An equation for geometric progression in Brooks-Dyar growth ratios

ABSTRACT

The Brooks-Dyar rule predicts a geometric progression in the growth increments of successive instars. Although these increments are readily calculated as ratios, analysis of the relationship between these ratios has been more problematic. In the early twentieth century, the use of logarithms was proposed as a method for calculating the equation of the line. This is as close to a standardized method of analysis that the ratios have come in entomology. It is argued here: (a) that a log scale is a misleading and misunderstood criterion in this context because growth increments do not span multiple orders of magnitude; and (b) a straight-line is an inappropriate mode of representation for growth patterns that are discontinuous. Moreover, the transformation does not provide any means for quantifying the degree of geometric progression. A simple exponential equation can be applied easily to any ratio data set to provide a more realistically curved plot of growth increments. A property of geometric progression sequences (b² = ac) can be utilized to provide an equation for evaluating the degree of geometric progression in development. Transformations are replaced by greater biological realism and a precise method of quantifying agreement with Brooks-Dyar’s rule.

KEYWORDS:

• Ecdysozoa
• exoskeleton
• instar
• Lepidoptera
• moulting
• ontogeny

Acknowledgments

Robert Robbins and Jim Miller are thanked for their correspondence on a few taxonomic queries; in particular, the latter, for sharing an updated, as yet unpublished, checklist of the Notodontidae. Any additional taxonomic errors are the author’s responsibility.

Disclosure statement

No potential conflict of interest was reported by the author.