## Some mathematical considerations in estimating daily ration in fish using food consumption models

Hartmut Richter, Christian Lückstädt, Ulfert Focken, Klaus Becker

Department of Aquaculture Systems and Animal Nutrition, Institute for Animal Production in the Tropics and Subtropics, University of Hohenheim, Fruwirthstraße 12, 70599 Stuttgart, GermanyReceived 9 December 2002; received in revised form 26 June 2003; accepted 11 July 2003

### Abstract

The field of stomach content modelling can be broadly divided into evacuation models, which are used to determine evacuation rates under carefully controlled conditions, and consumption models, which apply these evacuation rates to field data to estimate food consumption. In the past, four main forms of evacuation model have been investigated, namely (a) the linear, (b) square root, (c) surface area and (d) exponential forms, with other models related relatively closely to these. Four consumption models are considered in the present work, namely the Bajkov, Elliott-Persson, MAXIMS and Olson-Mullen models. It was attempted here to develop concise mathematical functions for all those of the 16 combinations of evacuation and consumption model for which this has not been done in the past. It was found that no arithmetic solution exists for the Elliott-Persson and MAXIMS models in conjunction with square root and surface area evacuation although in the case of the Elliott-Persson model, a converging process could theoretically be applied. The Olson-Mullen model presents difficulties unless exponential evacuation is assumed and the assumptions which would have to be made in order to implement the model may not be justified. It was concluded that the Bajkov model represents the most universally applicable model in mathematical terms. However, analyses based on simulated data sets highlighted severe problems when this model assumed forms of evacuation other than the exponential. These problems were associated with the fact that the linear, square root and surface area models allowstomach fullness to drop to zero, after which they have to be mathematically constrained to prevent the model from arithmetically assuming evacuation that does not realistically take place. If the fish species analysed shows diel feeding periodicity and the feeding times are known, the errors could be eliminated by basing the analysis only on that part of the feeding cycle when stomachs have at least some contents, providing the full feeding period is covered. In fish populations where fish with empty stomachs are likely to be found at any time of day, it is not possible to truncate the data, concentrating the analysis on those periods when all stomachs have at least some contents. A simple correction factor was devised whereby fish with empty stomachs are excluded from the analysis and the daily ration estimate is corrected for their omission. This factor may be applied regardless of the feeding behaviour of the fish species analysed.

** Keywords:** Fish feeding; Food consumption models; Stomach evacuation models; Trophic ecosystem models

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