J.A.Kaandorp - Fractal modelling. Growth and form in biology.Contents

Table of Contents

1 Introduction 1

1.1 Structure of the Book 5

2 Methods for Modelling Biological Objects 7

1. Reaction Diffusion Mechanisms 7

2. Iteration Processes and Fractals 9

3. Generation of Objects Using Formal Languages 13

4. Diffusion Limited Aggregation Models 18

5. Generation of Fractal Objects

Using Iterated Function Systems 23

2.6 Iterative Geometric Constructions 27

1. Geometric Production Rules in 2D Modelling 27

2. The Geometric Modelling System for 2D Objects 41

3. Modelling a Growth Process in 2D

with Iterative Geometric Constructions 44

2.7 A Review of the Methods 53

3 2D Models of Growth Forms 55

1. Modular Growth 55

2. Radiate Accretive Growth 56

3. Growth Forms of Modular Organisms

and the Physical Environment 58

3.4 Description of the Internal Architecture

of the Autotrophic Example: Montastrea annularis 64

3.5 Description of the Internal Architecture

of the Heterotrophic Example: Haliclona oculata 66

3.6 An Iterative Geometric Construction Simulating the Radiate Accretive Growth Process of a Branching Organism .... 68

XII Table of Contents

1. The Basic Construction: the generator 69

2. Modelling the Coherence of the Skeleton 74

3. Introduction of the Smallest Skeleton Element in the Model 77

4. Modelling the "Widening Effect" 78

5. Formation of New Growth Axes 79

6. Disturbance of the Growth Process, Formation of Plates . . 82

7. Additional Rules for the Formation of Branches and Plate s . 83

8. Formation of Branches 85

9. A Combination of the Previous Models 88

3.7 A Model of the Physical Environment 89

1. The Light Model 90

2. A Combination of the Geometric Model

and the Concentration Gradient Model 92

8. Conclusions and Restrictions of the 2D Model 98

9. List of Symbols Used in this Chapter 100

4 A Comparison of Forms 103

4.1 A Comparison of a Range of Forms 103

4.1.1 A Comparison of a Range of Actual Forms

and the Virtual Objects 105

2. A Comparison of the Growth Forms of Haliclona oculata Collected in Different Localities 110

3. Determination of the Fractal Dimensions

in a Range of Forms 110

4.2 An Experimental Verification of the Model 112

1. The Simulation Experiments 113

2. The Transplantation Experiments 114

3. Comparison of Growth Forms of the Transplants

and Simulation Experiments 118

4.3 Conclusions 126

5 3D Models of Growth Forms 129

5.1 Constructions in Space, a 3D Modelling System

for Iterative Constructions 129

5.2 Description of an Organism with Radiate Accretive Growth

and a Triangular Tessellation of the Surface 133

3. Representation of a Triangular Tessellation 134

4. Representation of a Multi-Layer Triangular Tessellation . . 140

Table of Contents XIII

5.5 The Lattice Representation of a Volume Tessellated

with Triangles 142

1. The Lattice Model 144

2. The Virtual Lattice, a Subdivision of Space 146

5.6 An Iterative Geometric Construction Simulating the Radiate Accretive Growth Process of a Branching Organism .... 148

1. The Initiator 148

2. The Basic Construction: the Generator 149

3. Isotropic Growth and the Inserti on of New Elements .... 150

4. Anisotropic Growth and the Insertion of New Elements . . 155

5. Formation of Branches 157

6. The Coherence Conserving Rules 164

7. More Evolved Branching Objects and Collision Detection . 165

8. A Model of the Influence of Light Intensity

on the Growth Process 172

5.6.9 A Model of the Influence of Nutrient Distribution

on the Growth Process 178

7. Conclusions and Restrictions of the Presented 3D Models . 182

8. List of Symbols Used in Sects. 5.3 to 5.7 184

6 Final Conclusions 189

1. The 2D and 3D Simulation Models 189

2. Application of the Simulation Models in Ecology 191

References 197

Index 205