Estimation of Parameters in a Bertalanffy Type of Temperature Dependent Growth Model Using Data on Juvenile Stone Loach
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Estimation of Parameters in a Bertalanffy Type of Temperature Dependent Growth Model Using Data on Juvenile Stone Loach (Barbatula barbatula) Johan Grasman • Willem B. E. van Deventer Vincent van Laar
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Received: 8 February 2012 / Accepted: 15 September 2012 / Published online: 28 September 2012 Springer Science+Business Media Dordrecht 2012
Abstract Parameters of a Bertalanffy type of temperature dependent growth model are fitted using data from a population of stone loach (Barbatula barbatula). Over two periods respectively in 1990 and 2010 length data of this population has been collected at a lowland stream in the central part of the Netherlands. The estimation of the maximum length of a fully grown individual is given special attention because it is in fact found as the result of an extrapolation over a large interval of the entire lifetime. It is concluded that this parameter should not at forehand be set at one fixed value for the population at that location due to varying conditions over the years. Keywords
Parameter sensitivity Method of least squares Adjusted Rsquare
1 Introduction Length growth of fish can be modelled in different ways. One of them, the Bertalanffy model (Von Bertalanffy 1938), is most frequently met in literature. It is based on the physiological processes of catabolism and anabolism. Other models take different starting points. Presently, modelling based on energy flow takes a prominent place (Kooijman 2000; West et al. 2001, Omori et al. 2009). In line with the phenomenological approach of Von Bertalanffy a group of growth models can be discerned. We mention the logistic equation (Ricker 1975) and Gompertz law J. Grasman (&) Biometris, Wageningen University and Research Centre, P.O.Box 100, 6700AC Wageningen, The Netherlands e-mail: [email protected] W. B. E. van Deventer V. van Laar Department of Urban Ecology, City of Amersfoort, P.O.Box 4000, 3800EA Amersfoort, The Netherlands
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(Gompertz 1825). The latter one is the limit case of the generalized logistic equation (Richards 1959). The Schnute-Richards model (Schnute and Richards 1990) constitutes a generalization at even a higher level. In Katsanevakis (2006) a comparison of these models is made using data from different fish populations. Furthermore in that study the final length parameter L1 is estimated by averaging the value of this parameter over the different models. The present study focusses on the estimate of model parameters from cohort data of a free living population. Since the length and age of individual fishes are not known one to one, we have to cope with various uncertainties. It is obvious that if the length of a cohort over the entire lifetime is known, a good estimate of L1 can be obtained from length measurements of fully grown individuals. However, if only partial information is at hand some type of extrapolation has to be applied. In this study mathematical and statistical techniques are presented that are useful for handling this problem. For the estimation of model
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