compartmental models 622.3. Representations based on Forrester diagrams 652.4. “Chemical-type” representation and multilinear differential models 662.4.1. General overview of the translation algorithm 672.4.2. Example of the logistic model 712.4.3. Saturation phenomena 732.5. Functional representations of models in population dynamics 762.5.1. Single population model 762.5.2. Models with two interacting populations 792.6. General points on functional representations and the interpretation of differential models 842.6.1. General hypotheses 842.6.2. Interpretation: phenomenological and mechanistic aspects, superficial knowledge and deep knowledge 852.6.3. Towards a classification of differential and integro-differential models of population dynamics  862.7. Conclusion 87Chapter 3. Growth Models – Population Dynamics and Genetics 893.1. The biological processes of growth 903.2. Experimental data 933.2.1. Organism growth data 933.2.2. Data relating to population growth 963.3. Models 983.3.1. Questions and uses of models 993.3.2. Some examples of classic growth models 1003.4. Growth modeling and functional representations 1043.4.1. Quantitative aspects 1063.4.2. Qualitative aspects: choice and construction of models 1073.4.3. Functional representations and growth models 1073.4.4. Examples of the construction of new models 1103.4.5. Typology of growth models 1153.5. Growth of organisms: some examples 1173.5.1. Individual growth of the European herring gull, Larus argentatus 1173.5.2. Individual growth of young muskrats, Ondatra zibethica 1183.5.3. Growth of a tree in a forest: examples of the application of individual growth models 1243.5.4. Human growth 1323.6. Models of population dynamics 1333.6.1. Examples of growth models for bacterial populations: the exponential model, the logistic model, the Monod model and the Contois model 1333.6.2. Dynamics of biodiversity at a geological level 1463.7. Discrete time elementary demographic models 1533.7.1. A discrete time demographic model of microbial populations 1533.7.2. The Fibonacci model 1553.7.3. Lindenmayer systems as demographic models 1573.7.4. Examples of branching processes 1643.7.5. Evolution of the “Grand Paradis” ibex population 1703.7.6. Conclusion 1723.8. Continuous time model of the age structure of a population 1733.9. Spatialized dynamics: example of fishing populations and the regulation of sea-fishing 1743.10. Evolution of the structure of an autogamous diploid population 1753.10.1. The Mendelian system 1763.10.2. Genetic evolution of an autogamous population 177Chapter 4. Models of the Interaction Between Populations 1834.1. The Volterra-Kostitzin model: an example of use in molecular biology. Dynamics of RNA populations 1844.1.1. Experimental data 1854.1.2. Elements of qualitative analysis using the Kostitzin model 1874.1.3. Initial data 1904.1.4. Estimation of parameters and analysis of results 1904.2. Models of competition between populations 1934.2.1. The differential system 1944.2.2. Description of competition using functional representations 1984.2.3. Application to the study of competition between Fusarium populations in soil 2034.2.4. Theoretical study of competition in an open system 2074.2.5. Competition in a variable environment 2104.3. Predator–prey systems 2174.3.1. The basic model (model I) 2184.3.2. Model in a limited environment (model II) 2224.3.3. Model with limited capacities of assimilation of prey by the predator (model III) 2274.3.4. Model with variable limited capacities for assimilation of prey by the predator 2334.3.5. Model with limited capacities for assimilation of prey by the predator and spatial heterogeneity  2344.3.6. Population dynamics of Rhizobium japonicum in soil 2374.3.7. Predation of Rhizobium japonicum by amoeba in soil 2394.4. Modeling the process of nitrification by microbial populations in soil: an example of succession  2414.4.1. Introduction 2414.4.2. Experimental procedure 2434.4.3. Construction of the model – identification 2444.4.4. Results 2484.4.5. Discussion and conclusion 2494.5. Conclusion and other details 251Chapter 5. Compartmental Models
2535.1. Diagrammatic representations and associated mathematical models 2565.1.1. Diagrammatic representations 2565.1.2. Mathematical models 2575.2. General autonomous compartmental models 2655.2.1. Catenary systems 2665.2.2. Looped systems 2675.2.3. Mammillary systems 2685.2.4. Systems representing spatial processes 2685.2.5. General representation of an autonomous compartmental system 2695.3. Estimation of model parameters 2725.3.1. Least squares method (elementary principles) 2725.3.2. Study of sensitivity functions – optimization of the experimental procedure 2745.4. Open systems 2745.4.1. The single compartment 2745.4.2. The single compartment with input and output 2755.5. General open compartmental models 2785.6. Controllabillity, observability and identifiability of a compartmental system 2805.6.1. Controllabillity, observability and identifiability 2805.6.2. Applications of these notions 2815.7. Other mathematical models 2825.8. Examples and additional information 2835.8.1. Model of a single compartment system: application to the definition of optimal posology 2835.8.2. Reversible two-compartment system 2875.8.3. Estimation of tracer waiting time in cellular structures 2935.8.4. Example of construction of the diffusion equation 300Chapter 6. Complexity, Scales, Chaos, Chance and Other Oddities 3056.1. Complexity 3076.1.1. Some aspects of word use for complex and complexity 3086.1.2. Biodiversity and complexity towards a unifying theory of biodiversity? 3256.1.3. Random, logical, structural and dynamic complexity 3286.2. Nonlinearities, temporal and spatial scales, the concept of equilibrium and its avatars 3316.2.1. Time and spatial scales 3356.2.2. About the concept of equilibrium 3376.2.3. Transitions between attractors: are the bifurcations predictable? 3426.3. The modeling of complexity 3446.3.1. Complex dynamics: the example of deterministic chaos 3446.3.2. Dynamics of complex systems and their structure 3526.3.3. Shapes and morphogenesis – spatial structure dynamics: Lindenmayer systems, fractals and cellular automata 3586.3.4. Random behavior 3696.4. Conclusion 3716.4.1. Chance and complexity 3716.4.2. The modeling approach 3756.4.3. Problems linked to predictions 378APPENDICES 383Appendix 1. Differential Equations 385Appendix 2. Recurrence Equations 465Appendix 3. Fitting a Model to Experimental Results 489Appendix 4. Introduction to Stochastic Processes 561Bibliography 597Index 617