Annals of Botany 79: 657-665, 1997
© 1997 Annals of Botany Company
Modelling Asymmetrical Growth Curves that Rise and then Fall: Applications to Foliage Dynamics of Sugar Beet (Beta vulgaris L.)
IACR Broom's Barn, Higham, Bury St Edmunds, Suffolk, IP28 6NP, U.K.
Received October 11, 1996 ; Accepted January 2, 1997
This paper discusses the derivation and fitting of three empirical models with turning points for describing the growth of plant components, such as shoot weight, leaf area and root length, that typically rise and then fall during the course of the growing season. The models (Models I, II and III) have analytical solutions and may be viewed as extensions of the Gompertz, Richards and Chanter growth equations. They differ by having an additional parameter which, following a sigmoidal rise of the dependent variable, determines subsequent net rate of decline. The models were fitted to sequential measures of foliage cover of sugar beet crops grown in the UK during 1980–1991. It was important that this could be done with relative ease using standard statistical procedures. Partial linear transformations of two of the models, with one non-linear parameter remaining, are described; these were useful for estimating initial values for the parameters. All three models described the data well, although the fitting of Model II invariably failed to converge. For Models I and III common and separate parameters, amongst years, were estimated relating to date of emergence, initial relative growth rate, maximum cover attained and rate of late season decline of foliage cover. The reduction in the residual mean square on fitting separate, rather than common, parameters was usually significant. The models accommodate several biological processes that yield similar shapes. This is demonstrated for Model I, in relation to its formulation and to effects of small perturbations in the values of the parameters on the shape of the curves. Model I, the simplest of the three models tested, has good fitting properties, and in practice was best suited to describing foliage cover dynamics of sugar beet.
Beta vulgaris L.; sugar beet; foliage cover; senescence; models; parameter estimation; growth functions