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A simulation model of climate effects on plant

productivity and variability in cauliflower

(

Brassica oleracea

L.

botrytis

)

J.E. Olesen

a,*

, K. Grevsen

b

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).

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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.

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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).

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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 ˆaE…Ws‡Wl†; (1)

whereaEis 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 timetwill emerge from the last age class (K) at timet‡s

on the average. The spread in developmental times around the average is described by an Erlang distribution with variances2/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 calledSl, 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 ofTbrˆ08C is used to describe the ageing of roots. The mean age of root dry matter is set toSrˆ3008Cd based on root turnover rates in cauliflower reported by Greenwood et al. (1982). The number of age classes is arbitrarily set toKrˆ20.

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

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The height is assumed to be proportional to the radius, i.e.

hcˆchrc; (3) wherechis a nondimensional constant.

Fresh weights of curds (Fc) and of leaves and stems (Fv) (g plantÿ1) are calculated using empirical relationships:

FcˆafWc‡bfVc; (4)

Fvˆcf…Ws‡Wl† ‡efR; (5) whereaf,cfandefare nondimensional constants, andbf is a constant (g mmÿ3).

2.3. Crop development

The hourly crop developmental rate (d/dt) is defined as (Grevsen and Olesen, 1994a):

where fd1, fd2 and fd3 are temperature response functions for the juvenile, curd induction and curd growth phases, respectively, andThis 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 ofTdb2ˆ5.18C andTdo2ˆ15.58C, 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.

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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):

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 (m

2

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), andal 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:

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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 ˆxVc‰1ÿ …1ÿV†bcŠ; (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 mÿ2dÿ1) is composed of five demands

DˆDr‡Ds‡Dl‡Dc‡DE; (10) where Dr is the demand rate for growth of structural dry matter in roots (g mÿ2dÿ1), Ds the demand rate for growth of structural dry matter in stems (g mÿ2dÿ1),Dl is the demand rate for growth of structural dry matter in leaves including stalks and midribs (g mÿ2dÿ1), Dc the demand rate for curd growth (g mÿ2dÿ1), and DE is the demand rate for growth of reserves (g mÿ2dÿ1). 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:

Dlˆd‰slAx‡aD…slxAÿWl†‡fp…Td†Š; (11)

whereslis the weight of new leaf area (g mÿ2),slxis the maximum weight of leaf

area (g mÿ2),aD is a constant ((8Cd)ÿ1),d the plant density (plant mÿ2),Axis

the maximum daily increase in leaf area (m2dÿ1plant-), andfPis 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 thataDˆ0.04 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:

DrˆdbD…cD…Wl‡Ws† ÿWr†‡; (12)

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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 ˆdbD…eD…Fv‡Fc† ÿWs†‡; (13)

whereeDis a nondimensional constant.

The demand rate for curd growth is assumed to be proportional to the maximum volume increase:

Dcˆd‰cc‡ec…3ÿ†‡ŠxVc; (14)

whereccandecare 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:

DEˆdmin‰bE…RcÿR†; cE…Wl‡Ws†Š; (15)

wherebEandcEare 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 mÿ2dÿ1) is calculated using the concept of a radiation conversion coefficient () (g MJÿ2) and an effect of sink limitation:

Pa ˆmin‰D; fP…Td†QŠ; (16)

where D is the demand rate for dry matter growth (g mÿ2dÿ1), Q the photosynthetic active radiation (MJ mÿ2dÿ1), and fPis 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 intensityQi (W mÿ

2

], which is calculated by dividingQby the day length:

ˆ1‡2Qi; (17)

where 1 and 2 are constants, which were estimated as 1ˆ5.44 g MJÿ 2

and

2ˆÿ0.123 g MJÿ2Wÿ1m2 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 mÿ2dÿ1) is the sum of the net assimilation and a contribution from the plant reserves (PE) (g mÿ2dÿ1):

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

PE ˆ …DÿPa† 1ÿexp ÿ

EdR

DÿPa

; (19)

whereEdefines the mobilisation rate of reserves which is set to 0.15 dÿ1in 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 ˆmin‰P=…Dÿ …1ÿeE†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 giveseEˆ0.1.

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.

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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 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

Dates of transplanting, curd initiation and final sample of cauliflower at Aarslev Research Centre in 1995

Planting Transplanting Curd initiation Last sample

1 21 April 24 May 11 July

2 16 May 7 June 29 July

3 7 June 30 June 9 August

4 30 June 25 July 7 September

5 1 August 28 August 23 October

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blocks (5 cm5 cm5 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 haÿ1and 200 kg K haÿ1 was applied to the experimental area. Each transplanting received 150 kg N haÿ1in 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 (108270E, 558180N) 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 ofaE,cf,ef,bEandcErequired 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 ofWl‡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…aÿbt†; (21)

wheretis number of days from onset of experiment,Axis the maximum leaf area

(m2plantÿ1), andaandbare constants.amay be substituted by ln((AxÿAo)/Ao),

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observed values ofAowere used, andAxwas set to 0.6 m

2

(Olesen and Grevsen, 1997).

Some model constants (sl,acandbc) 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 andKd2 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,bEandcE).

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.

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The ratio of curd height to curd radius Eq. (3) was estimated by linear regression aschˆ1.34 (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 cmÿ3using 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 estimateKd1andKd2under 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 settingKd1ˆKd2ˆ26.

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 mÿ2by 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 aseDˆ0.010 (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 andcEˆ0.12 dÿ1using data from Experiment 1 prior to curd initiation.

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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

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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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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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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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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 experimen-tally 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.

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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

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Appendix A

The variables used in the model are listed below

fraction of PAR intercepted by crop canopy (0±1)

radiation conversion efficiency (g MJÿ1PAR)

Ax maximum daily increase in leaf area (m

2

dÿ1plantÿ1)

Vc daily increase in curd volume (mm3dÿ1)

xVc maximum daily increase in curd volume (mm3dÿ1)

V vegetative supply demand ratio (0±1)

crop developmental stage (0±3)

A green leaf area (m2 plantÿ1)

d plant density (plants mÿ2)

D overall demand rate for dry matter growth (g mÿ2dÿ1)

Dc demand rate for growth of curd dry matter (g mÿ2dÿ1]

Dl demand rate for growth of structural dry matter in leaves (g mÿ2dÿ1)

Dr demand rate for growth of root dry matter (g mÿ2dÿ1)

Ds demand rate for growth of structural dry matter in stems (g mÿ2dÿ1)

DE demand rate for growth of reserves (g mÿ2dÿ1)

Fc curd fresh weight (g plantÿ1)

Fv fresh weight of leaves and stem (g plantÿ1)

hc curd height (mm)

P assimilates available for partitioning (g mÿ2dÿ1)

Pa dry matter assimilation (g mÿ2dÿ1)

PE mobilisation of reserves (g mÿ2dÿ1)

Q photosynthetic active radiation (MJ mÿ2d-1)

R reserves in leaves and stem (g plantÿ1)

Rc capacity for reserve storage in leaves and stem (g plantÿ1)

rc curd radius (mm)

t time (d)

Th hourly air temperature (8C)

Td daily mean air temperature (8C)

Vc curd volume (mm3)

Wc weight of structural dry matter in curds (g plantÿ1)

Wl weight of structural dry matter in leaves (g plantÿ1)

Wr weight of dry matter in roots (g plantÿ1)

Ws weight of structural dry matter in stems (g plantÿ1)

Appendix B

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Constants Value Source

Plant organs

afEq. (4) 10.4 C

aEEq. (1) 2.5 C

bfEq. (4) 0.077 g cmÿ3 C

cf Eq. (5) 10.7 C

ch Eq. (3) 1.34 C

ef Eq. (5) 4.5 C

Kl 30 4

Kr 20 G

Sl 5958Cd C

Sr 3008Cd 5

Tbl 1.98C 8

Tbr 08C G

Development

Kd1ˆKd2 26 C

Sd1 83.38C 6

Sd2 108.28C 7

Sd3 10508Cd 9

Tdb1 08C 6

Tdb2 5.18C 7

Tdb3 08C 9

Tdo2 15.58C 7

Leaf area expansion

al Eq. (7) 0.179 dÿ1 8

LxEq. (7) 6 m2mÿ2 3

Curd volume growth

acEq. (8) 0.0123 (8C)ÿ1 C

bcEq. (9) 2.9 C

Demands for dry matter growth

aDEq. (11) 0.04 dÿ1 G

bDEqs. (12) and (13) 1 dÿ1 G

bEEq. (15) 0.075 dÿ1 C

ccEq. (14) 0.005 g cmÿ3 C

cDEq. (12) .09 1

cEEq. (15) 0.12 dÿ1 C

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