Fractal Modelling: Growth and Form in Biology

ISBN-13: 9783642634024 / Angielski / Miękka / 2012 / 209 str.

Jaap A. Kaandorp; P. Prusinkiewicz

Opis

New developments in computer science, biology, mathematics and physics offer possibilities to obtain deeper understanding of growth and forms of organisms. It is now possible to carry out simulation experiments in which the growth process can be simulated in virtual computer objects. In this book, methods from fractal geometry are applied to model growth forms. As a case study, a type of growth process is used which can be found among various taxonomic classes of organisms such as sponges and corals. The growth of these organisms is simulated with 2D and 3D geometrical objects. The models presented in the book provide a rendering method for natural objects which is based on the actual growth process. The models can be used, for example, to understand the amazing variety of forms to be found in a coral reef. Models which mimic the growth of forms and the environmental influence on the growth process are also useful for ecologists. A combination of simulation models and the actual growth forms can be used to detect the effects of slow changes in the environment.

Spis treści

1 Introduction.- 1.1 Structure of the Book.- 2 Methods for Modelling Biological Objects.- 2.1 Reaction Diffusion Mechanisms.- 2.2 Iteration Processes and Fractals.- 2.3 Generation of Objects Using Formal Languages.- 2.4 Diffusion Limited Aggregation Models.- 2.5 Generation of Fractal Objects Using Iterated Function Systems.- 2.6 Iterative Geometric Constructions.- 2.6.1 Geometric Production Rules in 2D Modelling.- 2.6.2 The Geometric Modelling System for 2D Objects.- 2.6.3 Modelling a Growth Process in 2D with Iterative Geometric Constructions.- 2.7 A Review of the Methods.- 3 2D Models of Growth Forms.- 3.1 Modular Growth.- 3.2 Radiate Accretive Growth.- 3.3 Growth Forms of Modular Organisms and the Physical Environment.- 3.4 Description of the Internal Architecture of the Autotrophic Example: Montastrea annularis.- 3.5 Description of the Internal Architecture of the Heterotrophic Example: Haliclona oculata.- 3.6 An Iterative Geometric Construction Simulating the Radiate Accretive Growth Process of a Branching Organism.- 3.6.1 The Basic Construction: the generator.- 3.6.2 Modelling the Coherence of the Skeleton.- 3.6.3 Introduction of the Smallest Skeleton Element in the Model.- 3.6.4 Modelling the “Widening Effect”.- 3.6.5 Formation of New Growth Axes.- 3.6.6 Disturbance of the Growth Process, Formation of Plates.- 3.6.7 Additional Rules for the Formation of Branches and Plates.- 3.6.8 Formation of Branches.- 3.6.9 A Combination of the Previous Models.- 3.7 A Model of the Physical Environment.- 3.7.1 The Light Model.- 3.7.2 A Combination of the Geometric Model and the Concentration Gradient Model.- 3.8 Conclusions and Restrictions of the 2D Model.- 3.9 List of Symbols Used in this Chapter.- 4 A Comparison of Forms.- 4.1 A Comparison of a Range of Forms.- 4.1.1 A Comparison of a Range of Actual Forms and the Virtual Objects.- 4.1.2 A Comparison of the Growth Forms of Haliclona oculata Collected in Different Localities.- 4.1.3 Determination of the Fractal Dimensions in a Range of Forms.- 4.2 An Experimental Verification of the Model.- 4.2.1 The Simulation Experiments.- 4.2.2 The Transplantation Experiments.- 4.2.3 Comparison of Growth Forms of the Transplants and Simulation Experiments.- 4.3 Conclusions.- 5 3D Models of Growth Forms.- 5.1 Constructions in Space, a 3D Modelling System for Iterative Constructions.- 5.2 Description of an Organism with Radiate Accretive Growth and a Triangular Tessellation of the Surface.- 5.3 Representation of a Triangular Tessellation.- 5.4 Representation of a Multi-Layer Triangular Tessellation.- 5.5 The Lattice Representation of a Volume Tessellated with Triangles.- 5.5.1 The Lattice Model.- 5.5.2 The Virtual Lattice, a Subdivision of Space.- 5.6 An Iterative Geometric Construction Simulating the Radiate Accretive Growth Process of a Branching Organism.- 5.6.1 The Initiator.- 5.6.2 The Basic Construction: the Generator.- 5.6.3 Isotropic Growth and the Insertion of New Elements.- 5.6.4 Anisotropic Growth and the Insertion of New Elements.- 5.6.5 Formation of Branches.- 5.6.6 The Coherence Conserving Rules.- 5.6.7 More Evolved Branching Objects and Collision Detection.- 5.6.8 A Model of the Influence of Light Intensity on the Growth Process.- 5.6.9 A Model of the Influence of Nutrient Distribution on the Growth Process.- 5.7 Conclusions and Restrictions of the Presented 3D Models.- 5.8 List of Symbols Used in Sects. 5.3 to 5.7.- 6 Final Conclusions.- 6.1 The 2D and 3D Simulation Models.- 6.2 Application of the Simulation Models in Ecology.- References.