1. The researchers modeled the production of "calçots", onion resprouts harvested for food, using a modified Gompertz equation to predict yield over time. 2. They monitored the number of commercial-sized calçots from 100 onions in each of three populations over seven months. The model provided a good fit for individual plants and differentiated populations. 3. Population 1 produced earliest with the highest early yield. Population 2 produced later with the highest overall yield. Population 3 had the lowest yield but highest growth rate. The model can help design planting strategies to meet consumer demand.

1. 1
Modelling “calçots” production by means of Gompertz equation
M. Plans1*, J. Simó1 , F.Casañas1, J.Sabaté1
*
marcal.plans@upc.edu,1 Departament d’Enginyeria Agroalimentària i
Biotecnologia, Universitat Politècnica de Catalunya
Abstract
“Calçots”, the second-year onion resprouts, are produced from November to
May. Plants are harvested individually when an acceptable amount of “calçots”
reach the commercial size. In order to optimize the culture management a
modified Gompertz equation was used to model the commercial “calçots”
production. Prediction of plant yield evolution appeared enough good to
establish significant differences between the three populations checked.
Keywords: onion, sigmoid, yield.
1. Introduction
“Calçots” are the second-year onion resprouts of the “Ceba Blanca
Tardana de Lleida” landrace. In “calçots” production all the resprouts from one
onion are harvested at the same time when an acceptable amount of “calçots”
(≥50%) reach the commercial size (1.7 cm – 2.5 cm in diameter and 20 cm in
length, according to the Protected Geographical Indication “Calçot de Valls”
regulations). Each onion 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” never reach the commercial requirements.
The production lasts from mid-November to the end of April, and a more or
less constant release of marketable product is needed during this period. As
there is genetic variability in earliness, farmers use combinations of genotypes
and/or planting dates to adjust the production to the consumers demand but
these combinations are made quite inefficiently.
An optimum management of the crop would require a deep knowledge
and precise monitoring of the growth dynamics. Biological systems modelling
allows predicting development, to determine the critical points and to optimize
processes [1]. Our objectives are: i) to model the commercial “calçots”
production in a population and ii) to compare the checked populations, in order
to improve the culture management.
2. Material and methods
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 every two weeks.
XIII Conferencia Española y III Encuentro Iberoamericano de Biometría CEIB2011
7 a 9 de septiembre de 2011 Barcelona

2. 2 Template for the oral communications
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 growth was described by Gompertz and modified
by Zwietering (Table 1)[1].
Table 1. Original and modified Gompertz equations for bacterial growth.
Gompertz Equation Modified Gompertz Equation
   ·e 
y  a·exp   exp  b  c·x   y  A·exp   exp  max ·   t   1 
  A 
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 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.
3. 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.

3. 3
Table 2. Mean values (± SE of the mean) of the parameters for each population
Population λ (week) µmax (week-1) A R2
P1 4.55c * ± 0.39 2.25b ± 0.27 7.97ab ± 0.41 0.970 ± 0.002
P2 5.98b ± 0.33 1.87b ± 0.12 8.62a ± 0.38 0.978 ± 0.002
P3 7.50a ± 0.51 3.24a ± 0.60 6.87b ± 0.39 0.972 ± 0.003
*Mean values in a column followed by a different letter are significantly different (p≤0.05) with
the LSD test.
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 the Gompertz model [5].
In our case, population P1 would correspond to an early population,
starting to produce and reaching the maximum number of commercial
“calçots” earlier than P2 and P3 (Figure 1). Population P2 would represent a
late population, as the maximum number of commercial “calçots” appears 18
weeks after planting coinciding with the usually maximum consumers
demand. Furthermore, its average production of 8.62 commercial “calçots” is
very high. Population P3 showed the lowest yield and the highest growth rate.
Figure 1. Average curves of commercial “calçots” evolution for each
population
The biological reasons underlying the good adjustment of the model
would be a combination of genetic factors determining the potential number of
resprouts, onion size, earliness in sprouting and cold resistance, altogether with
environmental factors affecting the phenotypic expression of this traits, such
as temperature and water availability during the culture.

4. 4 Template for the oral communications
4. Conclusions
The modified Gompertz model fits properly (R2min,individual=0.79) to the
individual evolution of each plant and also suggests a biologic meaning for the
differences found between plants and populations.
As differences have been established between populations, the
information given by the model could be used to identify or create
complementary populations. This would be very useful to design a planting
strategy ensuring a “calçots” production parallel to the expected consumer’s
demand along the season.
5. Contact
Address:
Marçal Plans Pujolràs
Departement d’Enginyeria Agroalimentaria i Biotecnologia.
Universitat Politècnica de Catalunya
Avda. del canal Olímpic, S/N
08860 Castelldefels
Spain
Telephone: (+34) 93-5521226
e-mail: marcal.plans@upc.edu
www: http://www.esab.upc.edu/
6. Bibliography
[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.R-
project.org/package =agricolae
[5] Yin, X. et al. J. (2003). A flexible signoid function of determinate growth.
Annals of Botany, 91, 361-371.