This document describes modeling the production of "calçots", the second-year onion resprouts of a particular onion variety, using a modified Gompertz equation. The model was fit to data from three populations that varied in characteristics like earliness and yield. The model provided a good fit for individual plant growth curves and identified differences between populations. Using the model, farmers could design planting strategies to better match calçot production to consumer demand over the season by combining complementary populations.

1. Modelling “calçots” production by means of Gompertz equation
J. Simó, M. Plans*, F. Casañas and J.Sabaté
Dept. d'Enginyeria Agroalimentària i Biotecnologia, UPC, Avda. Canal Olímpic,
15, 08860 Castelldefels, Barcelona. *marcal.plans@upc.edu
INTRODUCTION In our case, population P1 would correspond to an early population, starting to
produce and reaching the maximum number of commercial “calçots” earlier than
“Calçots” are the second-year onion resprouts of the “Ceba Blanca Tardana de
P2 and P3 (Figure 1). Population P2 would represent a late population, as the
Lleida” landrace. In “calçots” production all the resprouts from one onion are
maximum number of commercial “calçots” appears 18 weeks after planting
harvested at the same time when an acceptable amount of “calçots” (≥50%) reach
coinciding with the usually maximum consumers demand. Furthermore, its
the commercial size (1.7 cm – 2.5 cm in diameter and 20 cm in length, according to
average production of 8.62 commercial “calçots” is very high. Population P3
the Protected Geographical Indication “Calçot de Valls” regulations). Each onion
showed the lowest yield and the highest growth rate.
yields between 1 and 20 “calçots”, but their thickness is negatively correlated with
the number of “calçots” per onion, so in the most productive onions many “calçots”
The biological reasons underlying the good adjustment of the model would be a
never reach the commercial requirements. The production lasts from mid-November
combination of genetic factors determining the potential number of
to the end of April, and a more or less constant release of marketable product is
resprouts, onion size, earliness in sprouting and cold resistance, altogether with
needed during this period. As there is genetic variability in earliness, farmers use
environmental factors affecting the phenotypic expression of this traits, such as
combinations of genotypes and/or planting dates to adjust the production to the
temperature and water availability during the culture.
consumers demand but these combinations are made quite inefficiently.
An optimum management of the crop would require a deep knowledge and precise CONCLUSIONS
monitoring of the growth dynamics. Biological systems modelling allows predicting
development, to determine the critical points and to optimize processes [1]. Our The modified Gompertz model fits properly (R2 min, individual=0.79) to the
objectives are: i) to model the commercial “calçots” production in a population and individual evolution of each plant and also suggests a biologic meaning for the
ii) to compare the checked populations, in order to improve the culture differences found between plants and populations.
management.
As differences have been established between populations, the information given
by the model could be used to identify or create complementary populations. This
MATERIAL AND METHODS would be very useful to design a planting strategy ensuring a “calçots” production
parallel to the expected consumer’s demand along the season.
One hundred onions of three different populations were monitored plant by plant.
During seven months, the number of commercial “calçots” in each onion was scored Table 1. Original and modified Gompertz equations for bacterial growth.
every two weeks.
The data recorded in the three populations suggested that the evolution of the
number of commercial “calçots” (y) can be described by a sigmoid function which
shows three phases corresponding to latency, growth and steady state phase. This
function requires three parameters: the lag time (λ), the maximum growth rate
(μmax) and the asymptotic value for long time (A), in the same way that bacterial Table 2. Mean values (± SE of the mean) of the parameters for each population
growth was described by Gompertz and modified by Zwietering (Table 1)[1].
Nonlinear least squares, determined using Gauss-Newton algorithm [2], were used to
estimate the parameters of modified Gompertz equation for each plant. A One-Way
ANOVA has been used in search of statistical significant differences between the
three populations for the three fitting parameters (λ, μmax, A ). Computations were *Mean values in a column followed by a different letter are significantly different (p≤0.05) with the LSD test.
carried out by R-program [3] and Agricolae packages [4].
Plants that did not reach four commercial “calçots” at the end of the season were
discarded as this number is not sufficient to show trends in the model. Anywhere, in
a next future such unproductive onions will not be present in the new varieties that
are being obtained by breeding.
RESULTS AND DISCUSSION
Significant differences occurred between populations referring to the mean values
for λ . For μmax population P1 and P2 were significantly different from P3, and for
parameter A population P2 was significantly different from P3 (Table 2).
The variation into population estimated by means of the standard deviation is due
to genetic and environmental differences between plants and are those expected
in a population of an allogamous open pollinated landrace. The new improved
varieties obtained from these and other populations will decrease their internal
variability as breeding processes tend to increase the frequencies of the
favourable alleles and concentrate the phenotypes around the mean.
The goodness of the model adjustment estimated in each population (R2) is
similar to the one reported by Yin when modelling the wheat grain filling, using
Figura 1. Average curves of commercial “calçots” evolution for each population
the Gompertz model [5].
[1] Zwietering, M. H. et al. (1990) Modeling of the bacterial-growth curve. Appl. Environ. Microbiol, 56, (6) 1875.
[2] Bates, D. M. and Chambers, J. M. (1992) Nonlinear models. Chapter 10 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole
[3] R Development Core Team (2011). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-Project.org/
[4] Felipe de Mendiburu (2010). Agricolae: Statistical Procedures for Agricultural Research. R package version 1.0-9. http://CRAN.Rproject.org/package =agricolae
[5] Yin, X. et al. J. (2003). A flexible signoid function of determinate growth. Annals of Botany, 91, 361-371.