Elementary functions modified for seasonal effects to describe growth in freshwater fish✰

Abstract

Two models were derived to describe fish growth while accounting for the effects of fluctuating water temperatures. The models were initially expressed in a rate:state form and subsequently integrated resulting in two analytical solutions, representing two distinct types of growth: exponential (Model 1) and asymptotic (Model 2). Both models share the assumptions that growth machinery works at a rate which varies with water temperature and that growth is irreversible. In addition, in Model 1 it is assumed that quantity of growth machinery is proportional to live body weight and substrate is non-limiting over the period of growth; whereas Model 2 is based on the assumption that quantity of growth machinery is proportional to available substrate. Effects of seasonal variations in water temperature on fish growth are represented in both models by a sinusoidal function. The potential of these models was investigated through their ability to describe growth in eight datasets encompassing three species: European bullhead (Cottus gobio), brown trout (Salmo trutta) and rainbow trout (Oncorhynchus mykiss). Models were evaluated using statistical measures of goodness-of-fit and through the analysis of residuals. Of the eight datasets, six displayed asymptotic growth while the other two exhibited exponential growth. Both models yield suitable simple growth functions with acceptable goodness-of-fit to fish growth curves under fluctuating water temperatures. However, Model 1, representing exponential growth, shows limited ability to predict fish size (length) when growth curves follow a clear asymptotic trend. This study enforces the idea that a given model is not always superior to another and that data structure and underlying model assumptions must be considered in model selection.

Keywords: Asymptotic; Ectotherm; Exponential; Growth function; Water temperature.