Shell coiling in some larger foraminifera: general comments and problems

The logistic model provides a useful geometric description of coiling for larger Foraminifera. This model represents a spiral in polar coordinates by means of a logistic equation. The model describes the spires of the genera Alveolina, Nummulites, and Assilina very well. The logistic model is a consequence of allometric growth in these genera. The spires of Nummulites do not fit the logarithmic spiral model well, because the model does not accommodate ontogenetic change of form. Computer simulations using the logistic model also show the very different morphologies of the genus Alveolina. The model allows study of geometrical constraints on test form and of pathways followed by the evolution of the genus (or genera). The logistic model describes ontogenetic change in the organism's functional requirements. The main physiological requirements are nourishment, general exchanges of matter and energy with the environment, and light for the photosynthetic symbionts that live in the cytoplasm of these larger Foraminifera. All these functions require suitable area/volume ratios, that mandate changes of form during growth. The possible biological meaning of the parameters of the model is explored. The use of the more meaningful parameters is proposed in statistical multivariate studies.