WILLIAMS, R.M.; PRICE, R.M.*; Fayetteville State University; University of North Carolina at Chapel Hill: A mathematical model for exploring the trade-offs in energy associated with growing gastropod shells

If the whorls in a gastropod shell overlap, as they do in a tightly-coiled shell, then the animal requires less material to grow its shell. Because the amount of material in a shell is correlated to the energy required to grow it, the animal can allocate more energy to other functions. However, a shell that is spiraled too tightly will be too small to house large reproductive and digestive systems, potentially lowering reproductive fitness. We are developing a model that will allow us to explore the trade-off between the amount of shell material deposited and the internal volume of the shell. Previous models assumed an infinitely thin shell and estimated the energy consumed during growth with surface area instead of volume. We correct this over-simplification by rotating the entire thickness of the aperture about a coiling axis. We model each whorl independently, and use only three parameters: the outlines of the inner and outer edges of the aperture (each described by 100 evenly-spaced landmarks) and the rate of whorl expansion. The outlines record the actual shape of the aperture, instead of a geometric shape like an ellipse. They also allow us to quantify the difference between the thick columellar edge and the relatively thin outer lip. The coordinates of the landmarks on the outlines are measured in pixels on digital photographs and then converted into centimeters. We tested the accuracy of the model by comparing our predicted volumes to measured volumes; our best preliminary result predicts 60% of the measured internal volume and only 52% of the measured volume of shell material. The model depends on accurate estimates of whorl expansion and the conversion between pixels and centimeters, so we will improve our predictions with more accurate input.