Directory UMM :Data Elmu:jurnal:S:Scientia Horticulturae:Vol83.Issue2.Febr2000:

Scientia Horticulturae 83 (2000) 83±107

A simulation model of climate effects on plant
productivity and variability in cauliflower
(Brassica oleracea L. botrytis)
J.E. Olesena,*, K. Grevsenb
a

Department of Crop Physiology and Soil Science, Research Centre Foulum,
PO Box 50, 8830 Tjele, Denmark
b
Department of Fruit, Vegetable and Food Science, Kirstinebjergvej 6, 5792 Aarslev, Denmark
Accepted 5 May 1999

Abstract
The paper describes a model of cauliflower (Brassica oleracea L. botrytis) growth and
development under conditions where water and nutrients are not limiting. The model consists of
four linked processes: crop development, leaf area expansion, increase in curd diameter and growth
of dry matter. The model aims to describe variability between plants in time of curd initiation and
harvest. Crop development is described as a function of temperature only. Leaf area expansion is
described by a logistic function where growth rate depends on temperature. Temperature and

available carbohydrates control the rate of increase in curd diameter. Dry matter assimilation is a
function of intercepted radiation and of demand for dry matter growth. The assimilates are
distributed among the organs in proportion to their demand. The model was calibrated for the
cultivar Plana using data from a growth chamber experiment and from a field experiment. A
verification of the model against field data showed that the model was able to reproduce variability
in curd diameter, indicating that variability in curd size is caused by variability in time of curd
initiation. # 2000 Elsevier Science B.V. All rights reserved.
Keywords: Brassica oleracea L. botrytis; Growth; Simulation model; Plant variability; Harvest
duration

* Corresponding author. Tel.: +45-89991659; fax: +45-89991619.
E-mail address: jorgene.olesen@agrsci.dk (J.E. Olesen).
0304-4238/00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 4 - 4 2 3 8 ( 9 9 ) 0 0 0 6 8 - 0

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1. Introduction

Growth and development of cauliflower (Brassica oleracea L. botrytis) are
strongly influenced by environmental conditions such as temperature and
radiation (Salter, 1960; Wurr et al., 1990a). CO2 concentration also affects dry
matter accumulation and hence curd weight (Wheeler et al., 1995).
The time from transplanting to harvest can be divided into three phases (Wiebe,
1972a, b; Wurr et al., 1981a): a juvenile phase, a curd induction phase and a curd
growth phase. Environmental variables, especially temperature, influence growth
and development differently in these phases. Consequently the effect of climate
variability on cauliflower production cannot be predicted without first quantifying
and integrating the impact of the environment on both development and growth.
The duration of the various phases of cauliflower development and growth has
previously been described using simple temperature driven models (Wurr et al.,
1990b, 1994; Grevsen and Olesen, 1994a; Pearson et al., 1994). A simple model
for effects of temperature and radiation on leaf area expansion and dry matter
growth in cauliflower was presented by Olesen and Grevsen (1997).
The variability in growth and development of individual plants (plant
variability) is an important aspect of many horticultural crops including
cauliflower as it affects the product quality and the harvesting process. However,
no attempts have been made to describe effects of climate on plant variability
through the use of dynamic simulation modelling, for example on length of the

harvest period in cauliflower. The length of the cutting period is one of the major
aspects of cauliflower production as it directly influences the costs of the
harvesting operation, and the harvesting costs constitute a large fraction of the
total costs of production (Wheeler and Salter, 1974). A large proportion of the
variation in duration of the harvest period can be attributed to variation in the
duration of curd induction and in temperature during curd growth (Booij, 1990).
Attempts have been made to reduce the length of the harvest period by cold
treatment before transplanting with variable success (Salter and Ward, 1972,
1974; Wiebe, 1975; Wurr et al., 1981b, 1982). The different effects of cold
treatment may be due to the timing of the treatments relative to the crop growth
phases.
A model incorporating effects of environment on both crop growth and
development including plant variability may be used for assessing the
performance of crops under changing environmental conditions such as those
implied by the enhanced greenhouse effect. Cauliflower is a short rotation crop
that is grown throughout the season in many parts of Europe. Such a model may
therefore also be suited for analysis and prediction of crop responses to current
seasonal variation in temperature and radiation.
This paper describes a model for cauliflower growth and development. Growth
in field conditions where water and nutrients are not limiting are simulated,

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85

because cauliflower is a high value crop, which in many European countries is
grown under conditions of good water and nutrient supply. The model is
calibrated for the cultivar Plana using data from controlled environments and
from field experiments.

2. Materials and methods
2.1. Model description
The model simulates crop growth and development in a dynamic way, and also
treats variability between plants in the time of curd initiation and harvest. The
model includes four main processes: crop development, leaf area expansion,
increase in curd volume and increase of dry matter. The links between these
processes are illustrated in Fig. 1. The model follows the supply-demand
approach (Gutierrez, 1996) and is structured into a hierarchy of metabolic pools
(Holst et al., 1997). The current state of the crop sets its demands for resources
(carbohydrates) as a function of growth stage and rate of area expansion. The

demand, together with resource availability, sets the supply.
Simulations are started at transplanting, and the model requires initial
information on date of transplanting and on initial leaf area, top dry weight
and root dry weight. In addition plant density must be defined. The model uses
daily minimum and maximum air temperature and global radiation. The
minimum and maximum temperatures are converted to hourly temperatures
assuming a sinusoidal diurnal variation (Allen, 1976).

Fig. 1. Outline of model structure. Model states are indicated by rectangular boxes, processes and
rates by ovals and effects of temperature (T) and global radiation (Q) by circles. Arrows with full
lines indicate flow of assimilates, and arrows with broken lines indicate direction of information
flow.

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A list of model variables is given in Appendix A, and the model parameters for
cv. Plana are listed in Appendix B.

2.2. Plant organs
Four separate plant organs are considered in the model: roots, stems, leaves and
curds (Fig. 1). All organs are assumed to have a pool of structural dry matter: Wr,
Ws, Wl and Wc (g plantÿ1) for roots, stems, leaves and curds, respectively. In
addition stems and leaves are assumed to share a pool of reserves (R) (g plantÿ1).
These organs are assumed to have a capacity for reserve storage (Rc) (g plantÿ1)
which is set proportional to the weight of structural components:
Rc ˆ aE …Ws ‡ Wl †;

(1)

where aE is a nondimensional constant.
Leaves also have an area (A) (m2 plantÿ1). Only active green leaves are
considered in the model. Leaves and roots are assumed to have a turn over of dry
matter, whereas dry matter simply accumulates in stems and curds. The ageing of
leaves and roots is described by the distributed delay procedure (Manetch, 1976),
where mass or area flow through a number of age classes from young to old.
Mass entering class 1 at time t will emerge from the last age class (K) at time t‡s
on the average. The spread in developmental times around the average is
described by an Erlang distribution with variance s2/K.

The ageing of the leaf area and of leaf mass is described by distributed delay
procedures with identical parameters. A temperature sum with a base temperature
of Tblˆ1.98C is used. This is the base temperature for leaf appearance in
cauliflower (Olesen and Grevsen, 1997). The mean age of a leaf at senescence is
called Sl, and the number of classes in the distributed delay procedure is set to
Klˆ30 as suggested by Graf et al. (1990). Leaf death is assumed only to affect
structural leaf dry matter and not the pool of reserves.
Root mass is also handled by a distributed delay procedure. A temperature sum
with a base temperature of Tbrˆ08C is used to describe the ageing of roots. The
mean age of root dry matter is set to Srˆ3008Cd based on root turnover rates in
cauliflower reported by Greenwood et al. (1982). The number of age classes is
arbitrarily set to Krˆ20.
A cauliflower curd is assumed to have the shape of a half sphere with radius
(rc) (mm) and height (hc) (mm) (Kieffer et al., 1998). The curd height is not
necessarily identical to the radius, and the curd shape is therefore not strictly a
hemisphere. The volume of the curd (Vc) (mm3) is then calculated as
Vc ˆ 23 rc2 hc :

(2)

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87

The height is assumed to be proportional to the radius, i.e.
hc ˆ ch rc ;

(3)

where ch is a nondimensional constant.
Fresh weights of curds (Fc) and of leaves and stems (Fv) (g plantÿ1) are
calculated using empirical relationships:
Fc ˆ af Wc ‡ bf Vc ;
Fv ˆ cf …Ws ‡ Wl † ‡ ef R;

(4)
(5)
ÿ3

where af, cf and ef are nondimensional constants, and bf is a constant (g mm ).

2.3. Crop development
The hourly crop developmental rate (d /dt) is defined as (Grevsen and Olesen,
1994a):
8
f …T †  < 1
d < d1 h
ˆ fd2 …Th † 1   < 2
(6)
:
dt
fd3 …Th † 2  

where fd1, fd2 and fd3 are temperature response functions for the juvenile, curd
induction and curd growth phases, respectively, and Th is hourly temperature (8C).
 is the current developmental stage of the crop, ranging from 0 at transplanting
to 3 at end of curd growth.  is 1 at end of juvenility, and 2 at time of curd
initiation.
Crop development during the juvenile phase (fd1) is described by a simple
temperature sum with a base temperature of Tdb1ˆ08C and a requirement of
Sd1ˆ83.38Cd in the cv. Plana (Grevsen and Olesen, 1994a). Crop development

during the curd induction phase (fdb2) is described by symmetrical linear
responses to temperature below and above an optimum temperature (Grevsen and
Olesen, 1994a). Grevsen and Olesen (1994b) estimated base and optimum
temperatures of Tdb2ˆ5.18C and Tdo2ˆ15.58C, respectively, and a requirement of
Sd2ˆ108.28Cd in Plana. These values are close to those found by Wheeler et al.
(1995). The duration of the curd growth phase (fd3) is described by a temperature
sum with a base temperature of Tdb3ˆ08C and a requirement of Sd3ˆ10508Cd.
These values were taken from Pearson et al. (1994) as the average of estimates for
three cultivars.
The ageing of plants from transplanting to curd initiation is described by the
distributed delay procedure. Separate delay procedures are used for the juvenile
and the curd induction phases with Kd1 and Kd2 age classes in the delay
procedures, respectively. Plant variability before curd initiation is thus handled
through the use of distributed delay procedures. After curd initiation the
variability is handled through the use of a number of curd cohorts. The growth

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and development of each cohort is treated separately, whereas before curd
initiation only variability in development is handled in the distributed delay
procedures.
Crop development is updated in hourly steps, and the number of plants
reaching curd initiation is accumulated. If this number exceeds 0.2% of the total
number of plants then a new cohort of curds is generated with an initial curd
diameter of 0.6 mm (Salter, 1969; Wiebe, 1972c). The curd cohorts are all
handled separately.
2.4. Leaf area expansion
The leaf area includes both leaf blades, stalks and midribs although the last two
components do not contribute fully to the photosynthetic active area. Stalks and
midribs constitute about 9% of the total leaf area (Olesen and Grevsen, 1997).
The expansion rate of leaf area (dA/dt) (m2 plantÿ1 dÿ1) is assumed to be
described by a logistic equation scaled by the vegetative supply demand ratio
(V):
…Lx =d ÿ A†‡
dA
ˆ al A
fA …Td †V ;
dt
Lx =d

(7)

where al is the maximum relative expansion rate (dÿ1), Td the mean daily air
temperature (8C), fA a function of daily mean temperature (0±1), d the plant
density (plants mÿ2), and Lx the maximum leaf area index (m2 mÿ2) which
depends on available nitrogen (Grindlay, 1997; Booij et al., 1996). The suffix ‡
denotes that only positive contributions are considered. Lx is set to 6 (van den
Boogard and Thorup-Kristensen, 1997). The effect of temperature on leaf area
expansion rate (fA) is described by the function estimated for Plana by Olesen and
Grevsen (1997), and al is set to 0.179 (Olesen and Grevsen, 1997). The leaf area
is updated in daily time steps using Euler integration.
2.5. Curd volume growth
The maximum growth rate of the curd radius (drc/dt)x is assumed to decline
linearly with crop age during the curd growth phase (Pearson et al., 1994) and
increase linearly with temperature:
 
drc
ˆ ac …3 ÿ  †‡ …Th ÿ Tb3 †‡ rc ;
(8)
dt x
where ac is the maximum relative radius growth rate ((8Cd)ÿ1). Knowing the
maximum radius growth rate, the maximum daily increase in volume (
xVc) may
be calculated using Euler integration in hourly time steps.

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The actual growth of curd volume is detemined by the vegetative supply
demand ratio (V) which is identical to the supply demand ratio for growth of
curd dry matter:

Vc ˆ
x Vc ‰1 ÿ …1 ÿ V †bc Š;

(9)

where bc is a nondimensional constant. The actual growth in radius is then
calculated by converting the volume growth to radius growth.
2.6. Demands for dry matter growth
The demand rate for growth (D) (g mÿ2 dÿ1) is composed of five demands
D ˆ Dr ‡ Ds ‡ Dl ‡ Dc ‡ DE ;

(10)

where Dr is the demand rate for growth of structural dry matter in roots
(g mÿ2 dÿ1), Ds the demand rate for growth of structural dry matter in stems
(g mÿ2 dÿ1), Dl is the demand rate for growth of structural dry matter in leaves
including stalks and midribs (g mÿ2 dÿ1), Dc the demand rate for curd growth
(g mÿ2 dÿ1), and DE is the demand rate for growth of reserves (g mÿ2 dÿ1).
Conversion costs are ignored as these are assumed to be approximately identical
for all demand types. Respiration costs are also ignored as the ratio of respiration
to photosynthesis has been found to be constant over a wide range of
temperatures (Gifford, 1995; Dewar, 1996).
The demand rate for growth of leaves, stalks and midribs is assumed to have
two components. One component is related to growth of new leaf area, and one is
related to growth of secondary structures in the leaves:
Dl ˆ d‰sl
Ax ‡ aD …slx A ÿ Wl †‡ fp …Td †Š;

(11)

where sl is the weight of new leaf area (g mÿ2), slx is the maximum weight of leaf
area (g mÿ2), aD is a constant ((8Cd)ÿ1), d the plant density (plant mÿ2),
Ax is
the maximum daily increase in leaf area (m2 dÿ1 plant-), and fP is the function of
mean daily temperature (0±1) which is also used to adjust assimilation (Eq. (16)).
slx is assumed to be twice the weight of new leaf area (sl), and this weight is
assumed to be attained in 25 days at optimal temperature for assimilation. This
implies that aDˆ0.04 dÿ1.
The mass of above and below ground vegetative organs are assumed to be in
balance, and the demand rate for root growth is calculated such that this balance
is maintained:
Dr ˆ dbD …cD …Wl ‡ Ws † ÿ Wr †‡ ;

(12)

where bD is a constant which is arbitrarily set to 1. The ratio of root to leaf weight
is set to cDˆ0.11 based on data from Bligaard (1996).

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Stem structural dry matter is assumed to be proportional to plant top fresh
weight giving the following demand rate for stem growth:
Ds ˆ dbD …eD …Fv ‡ Fc † ÿ Ws †‡ ;

(13)

where eD is a nondimensional constant.
The demand rate for curd growth is assumed to be proportional to the
maximum volume increase:
Dc ˆ d‰cc ‡ ec …3 ÿ  †‡ Š
x Vc ;

(14)

where cc and ec are constants (g mmÿ3).
The demand rate for growth of reserves depends on the remaining capacity for
reserve storage and on the size of the stem and leaf organs:
DE ˆ d min‰bE …Rc ÿ R†; cE …Wl ‡ Ws †Š;

(15)

where bE and cE are constants (dÿ1).
2.7. Dry matter assimilation
The fraction of intercepted PAR () is calculated using an extinction coefficient
which decreases with increasing plant leaf area (Olesen and Grevsen, 1997).
Daily net assimilation (Pa) (g mÿ2 dÿ1) is calculated using the concept of a
radiation conversion coefficient () (g MJÿ2) and an effect of sink limitation:
Pa ˆ min‰D; fP …Td †QŠ;

(16)

where D is the demand rate for dry matter growth (g mÿ2 dÿ1), Q the
photosynthetic active radiation (MJ mÿ2 dÿ1), and fP is a function of daily mean
temperature (0±1) which is taken to be the function estimated by Olesen and
Grevsen (1997) for effect on radiation conversion efficiency.  is assumed to be a
linear function of mean daily PAR intensity Qi (W mÿ2], which is calculated by
dividing Q by the day length:
 ˆ 1 ‡ 2 Qi ;

(17)

where 1 and 2 are constants, which were estimated as 1ˆ5.44 g MJÿ2 and
2ˆÿ0.123 g MJÿ2 Wÿ1 m2 from data published by Olesen and Grevsen (1997)
for cauliflower corrected for an additional 9% to root dry matter.
The assimilates available for partitioning (P) (g mÿ2 dÿ1) is the sum of the net
assimilation and a contribution from the plant reserves (PE) (g mÿ2 dÿ1):
P ˆ Pa ‡ PE :

(18)

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91

The contribution of reserves to the growth of various organs is assumed to
depend on both supply and demand (Gutierrez and Baumgaertner, 1984):



E dR
;
(19)
PE ˆ …D ÿ Pa † 1 ÿ exp ÿ
D ÿ Pa
where  E defines the mobilisation rate of reserves which is set to 0.15 dÿ1 in line
with experience reported by Penning de Vries et al. (1989). The dry matter
growth is partitioned between the organs (roots, stems, leaves and curd) in
proportion to their demand. A fraction of the reserve demand (eEDE) is assumed
to be distributed with the same priority as the vegetative organs. Any assimilates
in excess of that are partitioned to reserves. All organs are thus considered as
being vegetative even if not conventionally so, and the vegetative supply demand
ratio (V) is thus calculated as:
V ˆ min‰P=…D ÿ …1 ÿ eE †DE †; 1Š:

(20)

In an experiment with defoliation of cauliflower plants, van den Boogard and
Thorup-Kristensen (1999) found that the total sugar and starch content of
cauliflower leaves, midribs and stems never became lower than 10%. Assuming
that the reserve demand on average is identical to the sum of all vegetative
demands, this gives eEˆ0.1.
Growth of dry matter and actual curd size is calculated in daily time steps using
Euler integration. Each curd cohort is handled separately and assumed to have its
own reserves, but the assimilation is calculated at the stand level.
2.8. Experimental data
Data from two experiments were used to calibrate the model.
2.8.1. Experiment 1
Olesen and Grevsen (1997) conducted an experiment on cauliflower in growth
chambers which provided control of air temperature, air humidity and light
intensity. Nine different treatments with varying temperatures and irradiances
were included in the experiment (Table 1). All treatments were conducted using a
16 h day and 8 h night.
Seeds of summer cauliflower cv. Plana F1 (Royal Sluis) were sown in nutrient
enriched peat soil in rectangular pots with a surface of 3040 cm and a depth of
40 cm. The soil was watered to full capacity at time of sowing and subsequently
watered every second day. After sowing all pots were placed under identical
conditions until the plants had reached about 10 initiated leaves. At this time the
plants were thinned to about 6±10 plants per pot. Five pots were then placed in
each chamber and the treatments were initiated.

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Table 1
Summary of treatments in the growth chamber experiment on cauliflower reported by Olesen and
Grevsen (1997). The durations from start of experimental treatment until curd initiation and until
the last sampling are shown
Number

Mean temperature
(8C)

Daily PAR
(mol mÿ2)

Duration (days)
Curd initiation

1
2
3
4
5
6
7
8
9

7.0
10.7
14.3
14.3
14.3
14.3
18.0
21.7
25.3

20.4
20.4
9.3
19.1
32.7
50.8
19.1
19.1
19.1

39
24
>20
21
22
21
26
18
>27

Number
of samples
Experiment
43
27
38
35
38
38
27
20
27

7
7
10
7
10
10
8
7
8

Samples were taken once or twice per week. This meant that the plant density
in the pots was successively reduced. The first sample was taken at onset of the
treatments. Each plant was dissected into stems, leaf blades, midribs and stalks.
Leaf area was measured, and the weight of the each organ was determined before
and after oven drying at 808C for 24 h. The apex diameter of each plant was
determined with a binocular microscope (magnification 50) after dissection.
Curd initiation was defined as the time when the apex had reached a diameter of
0.6 mm (Salter, 1969; Wiebe, 1972a) and determined by linear interpolation of
the logarithm of apex diameter against time. The duration of the experimental
treatment until curd initiation is shown in Table 1.
2.8.2. Experiment 2
Six transplantings were conducted in the field at Research Centre Aarslev in
Denmark in 1995 using cauliflower cv. Plana (Table 2). Seeds were sown in peat
Table 2
Dates of transplanting, curd initiation and final sample of cauliflower at Aarslev Research Centre in
1995
Planting

Transplanting

Curd initiation

Last sample

1
2
3
4
5
6

21 April
16 May
7 June
30 June
1 August
21 August

24 May
7 June
30 June
25 July
28 August
11 September

11 July
29 July
9 August
7 September
23 October
9 November

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93

blocks (5 cm  5 cm  5 cm) and nursery plants were raised under glasshouse
conditions with a minimum temperature of 128C. Transplants with about 10
initiated leaves were conditioned outside in sheltered frames for preferably 5±7
days before transplanting on a sandy loam at Aarslev. The plant density was 6.25
plants mÿ2. The plant density in transplanting 5 was reduced to approximately 4
plants mÿ2 due to severe attack of cutworms (Agrotis segetum Schiff.).
Fertilisation and irrigation followed guidelines for normal production. A basic
fertilisation of 40 kg P haÿ1 and 200 kg K haÿ1 was applied to the experimental
area. Each transplanting received 150 kg N haÿ1 in calcium ammonium nitrate at
planting followed by 100 kg N haÿ1 in urea 30±40 days after planting.
Samples of 10±40 plants were taken at weekly or biweekly intervals by cutting
plants at the soil surface. The samples were used to determine plant and curd
fresh weights. The dry matter content of subsamples was determined after oven
drying at 808C for 24 h. The area of leaf blades was measured on the subsamples.
Apex diameter was determined by dissection and measurement under stereo
microscope. Curd initiation was determined as in Experiment 1. The diameter and
height of the curd was measured. Meteorological data were taken from the
national meteorological station at Aarslev (108270 E, 558180 N) within 500 m of the
experimental site.
2.9. Parameter estimation
Some model constants (af, bf, cf, df, ch, eD) were estimated by multiple linear
regression using the procedure GLM with the no intercept option in the SAS
statistical package (SAS Institute, 1988).
Estimation of aE, cf, ef, bE and cE required observations of plant reserves using
data from Experiment 1. The reserve content (R) was calculated as the difference
between observed stem and leaf dry weights and model estimates of Wl‡Ws. Leaf
structural weight (Wl) was calculated from Eq. (11) using observed leaf area
fitted by a logistic function Eq. (21) to estimate increase in leaf area. It was
assumed that leaf area expansion was not restricted by assimilate supply. Stem
structural dry weight was assumed to be proportional to above ground fresh
weight Eq. (13).
The leaf area development in Experiment 1 was described by a logistic
equation fitted to the observed data (Thornley, 1990):
Aˆ

Ax
;
1 ‡ exp…a ÿ bt†

(21)

where t is number of days from onset of experiment, Ax is the maximum leaf area
(m2 plantÿ1), and a and b are constants. a may be substituted by ln((AxÿAo)/Ao),
where Ao is leaf area at onset of the treatment. The constants were estimated by
nonlinear regression using the NLIN procedure of SAS (SAS Institute, 1988). The

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observed values of Ao were used, and Ax was set to 0.6 m2 (Olesen and Grevsen,
1997).
Some model constants (sl, ac and bc) were estimated by running a reduced form
of the model and minimizing the squared difference between observed and
simulated values of either plant leaf area, plant dry weight or curd radius. A
square root transformation was applied to both observed and simulated values of
leaf area, dry weight or curd radius to obtain variance homogeneity. The
simulated values were calculated using subsets of the entire model. A grid
analysis and the downhill simplex method (Nelder and Mead, 1965) were used for
minimizing the difference.
The number of age classes in the distributed delay procedures for crop
development (Kd1 and Kd2) was estimated by adjusting these parameters to
match observed standard deviation in date of curd initiation. Kd1 and Kd2 were
assumed to be identical. Data from treatments 3, 5 and 6 in experiment 1
were used as these were conducted at identical temperatures from the same
set of nursery plants. The curd initiation date was estimated for each plant as the
time when curd diameter was 0.6 mm using the measured apex diameter (dc) and
a linear relationship between ln(dc) and number of days from onset of
experimental treatments (t). The following relationship was estimated using
mean sample apex diameter from the four plant samples embracing date of curd
initiation:
ln…dc † ˆ ÿ6:31 ‡ 0:260t:

(22)

Some model constants were subjectively estimated by either using the
characteristics of a few plant samples (Sl) or by visually fitting a line through
selected data points (cc, ec, bE and cE).

3. Results
3.1. Model calibration
The model parameter estimates are described here in the order in which they
were introduced in the model description section. An overview of all model
parameters is given in Appendix B.
The highest observed ratio of reserves to structural top dry matter was 2.4 in the
treatment with the highest irradiance in Experiment 1. The plants had clear
symptoms of reduced growth rate under these conditions indicating that the
capacity for reserve storage was reached. aE in Eq. (1) is thus set to 2.5.
The average age of the first true leaf was found to be 48 days at a mean daily
temperature of 14.38C in Experiment 1. This gives an average leaf area duration
in thermal time of Slˆ5958Cd.

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95

The ratio of curd height to curd radius Eq. (3) was estimated by linear
regression as chˆ1.34 (s.e.ˆ0.004) using all measurements of single curds from
Experiment 2.
The constants for estimating curd fresh weight Eq. (4) were estimated by
multiple linear regression as afˆ10.4 (s.e.ˆ0.33) and bfˆ0.077 (s.e.ˆ0.011)
g cmÿ3 using all measurements of single curds from Experiment 2.
The constants for estimating fresh weight of vegetative organs Eq. (5) were
estimated by multiple linear regression as cfˆ10.7 (s.e.ˆ0.19) and efˆ4.5
(s.e.ˆ0.45) using all mean sample values from Experiment 1 prior to curd
initiation.
The data from the four sample dates embracing date of curd initiation in
treatments 3, 5 and 6 in Experiment 1 was used to estimate Kd1 and Kd2 under the
assumption that these constants are identical. The measured standard deviation in
date of curd initiation was 2.59 (nˆ57). This variability was matched by the
distributed delay procedure by setting Kd1ˆKd2ˆ26.
The constants in Eqs. (8) and (9) for calculating curd expansion were estimated
using data from Experiment 2. The constants were estimated simultaneously as
acˆ0.0123 (8Cd)ÿ1 and bcˆ2.9 by minimizing the squared difference between
simulated and observed mean sample curd radius. Increase in curd radius was
simulated from the observed date of curd initiation using Eqs. (8) and (9). The
supply demand ratio (V) was calculated using Eq. (20), and using observed plant
dry matter interpolated linearly to estimate daily supply. The leaf area in Eq. (11)
was taken to be the observed plant leaf area fitted by a logistic equation Eq. (20).
The weight of new leaf area (sl) was estimated at 44.7 g mÿ2 by minimizing the
difference between observed and simulated leaf weight using data from the first
21 days of treatment 3 in Experiment 1. This treatment was carried out at low
irradiance and it was assumed that no reserves were accumulated during this
period. It was also assumed that assimilate supply did not restrict growth of
secondary leaf structures. Leaf dry weight was simulated by Eq. (11) using
observed leaf area fitted by a logistic function Eq. (21).
The ratio of stem dry weight to plant top fresh weight Eq. (13) was estimated
as eDˆ0.010 (s.eˆ0.00024) by linear regression of mean sample stem dry weight
against plant top fresh weight using data from Experiment 1.
The constants for calculating maximum curd density Eq. (14) were estimated
as acˆ0.005 and bcˆ0.12 g cmÿ3 by visually fitting a line through the highest
curd densities from Experiment 2 on a plot of mean sample curd density versus
accumulated temperature from curd initiation.
The constants for calculating reserve demand rate Eq. (15) were estimated as
bEˆ0.075 and cEˆ0.12 dÿ1 using data from Experiment 1 prior to curd initiation.
bE was estimated as the slope of a line visually fitted through the highest data
points on a plot of rate of increase of reserve dry matter to remaining capacity for
reserve storage (RcÿR). cE was estimated as the slope of a line visually fitted

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J.E. Olesen, K. Grevsen / Scientia Horticulturae 83 (2000) 83±107

through the highest data points on a plot of rate of increase of reserve dry matter
to structural dry matter in leaves and stem.
3.2. Model verification
The calibrated model was used to simulate crop growth and development for
the six plantings in Experiment 2. These experimental data were used to calibrate
the curd growth parameters only. Curd growth is affected by dry matter
accumulation in the model, but the reverse is not the case. The data may therefore
be used to verify the model's simulations of dry matter accumulation. These
experimental data were not used to estimate the parameters related to plant
variability and may therefore also be used to verify the model's simulation of
variability in curd diameter.
The simulation of curd growth is very sensitive to the simulated timing of curd
initiation. The simulation of crop development was therefore adjusted to match
observations, so that the verification of crop growth was not affected by errors in
simulation of development. This was done by adjusting the length of the juvenile
phase to fit the observed dates of curd initiation. The time course of simulated
and observed values are shown for each transplanting in Figs. 2±4 for plant leaf
area, top dry matter and standard deviation of curd diameter, respectively.
Figs. 2 and 3 shows a large variation in growth rate between the different
plantings related to time of year. There was for all plantings a good relationship
between observed and simulated leaf area and top dry matter.
Fig. 4 shows that simulated standard deviation of curd diameter increased with
time reaching a maximum and then declined again as curd relative growth rate
declined with age. There is a good agreement between observed and simulated
results, except for transplanting 4. The decline in standard deviation with age was,
however, not as clear in the observed data as in the simulated results.

4. Discussion
The model incorporates the development model previously described by
Grevsen and Olesen (1994a). The model uses a declining relative rate of increase
in curd diameter with accumulated temperature from curd initiation as suggested
by Pearson et al. (1994). This produces results which are comparable to the use of
a quadratic relationship between the logarithm of curd diameter and accumulated
temperature as suggested by Wurr et al. (1990b). Pearson et al. (1994)
additionally assumed that curd growth rate had an instantaneous optimum
temperature which was found to vary from 168C in cv. Jubro to 258C in cv.
Revito. The present model does not explicitly assume an optimum temperature
for curd growth. The increase in curd diameter is, however, reduced if the demand

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97

Fig. 2. Measured (points) and simulated (lines) time course of mean plant leaf area for each of the
six plantings at Aarslev. The length of the juvenile phase was adjusted to match the observed date of
curd initiation in each of the six plantings.

for growth of curd dry matter cannot be met by the supply. At high temperatures
this will result in reduction in rate of increase of curd diameter with increasing
temperature as assimilation does not increase with temperature for temperatures
above 148C.
The plant variability is simulated by introducing variability in the juvenile and
curd induction phases using distributed delay procedures. The growth of single
curd cohorts from curd initiation and onwards are simulated using deterministic
procedures with no new source of variability. The time course of simulated
standard deviation of curd diameter matched the observations with the exception
of transplanting 4 where simulated curd diameters also were higher than observed
(Fig. 4). In most cases the simulated standard deviation reached its maximum at a
curd diameter of about 10 cm.

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J.E. Olesen, K. Grevsen / Scientia Horticulturae 83 (2000) 83±107

Fig. 3. Measured (points) and simulated (lines) time course of mean plant top dry weight for each
of the six plantings at Aarslev. The length of the juvenile phase was adjusted to match the observed
date of curd initiation in each of the six plantings.

The simulated decline in standard deviation was not fully supported by the data
(Fig. 4). This may indicate that the simulated duration of the curd growth period
was too short. This growth period was not estimated for Plana, but taken from
other cultivars.
The agreement between simulated and observed variability of curd diameter
confirms the assumption that this variability is mainly caused by variation in time
of curd initiation. Booij (1990) found that the majority of the variation in duration
of the harvest period could be explained by the combined effect of variation in
duration of curd initiation and in temperature during curd growth.
The use of the distributed delay procedure provides a convenient way of
introducing variability in maturation rates into an otherwise deterministic model.
Plant and Wilson (1986) discussed the use of this and alternative formulations of
age structured populations. Sequira et al. (1993) also used the distributed delay

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99

Fig. 4. Measured (points) and simulated (lines) time course of standard deviation of curd diameter
for each of the six plantings at Aarslev. The length of the juvenile phase was adjusted to match the
observed date of curd initiation in each of the six plantings.

procedure to include variability in an existing model of cotton growth and
development.
The expansion of leaf area is assumed to be largely independent of the growth
of dry matter as found by Olesen and Grevsen (1997). This independency gives a
potential for large variations in specific leaf area which has been found to depend
on irradiance level (Olesen and Grevsen, 1997) and on CO2 concentration
(Wheeler et al., 1995). Some crop models (e.g. Spitters et al., 1989; Graf et al.,
1990) calculate the expansion of leaf area from increase in leaf dry matter by
assuming a constant or age dependent specific leaf area. This has been found to
give satisfactory descriptions in crops where branching depends on available
assimilates. Cauliflower does not form branches, and a large fraction of the
carbohydrate reserves are stored in the leaves. The use of a constant specific leaf

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J.E. Olesen, K. Grevsen / Scientia Horticulturae 83 (2000) 83±107

area in this crop may result in an unrealistic positive feedback between
assimilation and leaf area expansion.
The model uses a radiation conversion coefficient to convert solar radiation to
dry matter growth. The model does thus not specifically include the photosynthesis
and respiration processes, because several studies have shown that the ratio of
respiration to photosynthesis is constant in many crops (Gifford, 1995; Dewar, 1996).
Exceptions are crops where large changes in chemical composition occur during
development, e.g. from carbohydrate storage to storage of proteins or lipids
(Arkebauer et al., 1994). No such change occur in cauliflower, and the use of a simple
radiation conversion scheme can therefore be justified.
The value of the radiation conversion coefficient used is not assumed to be
constant. The coefficient declines with increasing light intensity as experimentally shown by Olesen and Grevsen (1997) for cauliflower and by Wheeler et al.
(1993) for lettuce. The obtained coefficients are similar to those typically found
under field conditions during summer in Northern Europe (Monteith, 1977;
Gallagher and Biscoe, 1978; Wheeler et al., 1995). The use of a dependency on
average daily light intensity will cause the estimated conversion coefficient to be
higher at higher latitudes and lower at lower latitudes as empirically shown for
wheat (Jamieson et al., 1998). The coefficient is also effectively reduced, if
calculated supply exceeds the demand Eq. (16). This may lead to an additional
downregulation at high irradiance.
The model gave a realistic simulation of leaf area expansion and dry matter
growth of six different transplantings in the field (Figs. 2 and 3). This shows that
model parameters and relationships obtained from controlled environments can
be successfully applied to field grown crops provided the crops in the controlled
environments are grown in semi-stands and under realistic light conditions.
The function of the demands is not only to regulate assimilation, but also to
serve as a mechanism for partitioning assimilates between the various organs.
Many crop models use a simple priority scheme or a set of age dependent
partitioning coefficients (e.g. Weir et al., 1984; Spitters et al., 1989). The use of
supply-demand ratios give a simple way of simulating plant functional responses
on assimilate partitioning (Graf et al., 1990; DeJong and Grossman, 1994).
The model assumes that all organs get a share of the supply in proportion to
their demand. The curd is thus not assumed to have a higher priority than
vegetative organs. Such a higher priority is usually assumed for reproductive
organs (Gutierrez, 1996). The curd is, however, not a true reproductive organ, but
rather an extension of the stem upon which the flowers develop. The change in
dry matter partitioning with time does thus not reflect a higher priority for
assimilates to the curd, but a dramatically increasing demand for growth of curd
dry matter as the curd diameter increases.
The structural leaf dry matter is assumed to be composed of primary and
secondary cell wall material. A similar model concept was used by Lainson and

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101

Thornley (1982) for modelling leaf expansion in cucumber. New leaf area will
thus generate a demand for primary structures, whereas thickening of existing
leaves contributes to the demand for secondary structures.
The demands for growth of root and stem dry matter are based on concepts of
balances between various organs. Root dry matter is thus assumed to be
proportional to leaf structural dry matter in leaves and stems. The rationale
behind this is a balance between above ground CO2 assimilation and below
ground uptake of water and nutrients which is reflected in similar sizes of the
assimilative organs, leaves/stems and roots. Stem dry matter is assumed to be
proportional to total top fresh weight, because stems must contain structures to
support this weight.
A fraction of the demand for reserves is distributed with the same priority as
the vegetative organs. The supply in excess of the vegetative demand is simply
accumulated in the pool of reserves. This approach follows concepts of
accumulation and of reserve formation suggested by Chapin et al. (1990).
Accumulation is the increase in compounds that do not directly promote growth,
whereas reserve formation is a metabolically regulated process of storage
formation from resources that might otherwise promote growth.
Vegetative top fresh weight was estimated to be a linear function of structural
dry matter and of reserves. The simulated dry matter contents may thus vary from
10.7% in plants with no reserves to 22.0% in plants saturated with reserves. The
curd fresh weight was estimated to be a linear function of curd dry matter and
curd volume. This results in a reduction in dry matter content of curds if the curd
demand for growth cannot be fulfilled.
The model may be used for predicting time of crop maturity and for design of
schedules for crop planting. Simpler models have already been developed for this
purpose (Wurr et al., 1990b; Pearson et al., 1994). The present model also
simulates growth of dry matter and fresh weight which are important elements of
crop yield and quality. The simulation of curd expansion and of dry matter growth
may form a conceptual basis for modelling quality defects, which depend on
some of the dynamic variables in the model (Olesen and Grevsen, 1993). The
model may also be used for evaluating effects of climate variability and climate
change on cauliflower production.

Acknowledgements
This work was funded under contract EV5V-CT93-0294 of DGXII of the
European Commission. The model is programmed in Pascal, and the source code
may be obtained from the first author or downloaded from the internet web site at
http://www.agrsci.dk/pvj/dyste/caulisim.

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J.E. Olesen, K. Grevsen / Scientia Horticulturae 83 (2000) 83±107

Appendix A
The variables used in the model are listed below



Ax

Vc

xVc
V

A
d
D
Dc
Dl
Dr
Ds
DE
Fc
Fv
hc
P
Pa
PE
Q
R
Rc
rc
t
Th
Td
Vc
Wc
Wl
Wr
Ws

fraction of PAR intercepted by crop canopy (0±1)
radiation conversion efficiency (g MJÿ1 PAR)
maximum daily increase in leaf area (m2 dÿ1 plantÿ1)
daily increase in curd volume (mm3 dÿ1)
maximum daily increase in curd volume (mm3 dÿ1)
vegetative supply demand ratio (0±1)
crop developmental stage (0±3)
green leaf area (m2 plantÿ1)
plant density (plants mÿ2)
overall demand rate for dry matter growth (g mÿ2 dÿ1)
demand rate for growth of curd dry matter (g mÿ2 dÿ1]
demand rate for growth of structural dry matter in leaves (g mÿ2 dÿ1)
demand rate for growth of root dry matter (g mÿ2 dÿ1)
demand rate for growth of structural dry matter in stems (g mÿ2 dÿ1)
demand rate for growth of reserves (g mÿ2 dÿ1)
curd fresh weight (g plantÿ1)
fresh weight of leaves and stem (g plantÿ1)
curd height (mm)
assimilates available for partitioning (g mÿ2 dÿ1)
dry matter assimilation (g mÿ2 dÿ1)
mobilisation of reserves (g mÿ2 dÿ1)
photosynthetic active radiation (MJ mÿ2 d-1)
reserves in leaves and stem (g plantÿ1)
capacity for reserve storage in leaves and stem (g plantÿ1)
curd radius (mm)
time (d)
hourly air temperature (8C)
daily mean air temperature (8C)
curd volume (mm3)
weight of structural dry matter in curds (g plantÿ1)
weight of structural dry matter in leaves (g plantÿ1)
weight of dry matter in roots (g plantÿ1)
weight of structural dry matter in stems (g plantÿ1)

Appendix B
The constants used in the model for cv. Plana is listed below along with their
estimated values and the source of this information

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J.E. Olesen, K. Grevsen / Scientia Horticulturae 83 (2000) 83±107

Constants

Value

Source

Plant organs
af Eq. (4)
aE Eq. (1)
bf Eq. (4)
cf Eq. (5)
ch Eq. (3)
ef Eq. (5)
Kl
Kr
Sl
Sr
Tbl
Tbr

10.4
2.5
0.077 g cm
10.7
1.34
4.5
30
20
5958Cd
3008Cd
1.98C
08C

C
C
C
C
C
C
4
G
C
5
8
G

Development
Kd1 ˆ Kd2
Sd1
Sd2
Sd3
Tdb1
Tdb2
Tdb3
Tdo2

26
83.38C
108.28C
10508Cd
08C
5.18C
08C
15.58C

C
6
7
9
6
7
9
7

Leaf area expansion
al Eq. (7)
Lx Eq. (7)

0.179 dÿ1
6 m2 mÿ2

8
3

Curd volume growth
ac Eq. (8)
bc Eq. (9)

0.0123 (8C)ÿ1
2.9

C
C

Demands for dry matter growth
aD Eq. (11)
bD Eqs. (12) and (13)
bE Eq. (15)
cc Eq. (14)
cD Eq. (12)
cE Eq. (15)
ec Eq. (14)

0.04 dÿ1
1 dÿ1
0.075 dÿ1
0.005 g cm ÿ3
.09
0.12 dÿ1
0.12 g cmÿ3

G
G
C
C
1
C
C

ÿ3

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J.E. Olesen, K. Grevsen / Scientia Horticulturae 83 (2000) 83±107

Constants

Value

Source

eD Eq. (13)
eE Eq. (20)
sl Eq. (11)
slx Eq. (11)

0.0100
0.1
44.7 g mÿ2
89.4 g mÿ2

C
2
C
G

Dry matter assimilation
 E Eq. (19)
1 Eq. (17)
2 Eq. (17)

0.15 dÿ1
5.33 g MJÿ1 PAR
ÿ0.123 g MJÿ1 Wÿ1 m2

10
8
8

C: Calibration
D: Definition
G: A priori guess
1: Bligaard (1996)
2: van den Boogard and Thorup-Kristensen (1997)
3: van den Boogard and Thorup-Kristensen (1999)
4: Graf et al. (1990)
5: Greenwood et al. (1982)
6: Grevsen and Olesen (1994a)
7: Grevsen and Olesen (1994b)
8: Olesen and Grevsen (1997)
9: Pearson et al. (1994)
10: Penning de Vries et al. (1989)

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