Abstract:

Aim: The shape of eggs is a subject of considerable interest among biologists and mathematicians, particularly regarding avian adaptation to diverse environments. The Carter-Morley Jones equation is a polar coordinate equation that describes the two-dimensional projection shape of bird eggs, typically with its model parameters estimated using multiple linear regression. However, multiple linear regression may result in poor fits due to the failure of its underlying assumptions. This study proposes a novel parameter estimation method for the Carter-Morley Jones equation based on nonlinear optimization theory.

Methods: We employed both multiple linear regression and nonlinear optimization techniques to fit actual egg-shape data obtained from 416 eggs spanning 51 species. The performance of each fitting method was assessed by evaluating the goodness of fit through root-mean-square error and analyzing the linear approximation behavior through nonlinear curvature measures. Additionally, we compared the egg volumes predicted by the Carter-Morley Jones equation with actual volumes measured using a graduated cylinder.

Results: The findings indicated that: (1) The nonlinear optimization method provided a superior goodness of fit compared to the multiple linear regression method; (2) No significant difference was observed in the linear approximation behavior between the two parameter estimation methods; and (3) There was no significant difference between the predicted egg volumes from the Carter-Morley Jones model and the actual volumes measured via the graduated cylinder.

Conclusion: This study offers a robust mathematical tool for applying the egg-shape equation in ecological research. Furthermore, the use of nonlinear curvature measures presents a fresh perspective for evaluating egg shape models in future investigations.

Key words: bird eggs, Carter-Morley Jones equation, nonlinear optimization, egg shape modeling, ecological adaptation