Modelling Asymmetrical Growth Curves that Rise and then Fall: Applications to Foliage Dynamics of Sugar Beet (Beta vulgaris L.)

This paper discusses the derivation and fitting of three empiricalmodels with turning points for describing the growth of plantcomponents, such as shoot weight, leaf area and root length,that typically rise and then fall during the course of the growingseason. The models (Models I, II and III) have analytical solutionsand may be viewed as extensions of the Gompertz, Richards andChanter growth equations. They differ by having an additionalparameter which, following a sigmoidal rise of the dependentvariable, determines subsequent net rate of decline. The modelswere fitted to sequential measures of foliage cover of sugarbeet crops grown in the UK during 1980–1991. It was importantthat this could be done with relative ease using standard statisticalprocedures. Partial linear transformations of two of the models,with one non-linear parameter remaining, are described; thesewere useful for estimating initial values for the parameters.All three models described the data well, although the fittingof Model II invariably failed to converge. For Models I andIII common and separate parameters, amongst years, were estimatedrelating to date of emergence, initial relative growth rate,maximum cover attained and rate of late season decline of foliagecover. The reduction in the residual mean square on fittingseparate, rather than common, parameters was usually significant.The models accommodate several biological processes that yieldsimilar shapes. This is demonstrated for Model I, in relationto its formulation and to effects of small perturbations inthe values of the parameters on the shape of the curves. ModelI, the simplest of the three models tested, has good fittingproperties, and in practice was best suited to describing foliagecover dynamics of sugar beet. Beta vulgaris L.; sugar beet; foliage cover; senescence; models; parameter estimation; growth functions