Fractal Modelling

In this chapter several methods for modelling biological objects are discussed. The methods described in this chapter have the potentiality to serve as morphological models of biological objects. In the first section a model for pattern formation, based on diffusing chemicals, is described. In Sect. 2.2 the iteration processes and fractals which form the general base of the methods described in the Sects. 2.3, 2.4, 2.5 and, 2.6 are discussed. In the last section of this chapter a review is given of the methods mentioned in the chapter and arguments are given as to which method is the most applicable for morphological models of growth processes.

This chapter discusses how a system of rules can be created, as shown in the section on the ramifying objects, suitable for the simulation of the growth process of various sessile marine organisms, for example sponges (Porifera) and corals (Scleractinia). A crucial difference with the example of the ramifying objects is that each modelling step is supposed to have a biological significance. An important reason to use sponges and corals as subjects of a case-study is that these organisms exhibit a relatively simple growth process, which makes it comparatively easy to design geometric production rules to simulate growth processes. Although the marine sessile organisms belong to many very different taxonomic groups, it is possible to distinguish several types of corresponding growth processes within these groups, in which a similar architecture emerges. First almost all groups belong to the large group of modular organisms. In this group there is a subset of organisms which are formed by one type of growth process, which will be discussed in particular in the next sections. This growth process can be found within sponges, stony corals and many other marine sessile organisms and will be indicated as radiate accretive growth. Modular growth and radiate accretive growth are the first subjects in this chapter, followed by a 2D model of radiate accretive growth.

In the first section of this chapter the simulated forms and the actual growth forms are compared to each other. A quantitative comparison of these forms is an essential step in testing the biological validity of simulation models. When both virtual and actual forms can be quantified, is is also possible to determine a relation between the model parameters and the observed forms.

In this chapter the development of a model of a growth in three dimensions is discussed. In the first section it is demonstrated how the modelling system for iterative geometric constructions (see also Sect. 2.6) can be extended to 3D. In Sect. 5.2, a discussion follows on the 3D structure of an organism with radiate accretive growth. In Sects. 5.3 and 5.4 it is discussed how this 3D structure can be represented in a model. The results of these sections, a suitable data representation for a 3D object developing in the radiate accretive growth process, is used in the final Sect. 5.6, in which the development of a model of a radiate growth process in three dimensions is presented. Most of the rules discussed in Sect. 3.6, on the iterative geometric constructions for simulating this growth process, will be extended to 3D. In Sect. 3.6 the biological examples used as a case-study were the sponge Haliclona oculata and the stony coral Montastrea annularis. For reasons which will become clear in the next sections, the sponge Haliclona simulans (see Fig. 3.15) is used as an example in this chapter. The biological significance of the rules will be indicated only briefly in this chapter, since most of them were already discussed in Sect. 3.6. The extension to 3D of the simulation model is an essential one, since many aspects of the growth process (e.g. a larger possibility for the branches to avoid each other, the formation of flattened forms influenced by the flow direction) can only be adequately described with a 3D model. For convenience the symbols used in the sections on the 3D model of radiate accretive growth (Sects. 5.3–5.7 are listed separately in Sect. 5.8).

In Chaps. 3,4 and 5 it was demonstrated that the radiate accretive growth can be simulated in 2D and 3D with a geometric model. In these models growth is described as an iterative process in which the growing object is represented by a geometrical object. In the iteration process a set of rules is applied which can be divided into two types: rules representing the internal properties of the growing object and rules which represent the influence of the environment on the growth process.