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Predicting the Growth Interactions between Plants in Mixed Species Stands using a SimpleMechanistic Model

S . E . P A R K{, L. R. BEN JAMIN *{, D . P. AIK MAN} and A. R. WATKINSO N{

{Schools of Biological and Environmental Sciences, University of East Anglia, Norwich, NR4 7TJ, UK,

{Horticulture Research International, Wellesbourne, Warwick, Warks., CV35 9EF, UK and}Arden Hill Cottage,

Hooknell, Norton Lindsey, Warks., UK

Received: 23 October 2000 Returned for revision: 30 November 2000 Accepted: 8 January 2001 Published electronically: 23 February 2001

The Conductance model is a simple mechanistic model used to predict the growth of species in monoculture ormixtures from parameter values derived from plants grown in isolation. In contrast to many mechanistic models thatrequire extensive parameterization, the Conductance model is able to capture the growth of a broad range of speciesusing a few simplied assumptions regarding plant growth and easily derived species-specic parameter values. Weexamine the assumptions within the Conductance model that total leaf area per plant is proportional to total plant

weight, and that an isolated plant has a projected crown zone area that is proportional to the 2/3 power of its weight.Power rather than linear relations were found between weight and leaf area for Brassica oleracea, Daucus carota,Matricaria inodora, Solanum nigrum, Stellaria media, Trifolium repens and Veronica persica. For all seven species, thevalue of the power was less than unity. All species also exhibited a power relation between crown zone area andweight, with the slope of this relation being less than 2/3 for B. oleracea, D. carota and S. media. Althoughmorphology type accounted for some of the variation in the parameter values relating to light interception, there wereconsiderable dierences between species within upright or prostrate foliage species groups. The Conductance modelwas used to predict yields of B. oleracea, S. nigrum and V. persica grown in both monoculture and binary weed-cropmixtures over a range of temporal and spatial scales. After calibrating the model to non-competing plants, the modelwas used to predict growth of the weed and crop species in contrasting densities and stand types. In some crop-weedcombinations, predicted crop and weed weights were within 17 % of observed values, with no systematic deviations.In others, systematic and large deviations occurred. # 2001 Annals of Botany Company

Key words: Brassica oleracea L., Daucus carota L., Matricaria inodora L., Solanum nigrum L., Stellaria media L.,Trifolium repens L., Veronica persica L., competition, growth, leaf area, crown zone area, light, shoot morphology,canopy architecture.

INTRODUCTION

The ecacy of weed management strategies will be

determined primarily by the ability to predict the mutual

competitive interactions between weed and crop species. By

predicting the growth of the crop and weeds at contrasting

crop and weed densities and establishment times, it will be

possible to evaluate dierent weed control strategies. Many

weed-crop models have been developed over the past two

decades to predict the dynamics of weeds within a crop (e.g.

Freckleton and Watkinson, 1998; Lintell Smith et al., 1999).

However, while the predictive value of many empirical,

phenomenological models may be high, they fail to oer

insight into the underlying processes of competition and

can be unreliable outside the range of data to which they

have been calibrated (Marcelis et al., 1998). At the other

extreme are explanatory models (e.g. Krop and Spitters,

1992), but these may be complex and based upon detailed

simulation of light interception, photosynthesis, respiration

and dry matter partitioning. Consequently, mechanistic

models, such as that proposed by Krop and Spitters

(1992), require determination of many species-specic

parameters.

In contrast, the Conductance model (Aikman and Scaife,

1993; Aikman and Benjamin, 1994; Benjamin and Aikman,1995) has a much simpler structure and is based uponindividual plants rather than mean crop values. Insummary, for each plant, leaf area is considered to be asimple time-independent function of total plant dry weight.

The model also considers the lateral spread of a plant'sshoot as viewed from above, and dened as the crown zonearea. Up to the time of canopy closure, crown zone area is afunction of total plant weight; thereafter crown zone area is

a complex function of plant weight and stand density.When the sum of crown zone areas equals the availablespace, those plants with a greater within-crown leaf areaindex continue to expand, consequently compressing thecrowns of neighbouring plants. Compression continues

until the within-crown leaf area indices of all individuals areequal. (The model does not consider the vertical distri-bution of foliage, except where each species entirelyoccupies an individual storey of the canopy.) The fraction

of incident light intercepted by the crown zone area is asimple function of the within-crown leaf area index, whichis calculated as the ratio of leaf area to crown zone area.Growth is estimated from the product of intercepted light

and the conversion eciency of light to dry matter. Hence,the Conductance model requires information on: (1) the

Annals of Botany 87: 523536, 2001doi:10.1006/anbo.2001.1369, available online at http://www.idealibrary.com on

0305-7364/01/040523+14 $35.00/00 # 2001 Annals of Botany Company

* For correspondence. E-mail [email protected]


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relation between total plant weight and leaf area; (2) therelation between total plant weight and crown zone area; (3)the extinction coecient for light; and (4) the conversion oflight to dry matter. This information can be obtainedrelatively simply by direct observations on well-spacedplants.

Despite the simplifying assumptions, close agreement hasbeen found between observed data and model predictionsfor the eects of density on the growth of Daucus carotaand Brassica oleracea in mixed-aged and mixed-speciesstands (Benjamin and Aikman, 1995). These predictionswere obtained after calibrating the model with data fromeven-aged monocultures. The model was also used topredict the growth of D. carota, Matricaria inodora andVeronica persica in binary mixtures after calibrating toobservations made on isolated plants (Aikman et al., 1995).

The objective of this paper is to explore more fully thepotential of the Conductance model to predict the growthof plants in mixed-species stands. The paper focuses on the

annual crop and weed species common to horticulturalproduction in the United Kingdom, as competitiveinteractions between these species are of commercial aswell as biological interest. In the rst experiment, thecritical assumptions regarding the relations between leafarea, crown zone area and plant weight are explored for twocrop species: Brassica oleracea (cabbage) and Daucus carota(carrot), and for ve weed species: Matricaria inodora(mayweed), Solanum nigrum (black nightshade), Stellariamedia (chickweed), Trifolium repens (clover) and Veronica

persica (speedwell). The species are grouped by shootmorphology to examine relationships between shootmorphology and the parameter values of the model, and

thus the potential for further simplication of the model.The second experiment explores the ability of two versionsof the Conductance model to predict the growth of a cropand weed species in binary mixtures using parameter values

obtained from isolated plants grown in expt 1. Modelpredictions are validated against observed data. In contrastto previous studies using the Conductance model (Aikmanet al., 1995; Benjamin and Aikman, 1995), the present studyutilises two densities of weed species and two sowing dates.

M A T E R I A L S A N D M E T H O D S

Both experiments were conducted at Horticulture Research

International, Wellesbourne, UK (52.128N, 1

.358W, NatGrid ref. SP271570) on a sandy loam of the Wick Series.

Prior to planting, all areas received a basal dressing ofphosphorus and nitrogen at rates of 250 kg ha1. Soilmoisture was kept sucient for crop growth throughout theduration of the experiment by using a sprinkler irrigationsystem on a daily basis after an absence of rain for a periodof 2 to 3 d. Except where specied, all seedlings were raisedin heated glasshouses in modules of Levington F1 compost(Scotts, Ipswich, UK) and fungal diseases were controlledby applications of Basilex (Scotts), Fubol (Novartis,Whittlesford, UK) and Rovral (Aventis, Ongar, UK).Seedlings were transplanted by hand into the eld. Insect

pests were controlled using applications of Disyston (Bayer,Bury St. Edmunds, UK), Yaltox (Bayer), Decis (Agrevo,

Ongar, UK), Ambush (Zeneca, Hazelmere, UK) andHostathion (Agrevo).

Experiment 1

Monocultures of the crop species Brassica oleracea (var.capitata `Myatts Oenham Compacta') and Daucus carota(`Marathon'), and the weed species Matricaria inodora,Solanum nigrum, Stellaria media, Trifolium repens (`Huia')and Veronica persica were established on either two or threedierent occasions during the growing season of 1994(Table 1). Prior to each planting, nitrogen was applied at arate of 100 kg ha1 in the form of Nitram (ammoniumnitrate) granules (34.5 % N) using an agrimono napsac forthe entire experiment, except for the areas in whichD. carota and T. repens were to be planted. No N wasapplied to these two species. Seed of B. oleracea andD. carota was supplied by Elsoms (Spalding, UK), while

seed of M. inodora, S. nigrum, S. media and V. persica wassupplied by Herbiseed (Wokingham, UK) and that ofT. repens by Agrovista (Stappleford, UK). Seed of D. carotawas sown directly into the eld by hand. All other specieswere initially raised from seed in a glasshouse beforetransplanting into the eld.

To emulate isolated plant growth, over 70 individuals ofeach species were planted at each sowing date into two rows0.56 m apart, with a within-row spacing of 0.65 m. Allseven beds (one per species) were 1.5 m wide and the totallength of bed occupied by a single sowing was 25 m. Foreach species at each sowing date, between ve and sevenharvests were taken at approx. 10 d intervals. At each

harvest between four and ten plants were selected atrandom and the following measurements were made per

TA B L E 1. Sowing and dates of transplanting in 1994 and thenumber of harvests for the crops Brassica oleracea andDaucus carota and the weeds Matricaria inodora, Solanumnigrum, Stellaria media, Trifolium repens and Veronica

persica in expt 1

Sowing dateTransplanting

dateNumber of

harvests

B. oleracea 9 May 7 June 7

2 June 14 July 728 July 31 August 5

D. carota 20 May 717 June 726 July 5

M. indora 3 June 28 June 62 July 14 July 6

28 July 31 August 5S. nigrum 3 June 28 June 6

28 July 31 August 5S. media 9 May 7 June 7

28 July 31 August 5T. repens 9 May 7 June 7

28 July 31 August 5V. persica 9 May 7 June 7

21 June 14 July 7

28 July 31 August 5

524 Park et al.Predicting Growth Interactions using a Simple Mechanistic Model


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plant: (1) the area of ground covered by the shoot crownwhen viewed from above; (2) the area of the smallestvertically projected circle to encompass all the leaves of aplant when viewed from above (crown zone area); (3) totalplant leaf area; and (4) total plant weight after drying at808C. Plant weight included roots that were still attached to

a plant after lifting with a trowel or garden fork. Groundcover and crown zone area were determined by capturingimages of the shoots immediately before harvest. This wasdone by placing white card around the base of the plants toallow easy discrimination between leaves and the back-ground. Images were captured using a video camera heldvertically above the plants. Leaf area measurements weretaken by removing all the leaves from an individual plant,laying them on a at surface and capturing the image usinga video camera. An object of known dimensions was placedin the eld of view of all images to allow area measurementsto be calibrated and determined. All image analysis wasconducted on a PC using VISILOG, supplied by Noesis,

France.

Experiment 2

Experiment 2 consisted of growing binary mixtures of thecrop B. oleracea with either S. nigrum or V. persica. ForS. nigrum, all combinations of two densities and twotransplantation dates were used. For V. persica, only onedensity and transplanting date was used. All three specieswere initially raised from seed in a glasshouse (Table 2).S. nigrum was sown into compost treated with Quintozene

(Aventis) to reduce the incidence of fungal infections.The experiment was a randomized split plot design

allowing for 11 treatment combinations (Table 2). The sitewas 52 7.5 m and divided into three replicate blocks.Each block contained one plot per treatment. There were 33plots in the experiment. Plots were split into either eight(pure B. oleracea) or four (pure weed and mixed crop-weed)harvest subplots. Each replicate was surrounded by a 1.5 mguard area. Within each treatment plot, three rows of plantswere used as a guard between harvest subplots. On eachside of a subplot there were two guard rows of plants. Therewere three additional guard rows at the end of each plot.Plots varied in length and width according to the treatmentthey contained.

A stale seedbed was produced by power harrowing thesite on 10 Jun. 1996 to stimulate weed growth and again on16 July to eradicate the native weed infestation. Experi-mental seedlings were transplanted into the eld accordingto the schedule outlined in Table 2. Plants were arranged ina square grid design within the plots, with mixed crop and

weed plots being planted in an additive, chequer boarddesign (Fig. 1). Plants were spaced at a distance of either 7or 14 cm, producing densities of 204 and 51 plants m2,respectively. B. oleracea was transplanted at a constantdensity of 51 plants m2 throughout the experiment.S. nigrum was transplanted at the two densities (referredto as high and low) 1 and 2 weeks after transplantingB. oleracea (referred to as early and late). The 2 2factorial treatment for S. nigrum was repeated in mono-culture and mixed stands with B. oleracea. V. persica waspresent in only two treatments: transplanted late at lowdensity in pure and mixed stands with B. oleracea. Anyplants failing to establish after being transplanted were

immediately replaced with spare plants.Nitrogen was applied on 16 July at 100 kg ha1 in theform of Nitram (ammonium nitrate) granules (34.5 % N)using an agrimono napsac. The site was kept free of non-transplanted weeds by hand-weeding.

B. oleracea was harvested 21, 28, 35, 42, 48, 56, 62 and70 d after transplanting. S. nigrum was harvested at 21, 35,48 and 62 d, and V. persica was harvested 28, 42, 56 and70 d after transplanting B. oleracea. Plots were harvestedsequentially along their length. At all harvests, 12 plants ofeach species were removed from the middle rows of the plotand their total weight (including roots as described for expt1) was determined after drying for a period of 48 h at 808C.

Total daily solar radiation was recorded at a weather station0.75 km from the experimental site.

ANALYS IS

Experiment 1

The Conductance model (Aikman and Benjamin, 1994)simulates the light interception of individual plants bycalculating crown zone area, az , and the within-crown leafarea index, l, from total plant dry weight, w. The keyrelations for the light interception component are (1)

TA B L E 2. The species, stand type, sowing and transplanting dates in 1996, together with the spacing and plot length for the 11treatments in expt 2

Treatment Species Stand type Sowing date Transplanting date Spacing (cm) Plot length (m)

1 B. oleracea Pure 12 June 16 July 14 3.502 S. nigrum Pure 13 June 23 July 7 1.193 S. nigrum Pure 27 June 30 July 7 1.194 S. nigrum Mixed with B. oleracea 13 June 23 July 7 2.385 S. nigrum Mixed with B. oleracea 27 June 30 July 7 2.386 S. nigrum Pure 13 June 23 July 14 2.947 S. nigrum Pure 27 June 30 July 14 2.948 S. nigrum Mixed with B. oleracea 13 June 23 July 14 3.229 S. nigrum Mixed with B. oleracea 27 June 30 July 14 3.22

10 V. persica Pure 4 July 30 July 14 2.94

11 V. persica Mixed with B. oleracea 4 July 30 July 14 3.22

Park et al.Predicting Growth Interactions using a Simple Mechanistic Model 525


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between total plant weight, w, and leaf area, al , and (2)

between total plant weight, w, and crown zone area, az .

Crown zone area before canopy closure is calculated as the

smallest circle in plan view that encompasses all the leaves

of an individual plant, and after canopy closure as the

amount of space per plant determined by stand density. All

parameters and units used in the Conductance model are

given in Table 3.

The parameters of the allometric relationship between aland w:

al Fwy 1

where F and y are constants, and between az and w:

az Awj 2

where A and j are constants, were estimated from linear

regression following logarithmic transformation of the

data. Regression analyses were carried out on the data for

each of the seven species individually, pooled data for

species with either upright or prostrate growth habits, and

pooled data for all seven species (Table 4).

Experiment 2

Separate analyses of variance (ANOVA) were conducted

for each species on the logarithmically transformed plant

dry weights of B. oleracea and S. nigrum at 62 d after

transplanting B. oleracea, and on V. persica at 70 d after

transplanting B. oleracea. There were six treatment factors

for B. oleracea, consisting of treatments 1, 4, 5, 8, 9 and 11

(Table 2). There were eight treatment factors for S. nigrum,

consisting of treatments 29. There were two treatmentfactors for V. persica, consisting of treatments 10 and 11.

F I G. 1. The arrangement of individual crop (d) and weed plants (s) inexpt 2 for mixed species stands when the weed density was (A) low and(B) high. All plants within the rectangles were lifted at harvest. Note

that at low density (A), all plants contained within a plot are shown,whereas at high weed density (B), only part of the plot is illustrated.

TA B L E 3. Parameters and units used in the Conductancemodel

Symbol Unit Description

a m2 Area

Aj m2 gj

oAllometric constants for crown zoneareaplant weight relation

b g MJ1 Conversion of total solar radiation tototal plant dry matter

Fy

m2 gyo

Allometric constants for the total leafareaplant weight relation

I MJ m2 d1 Incident ux of total solar radiation

k Light extinction coecient of foliage

w g Dry weight per plant

Subscriptsg Groundl Leaf

z Crown zone



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TABLE4.Valuesofthemodelparametersforthesevenspeciesinexpt1eitherindividually,group

edbyshootmorphologyorforallspecies

Species

Morphology

Leafareaweightrelation

Zoneareaweightrelation

Extinctioncoecient

Conversioneciency

ln(F)

y

n

r2

ln(A)

j

n

r2

k

n

b

n

B.o

leracea

Upright

4.15

0.840

91

0.99

3.46

0.529

86

0.93

0.71

86

1.40

67

D.c

arota

Upright

4.63

0.801

87

0.96

3.67

0.591

75

0.88

0.65

75

1.23

58

M.inodora

Upright

3.99

0.890

78

0.98

3.69

0.623

61

0.87

0.59

61

1.63

41

S.nigrum

Upright

4.07

0.866

51

0.99

3.63

0.747

47

0.96

0.70

47

1.20

33

S.m

edia

Prostrate

4.06

0.849

82

0.99

3.09

0.585

58

0.88

0.71

58

0.90

44

T.repens

Prostrate

4.15

0.916

61

0.98

3.60

0.692

57

0.85

0.54

57

1.13

43

V.p

ersica

Prostrate

4.12

0.869

91

0.99

3.72

0.696

85

0.90

0.53

85

1.33

65

rms

(se)

0.040

0.0136

0.069

0.0263

0.029

0

.036

Uprightspecies

4.14

0.88

256

0.98

3.64

0.60

222

0.90

Prostratespecies

4.27

0.85

285

0.96

3.52

0.68

247

0.87

All

sevenspecies

4.21

0.86

541

0.97

3.59

0.63

469

0.88

Parametervaluesareprovidedforthelinear

regressionofleafareaweightallometric

relation[eqn(1)],zoneareaweightallometricrelation[eqn(2)],extinctioncoecientk[eqn(5)]andthe

conversioneciencyoflighttodrymatterb[eqn(3)].T

hesummarystatisticforindividualspeciesistherootmeansquareofthesta

ndarderrors[rms(se),t

hesquarerootofthequotientofsumof

the

squaredstandarderrorsandnumberofspecies].



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Model tting

A simplied version of the Conductance model (Aikmanand Scaife, 1993; Aikman and Benjamin, 1994; Benjaminand Aikman, 1995) is used in the present study in whichlight is assumed to be the sole factor limiting growth. If thefraction of incident light intercepted by the shoots increases

exponentially with within-crown leaf area index (al/az), thenthe Conductance model simplies to give absolute growth,dw/dt (g d1)

dw

dt bIaz1 expkalaaz 3

where b is the conversion eciency of solar radiation to drymatter, I is the incident total solar radiation and k is theextinction coecient for light.

The value of k was estimated from direct observations ofal , az and the ground cover ag for individual plants in expt1. If leaves do not transmit light and are arranged randomly

within the crown zone, then for vertical radiation:ag az1 expkalaaz 4

Rearranging eqn (4) and taking logarithms allows k to becalculated thus:

k azaal log1 agaaz 5

The value of b in eqn (3) was estimated for each speciesseparately by minimizing the dierence between observedand tted total plant dry weights in expt 1 using the leastsquares (leastsq) procedure in MATLAB (Mathworks,1996) and a fourth-order Runge-Kutta method of inte-

gration (Press et al., 1989). All other parameters in eqn (3)were held at the mean values estimated from directobservations for each species at each planting date. Foreach planting, the mean weight of plants at the rst harvestwas used as the starting weight. The observed daily totalsolar radiation (MJ m2) was used as the driving variable.Standard errors associated with b were estimated from theslope of the response surface of the sum of squares tochanges in the value of b (Nelder and Mead, 1965).

To illustrate any bias between observed and predictedweights in expt 2, these data were plotted against timeseparately for each treatment. To demonstrate the goodnessof t, the variance of the dierence between predicted and

observed weights was calculated for the mean dierenceaveraged over the second, third and fourth harvests. Therst harvest was excluded because the observed data wereused as the starting weights of the simulation. At any oneharvest, observed weights were averaged over the threereplicates. The goodness of t was estimated as the ratio ofthe variance of the mean dierence to the variance of thepure experimental error. A value of unity indicates thatthe model's predictions deviate from the observed data tothe same extent as the experimental error. Experimentalerror was estimated from the residual mean square aftertting an ANOVA to the total plant dry weights. Theanalysis of variance had the four harvests as a factor and

was tted separately to the dierent treatments. To becomparable to the variance of the dierences, the residual

mean square was divided by three so that the mean valuefor the three replicates was estimated.

RES ULTS

Experiment 1

A simple allometric equation [eqn (1)] can be used todescribe the relation between leaf area and plant dry weight(Table 4) for B. oleracea (Fig. 2A), M. inodora (Fig. 2C),S. nigrum (Fig. 2D), S. media (Fig. 2E) and V. persica(Fig. 2G). However, the distribution of the residuals for theother two species indicated signicant deviations fromlinearity, and in each of these cases the inclusion of aquadratic term provided a signicantly better t to thisrelationship: D. carota (Fig. 2B) (F1,84 52

.9, P 5 0.001)and T. repens (Fig. 2F) (F1,58 15

.1, P 5 0.001). In allseven species, the slope of the simple allometric relation was

less than unity (P 5 0.001), contrary to the originalsuggestion for the Conductance model by Aikman and

Benjamin (1994).Allowing separate relations for the two dierent shoot

morphology types gave a better t than a single relation forall the data (F2,481 14

.3, P 5 0.001). Allowing separaterelations for each species gave a signicantly better t to thedata than allowing separate relations for shoot morphologytype (F10,471 30

.5, P 5 0.001). The residual mean squareerror after linear regression for all the data pooled was0.1555, which was reduced to 0.1474 when allowing fordierent intercepts and slopes for shoot morphology. Thiswas further reduced to 0.0914 when allowing for dierent

intercepts and slopes for the dierent species.A simple allometric equation [eqn (2)] can also be used todescribe the relation between crown zone area and plant dryweight for B. oleracea (Fig. 3A), M. inodora (Fig. 3C),S. nigrum (Fig. 3D) and T. repens (Fig. 3F). The distri-bution of residuals for the other three species indicatednon-linearity. A quadratic term was included as a tool totest the signicance of this non-linearity. It improved the tof the model for these three species: D. carota (Fig. 3B)(F1,77 5

.5, P 5 0.05), S. media (Fig. 3E) (F1,55 16.0,

P 5 0.001), and V. persica (Fig. 3G) (F1,83 72.1,

P 5 0.001). For three species, the power on the allometricrelation between crown zone area and plant weight was

signicantly lower than the value of 2/3 postulated byAikman and Benjamin (1994): B. oleracea (t85 3.97,

P 5 0.001), D. carota (t78 3.09, P 5 0.01) and S. media

(t56 2.88, P 5 0.01) (Table 4).

Allowing separate relations for shoot morphology typegave a better t than a single relation for all the data(F2,448 6

.0, P 5 0.001). Allowing separate relations for

each species gave a signicantly better t to the data thanallowing separate relations for morphology type(F10,438 5

.1, P 5 0.001). The residual mean square errorafter linear regression for all the data pooled was 0 .4882,which was reduced to 0.4777 when allowing for dierentintercepts and slopes for shoot morphology, which was

further reduced to 0.4379 when allowing for dierentintercepts and slopes for the dierent species.



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Modelling growth in monoculture

In general, there was reasonable agreement between the

temporal changes in weight predicted by the Conductance

model and the observed values across all seven species

(Fig. 4). A common parameter value for b (Table 4) was

used for the separate sowings of any one species, but

separate values for the dierent sowings were used for

parameters F, y, A, j and k (values not shown) to minimizethe eects of any non-random residuals in the allometric

relations. There was a slight over-prediction of plant weightfor the rst planting and an under-prediction of plantweight for the second planting for S. media, M. inodora andV. persica (Fig. 4C, E, G).

Experiment 2

The dry weight of individual B. oleracea plants increasedwith time (Figs 57). When B. oleracea was grown in mixed

A B

DC

E F

G

0.1 1 10 100 1000

10

1

0.1

0.01

0.001

0.0001

L

eafarea(m2)

Plant dry weight (g)

0.1 1 10 100 1000


0.1 1 10 100


0.1 1 10 100 1000Plant dry weight (g)

0.01

0.1 1 10 100 1000


0.01

0.1 1 10 100 1000


0.1 1 10 100 1000Plant dry weight (g)

10

1

0.1

0.01

0.001

0.0001L

eafarea(m2)

10

1

0.1

0.01

0.001

0.0001

Leafarea(m2)

10

1

0.1

0.01

0.001

0.0001Leafarea(m2)

10

1

0.1

0.01

0.001

0.0001

Leafarea(m2)

10

1

0.1

0.01

0.001

0.0001

Leafarea(m2)

10

1

0.1

0.01

0.001

0.0001

Leafarea(m2)

0.00001

0.01

0.01

0.00001

F I G . 2. Relationship between total plant dry weight and leaf area for B. oleracea (A), D. carota (B), M. inodora (C), S. nigrum (D), S. media (E),T. repens (F) and V. persica (G) in expt 1. Symbols are for individual plants and refer to early (d), middle (j) and late (m) sowings, respectively.

Details of the regression analysis are given in Table 4.



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stands with S. nigrum, the weight ofB. oleracea at 62 d after

transplanting was signicantly lower than in pure stands

(F4,8 79.6, P 5 0.001), with the lowest weight observed

when S. nigrum was transplanted early at high density

(Figs 5 and 6, Table 5). In contrast, V. persica had no eect

on the weight of B. oleracea (F1,14 2.61, P 4 0.05)

(Fig. 7).

The dry weight of individual S. nigrum plants increased

with time (Figs 5 and 6). Sixty-two days after transplantingB. oleracea, early transplants of S. nigrum in mixed stand

had a weight approximately one-third of those in pure

stand, whereas late transplants of S. nigrum in mixed stand

had a weight approximately one-eighth of those in pure

stand (F1,14 21.2, P 5 0.001) (Table 6). Similarly, 62 d

after transplanting B. oleracea, low density transplants of

S. nigrum in mixed stand had a weight approximately half

that of plants in pure stand, whereas high density

transplants of S. nigrum in mixed stand had a weight

approximately one-ninth that of plants in pure stand(Table 6). Interestingly, there was no dierence in the

A

C

B

D

E F

G

0.1 1 10 100 1000


1

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0.001

Zonearea(m2)

0.1 1 10 100


0.1 1 10 100


0.1 1 10 100


0.1 1 10 100


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0.01

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onearea(m2)

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1

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F I G. 3. Relationship between total plant dry weight and crown zone area for B. oleracea (A), D. carota (B), M. inodora (C), S. nigrum (D), S. media(E), T. repens (F) and V. persica (G) in expt 1. Symbols are for individual plants and refer to early (d), middle (j) and late (m) sowings,

respectively. Details of the regression analysis are given in Table 4.



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weight of S. nigrum in mixed stands between the two

density treatments, whereas in pure stand, those spaced

14 cm apart weighed approximately twice as much as those

spaced 7 cm apart. This interaction was statistically

signicant (F1,14 37.7, P 5 0.001). The three-way inter-

action between stand type, transplanting date and densitywas not statistically signicant (F1,14 0

.4, P 4 0.05).

The dry weight of individual V. persica plants increasedwith time in pure stands, although in mixed stands with

B. oleracea, the weed displayed virtually no increase in size(Figs 5 and 7).

Modelling growth in pure stands

The model predicted the growth of cabbage well, with thedeviation of predicted values from the mean observed at

any one harvest never exceeding 14 % (Fig. 5A). However,when a mean was calculated for the observed dry weight

150 200 250 300

100

1000

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0.1

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ryweight(g)

Day of year

A B

DC

E F

G

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ryweight(g)

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F I G . 4. Relationship between total plant dry weight and time for B. oleracea (A), D. carota (B), M. inodora (C), S. nigrum (D), S. media (E),T. repens (F) and V. persica (G) in expt 1. Symbols are for individual plants and refer to early (d), middle (j) and late (m) sowings, respectively.

Lines were generated using eqn (3) with the mean weight at rst harvest from each planting and daily solar integral as inputs.



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A B

DC

20 30 40 50 60 70

Days after transplantingB. oleracea

20 30 40 50 60 70


20 30 40 50 60 70


20 30 40 50 60 70


S. nigrum Late Transplants V. persica

S. nigrum Early TransplantsB. oleracea25

20

15

10

5

Plantdryweig

ht(g)

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5

Plantdryweight(g)

2015

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1

1

1

F I G. 5. Observed and predicted growth of B. oleracea, S. nigrum and V. persica in pure stands in expt 2. A, B. oleracea; B, early transplantedS. nigrum at low (d) and high (j) density; C, late transplanted S. nigrum at low (d) and high (j) density; D, V. persica. Symbols represent the

mean weight of a replicate at each harvest. Lines represent the model prediction.

TA B L E 5. The eect of monocropping and mixed croppingand time ofS. nigrum transplanting and spacing on the mean

total plant dry weight (g) of B. oleracea 62 d aftertransplanting B. oleracea (expt 2)

Weeksbetween S. nigrumand B. oleracea

transplanting

S. nigrumspacing

(cm)

Weight ofB. oleracea

(g)

Monocrop 18.1

Mixed 1 7 6.71 14 14.32 7 13.62 14 12.0

LSD (P 0.05) 1.52

LSD given for the weight of B. oleracea across all treatments.

TA B L E 6. The eect of monocropping and mixed croppingand time of transplanting and spacing on the loge mean total

plant dry weight ( g) ofS. nigrum at 62 d after transplantingB. oleracea (expt 2)

Weeks between S. nigrumand B. oleracea

transplanting S. nigrum spacing (cm)

1 2 7 14

Monocrop 1.979 (7.24) 1.796 (6.03) 1.197 (3.31) 2.578 (13.17)Mixed 0.931 (2.54) 0.240 (0.79) 0.313 (1.37) 0.378 (1.46)

LSD (P 0.05) 0.3252 0.3252

Figures in parentheses show untransformed values. LSD given forthe weight of S. nigrum across treatments.



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over three replicates, with the exception of the last harvest,the predicted weights were consistently lower than theobserved data. The ratio of the variance of the mean dier-ence between the predicted and observed values to the pureerror was only 1.2, indicating that the lack of t of the model

can be mostly ascribed to the `noise' of experimental error.For S. nigrum, predicted weights tended to be lower thanthose observed, but the deviation of the predicted valuesfrom the mean of three replicates at each harvest neverexceeded 30 % (Fig. 5B and C). The ratio of the variance tothe mean dierence between the predicted and observedvalues to the pure error ranged from 2.7 t o 1 4.4 forS. nigrum in pure stand treatments. This indicates that thelack of t of the model cannot be ascribed to the `noise' ofexperimental error for this species.

Similarly, for V. persica, predicted weights were lowerthan those observed, with the deviation of the predictedvalues from the mean of three replicates at each harvest

ranging from 24 to 32 % (Fig. 5D). The ratio of thevariance to the mean dierence between the predicted and

observed values to the pure error was 51, indicating that thelack of t of the model was not due to the `noise' ofexperimental error.

Modelling growth in binary mixtures

When the similar height version of the Conductancemodel was used simultaneously to predict the growth ofB. oleracea and S. nigrum in mixed stands over the range oftreatments, the level of accuracy varied across the weedtreatments. This version of the Conductance model was runbecause the two species typically form a single mixed layercanopy. The predicted total plant dry weight of B. oleraceawas often close to the observed values (Fig. 6), with six ofthe 12 predictions (three harvests two weed densi-ties two weed transplanting times) falling within 10 %of the observed values. The best prediction of B. oleracea inmixed stands was with S. nigrum transplanted early at low

density, where the deviation was always less than 17 % ofthe mean and there was no evidence of systematic

6050403020


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Early Transplants (7 cm) Late Transplants (7 cm)

Late Transplants (14 cm)Early Transplant (14 cm)

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ht(g)

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1Plantdryweight(g)

0.3

0.3

A B

DC

F I G . 6. Observed and predicted growth of B. oleracea and S. nigrum in mixed stands in expt 2. A, Early transplanted S. nigrum at low density; B,late transplanted S. nigrum at low density; C, early transplanted S. nigrum at high density; D, late transplanted S. nigrum at high density. Symbols

represent the mean weight of a replicate at each harvest for B. oleracea (d), and S. nigrum (j). Lines represent the model prediction.



12/14

deviations. The ratio of the variance to the mean dierencebetween the predicted and observed values of the pure errorwas 2.0 for this treatment, indicating that even in this best

predicted treatment, the lack of t of the model cannot beentirely ascribed to the `noise' of experimental error.

Growth of S. nigrum was predicted with less accuracythan that of B. oleracea (Fig. 6). Only three of the 12

predicted weights were within 10 % of the observed values.The best prediction was for S. nigrum transplanted early athigh density, where the deviation was always less than 17 %

and there was no systematic deviation. The variance ratio ofthe mean dierence between observed and predicted valuesto pure error was 1.8. However, in the worst predictions, the

predicted weights were between 83 and 122 % greater thanthe observed values (Fig. 6B) and the variance ratio of themean dierence between observed and predicted values to

pure error was 154.

The separate height version of the model was used tosimulate the growth of B. oleracea and V. persica in binarymixtures. This version of the model was chosen because for

this species mixture, B. oleracea shoots typically formed an

over-storey above shoots ofV. persica. Weight ofB. oleraceaplants was under-predicted by 10 % at the second harvest,

but over-predicted by 28 % at subsequent harvests (Fig. 7).The variance ratio of the mean dierence between observedand predicted values to pure error was 22, indicating that

the deviation of predicted to observed weights was muchgreater than would be expected from the experimentalerror. Weight of V. persica plants was under-predicted by

851 %. The variance ratio of mean dierence betweenobserved and predicted values to pure error was only 1.25,indicating that the predictions made by the model for

V. persica were close to what could be explained byexperimental error. The weight of V. persica was over-

predicted three-fold when the similar height version of themodel was used (data not shown).

DIS CUS S ION

The growth of B. oleracea and S. nigrum in the presentstudy was similar to the growth of Beta vulgaris and

Chenopodium album in mixed stands (Krop et al., 1992).In both studies, there was little indication of a reduction inthe absolute growth rate of either the crop or the weed over

time as the plants matured. Also in the present study, as inthat of Krop et al. (1992), a delay in the time of

establishment of the weed relative to that of the crop byapproximately 1 week resulted in a reduction in weed

growth and an increase in crop growth. Increasing thedensity of C. album from 5.5 to 22 plants m2 had little

eect on the growth of B. vulgaris in the rst 50 d, butsubsequently caused large reductions (Krop et al., 1992).

In the present study, the absolute growth rate of B. oleraceawas reduced throughout the experiment in the presence of a

high density of S. nigrum.

The Conductance model makes a range of assumptionsregarding the growth of plants in monoculture and mixed

stands. In this study we specically examine two of theseassumptions: that total leaf area per plant is proportional to

plant weight, and that an isolated plant has an allometricrelation between projected crown zone area and total plantdry weight with a slope of 2/3.

The relation between leaf area and leaf dry weight has

been found to vary during the growth of a plant in a widerange of plant studies (e.g. Kreusler et al., 1877; Briggs et al.,

1920). In the early stages of growth, the relation betweenthe leaf area and the dry weight of a seedling approximates

to linearity. However, as plant size increases, fewerphotosynthates are allocated to the further development

of leaf area and more biomass is diverted towards structuralsupport (Evans, 1972). In the present studies, the value ofy

was always less than unity, which is consistent with the leafarea ratio (LAR, ratio of leaf area to total plant weight)

decreasing with plant weight. In all seven species, a linearregression of the logarithm of leaf area on the logarithm oftotal plant dry weight had r2 values ranging from 0.96 to

0.99. Linear relations between total leaf area and plantweight have also been observed in Plantago lanceolata L.

(Hinsberg et al., 1997). For B. oleracea, D. carota, S. media,T. repens and V. persica, the predicted leaf area forindividual plants weighing approx. 0.1 g consistently fell

below the tted values. Low values of LAR have been

observed in forage grasses (Carlen et al., 1999), and inAbutilon theophrasti, Chenopodium album and Polygonumpensylvanicum (McConnaughay and Coleman, 1999). The

discrepancy between observed and predicted leaf area in theearly stages of growth may be attributed to the dierence in

morphology between the cotyledons or coleoptile and thetrue leaves.

Many mechanistic models use complex functions to

describe leaf area development based upon a combinationof leaf emergence rate, rate and duration of leaf expansion,

and duration of individual leaves (Marcelis et al., 1998).The results of the present study imply that these processesand those determining partitioning of dry matter to the leaf

and specic leaf area, are all in a species-specic dynamicequilibrium. The presence of the allometric relation

7060403020 50

20

10

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3

1

0.3

0.1

Days afterB. oleracea transplanting

Plantdryweig

ht(g)

F I G. 7. Observed and predicted growth ofB. oleracea and V. persica inmixed stands in expt 2. Symbols represent the mean weight of areplicate at each harvest for B. oleracea (d) and V. persica (j). Lines

represent the model prediction.



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between leaf area and plant weight in all seven species of

this study is surprising given their dierent shoot

morphologies and that M. inodora, S. nigrum, S. media

and V. persica owered for much of the duration of expt 1,

whereas the other species remained vegetative throughout.

This simple allometric relation implies that the rate of

increase in leaf area is closely linked to the rate of increasein total plant weight. This close linkage suggests the

presence of a dynamic equilibrium arising from a level of

control of leaf area production at the whole plant level that

appears to have been overlooked previously. Where the

rates of leaf area and plant weight increase are out of kilter,

then curvature would occur on the log-log plots of these

two variables. There was some evidence that the dynamic

equilibrium did not hold perfectly for D. carota and

T. repens. For these species, there was a signicantly better

t of a quadratic rather than a linear equation relating the

logarithm of leaf area to the logarithm of plant weight. It

may be that these two species suered mild stress despite

attempts to provide ideal growing conditions. Indeed, noadditional N was applied to these two species in expt 1

because it was anticipated that their N requirements wouldbe met by native soil N. Reduced mineral nutrient supply

increases LAR in A. theophrasti, C. album, P. pensylvaticum

and Brassica campestris (Li et al., 1999; McConnaughay

and Coleman, 1999).

Whatever the underlying mechanism, the presence of an

allometric relation between leaf area and plant weight

obviates the need to simulate the leaf area dynamics with a

string of interlinked sub-models when seeking to predict the

competitive interactions between plants in fertilized, irri-

gated conditions of commercial horticulture. The allometric

relation between plant weight and leaf area can also be usedwhere competing plants are of very dierent stature, by

assuming that the separate species occupy dierent layers in

a multi-storey crop canopy (Benjamin and Aikman, 1995).

The second assumption examined was that an isolated

plant has a projected crown zone area that is, on an

allometric basis, proportional to the 2/3 power of the plant

weight. The value of the power term is based upon a

dimensional argument: area has two dimensions, whilst

weight has three dimensions.

The linear regression of logarithm of crown zone area on

logarithm of plant dry weight had r2 values that varied from

0.85 to 0.96, indicating that a simple allometric equation is

adequate to describe the crown zone areaplant weight

relation. For three species, a quadratic equation gave a

better t than a linear equation to the relation between

logarithms of crown zone area and plant weight. This may

be a consequence of experimental error leading to the

under-estimation of crown zone area on small plants.

Where a plant has only two or three leaves, a chance

movement or deformity of a single leaf would cause a large

reduction in the estimated crown zone area.

For the allometric relation between crown zone area and

plant weight, only four of the seven species had a power

term value equal to 2/3. Hence, the power term in this

allometric relation is species specic. Similarly, Li et al.(1996) found that the value of the exponent of the

allometric relation for crown zone area and weight wassignicantly lower than 2/3 for carrot.

It was anticipated that species of similar shootmorphology would have similar parameter values for therelations between leaf area and plant weight, and betweencrown zone area and plant weight. However, there was no

evidence for such a grouping of parameter values accordingto shoot habit for the species considered in this study.Therefore, the Conductance model cannot be simpliedfurther and parameter values must be provided for eachspecies individually.

There is no attempt to simulate the eects of the spatialpattern of neighbours in the Conductance model. Despitethis lack of detail and the calibration of the model towidely-spaced non-competing plants, the model showed nodiscernible dierence in its ability to predict the growth of aspecies at low density using a simple planting pattern, tostands containing a high density of plants arranged in arelatively more complex spatial pattern.

Whilst light was assumed to be the sole determinant ofgrowth in this study, the model has been found to give goodpredictions of growth across a range of species in responseto light and temperature (Aikman and Scaife, 1993;Benjamin and Aikman, 1995; Tei et al., 1996). A reviewof previous studies using the similar height version of theConductance model (Aikman et al., 1995; Benjamin andAikman, 1995; Tei et al., 1996) reveals that whilst the modelover- or under-predicted the growth of a number of species,there were no systematic deviations with stand type ordensity across the studies. Encouragingly, the number ofspecies, diering stand types and the density rangesincluded in present studies demonstrate the model to be

reasonably robust across the extent of its application in wellfertilized, irrigated conditions to date. Nevertheless, therewere systematic deviations between observed and predictedweights in some of the pure and binary species mixtures.The causes of these are not readily explained, butassumptions of non-limiting mineral nutrients and soilwater may have been incorrect. The framework suggestedby Aikman and Benjamin (1994) allows for dierentenvironmental factors to be jointly limiting.

Regardless of the numerous assumptions that remain tobe tested, the simplied Conductance model would appearto have provided a useful and satisfactory approximation toreality for commercial conditions. Most importantly, whilst

the extensive parameterization requirements of manymechanistic models render them impractical for appli-cation, the Conductance model requires a relatively limitednumber of easily estimated parameters and can therefore beconsidered a useful tool in predicting the competitive eectsof weeds on the growth of a crop over time. Further work isrequired to validate the full version of the Conductancemodel in conditions in which mineral nutrients and soilwater are also limiting.

ACKNOW LEDGEM ENTS

We thank the Ministry of Agriculture, Fisheries and Food,UK for nancial support, and L. Peach, A. Hellemans,



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B. Auwalu and V. Tenebe for technical assistance. S. Parkreceived a BBSRC CASE studentship.

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