Information-theory approach to allometric growth of marine organisms

Abstract

Allometric growth investigations are usually conducted by fitting the allometric model (L) $y = ax^{b} \Leftrightarrow \log y = \log a + b\log x$ (y, x are morphometric characters and b the allometric exponent), which is quite simple both conceptually and mathematically, and its parameters are easy to estimate by linear regression. However b is not necessarily constant and it may change either continuously or abruptly at specific breakpoints; thus, the simple L model quite often fails to describe allometric growth successfully. In the current context, a better alternative is proposed, based on Kullback–Leibler (K-L) information theory and multi-model inference (MMI). Allometric growth was investigated in eight marine species: the bivalves Pecten jacobaeus and Pinna nobilis, the squids Todarodes sagittatus and Todaropsis eblanae, the crab Pachygrapsus marmoratus (females), the ghost shrimp Pestarella tyrrhena (males), and the fishes Trachurus trachurus and Sparus aurata. In each of the eight species, a pair of body parts was measured and the allometric growth of one body part in relation to the other (reference dimension) was studied, by fitting five different candidate models including: the simple allometric model, two models assuming that b changed continuously and two other assuming that b had a breakpoint. For each species, the ‘best’ model was selected by minimizing the small-sample, bias-corrected form of the Akaike Information Criterion. To quantify the plausibility of each model, given the data and the set of five models, the ‘Akaike weight’ w_i of each model was calculated; based on w_i the average model was estimated for each case. MMI is beneficial, more robust, and may reveal more information than the classical approach. As demonstrated with the given examples, estimation of b from the linear model, when it was not supported by the data, revealed some characteristic pitfalls, such as concluding positive allometry when there is actually negative or vice versa, or reporting allometry when the data in reality support isometric growth or vice versa.