A MATRIX POPULATION MODEL FOR GROWTH OF THE ECHINOID SKELETON

The skeleton of an echinoid comprises hundreds to thousands of individual calcite plates. The collection of plates is analogous to a population of organisms. Traditional structured population models in biology treat populations of organisms as individuals (or cohorts of individuals) with specific birth and death rates that lead to stable age distributions. The simplest models can be represented as Leslie projection matrices, based on age-specific survivability and fecundity rates. The conceptual leap to representation of the echinoid skeleton as a population of plates is aided by consideration of a life cycle graph. For example, take a stylized life cycle graph of a single column of plates in a sea urchin test, representing each growth stage with a subscripted P_i , where P₀ represents the plate nucleation point with an inherent fecundity of 0. Addition of new plates (fecundity) is density dependent. Growth rates of individual plates are time-dependent. A realistic system will reach a stable plate size distribution representing a plate column of an adult urchin. A complete representation must take into account the fact that growth in urchins proceeds not by a single column, but in column pairs of alternating plates. Additionally, growth differs between ambulacral paired columns and interambulacral paired columns. Multiple interdependent life cycle graphs are needed to represent growth, requiring multi-dimensional population matrices and tensor algebra to calculate solutions. Geometric models derived from the plate population matrices are developed on functional surfaces representing gross morphology of various echinoid classes and families, using Voronoi polygonalization to constrain plate morphology. Models are validated from 3-dimensional plate configurations derived from computed tomography of actual echinoid skeletons. Results indicate that phylogenetically significant characters can be mathematically derived from the parametric structure of the plate population matrices. It is the nature of such parametric models that under certain conditions the models becomes unstable, becoming unbound or oscillating between two or more attractors. Failure in these cases to converge to stable growth configurations could explain why much of the theoretical morphospace of echinoids is unrealized in nature.