Optimal temperature regimes for a greenhouse crop with a carbohydrate pool: A modelling study

Optimal temperature regimes for a greenhouse crop with a carbohydrate pool: A modelling study

ELSEV [ ER Scientia Horticulturae 60 (1994) 55-80 Optimal temperature regimes for a greenhouse crop with a carbohydrate pool: A modelling study Ido ...

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Scientia Horticulturae 60 (1994) 55-80

Optimal temperature regimes for a greenhouse crop with a carbohydrate pool: A modelling study Ido Segine#'*, Christian Gary b, Marc Tchamitchian b aAgricultural Engineering Department, Technion, Haifa 32000, Israel DINRA, Unite de Bioclimatologie, Centre de Recherchesd'Avignon, BP 91, 84143 Montfavet Cedex, France

Accepted 3 June 1994

Abstract

A simple crop model with two state variables, namely structural biomass and carbohydrate pool, was used to explore the effect of alternative temperature regimes on greenhouse crop production. Assuming a repeated environmental cycle, certain qualitative predictions could be made. ( 1 ) The smaller the plants and the higher the light integral and CO2 enrichment, the higher are the temperatures which lead to maximum production. (2) Day temperatures higher than night temperatures usually lead to higher production. On winter days, however, an inverse temperature regime may result in energy saving without loss of production. (3) Temperature variations may often be tolerated, provided that the mean temperature (temperature integral) is maintained at the level appropriate for maximum production. A limited amount of published experimental data was used to fit the model, leading to a satisfactory agreement. Keywords: Carbohydrate pool; Crop model; Greenhouse crops; Lycopersicon esculentum Mill.;

Temperature regimes

1. Introduction Greenhouses are designed to increase crop productivity by controlling the plant environment. Heating to increase greenhouse temperature in cold climates has a major effect on the cost of operation, as well as on productivity. This has moti* Corresponding author. 0304-4238/94/$07.00 © 1994 Elsevier Science B.V. Allrightsreserved SSD10304-4238 (94) 00694-B

L Seginer et al. / Scientia Horticulturae 60 (1994) 55-80

56 List of symbols

Main symbols a c e f I K L P p R s T t U w a fl 7 ¢ ( ~/ 0 /z v H co

coefficient in Eqs. ( 1 ), (3) and (6) carbohydrate pool size overall growth efficiency ( Y in Thornley and Johnson, 1990) respiration response to temperature flux of light (PAR) coefficient in Eq. ( 1 ) leaf area index (LAI) gross photosynthesis rate cumulative gross photosynthesis respiration rate structural biomass temperature time defined in Eq. ( l 0) total biomass, s-F c coefficient in Eq. (40) coefficient in Eq. (40) coefficient in Eq. (4) coefficient in Eq. (6) light period dc/dt dark period - dc/dt coefficient in Eq. (3) coefficient in Eq. (39) coefficient in Eq. (39 ) coefficient in Eq. ( 1 ) leaf area ratio (LAR), L / s

m 2 ground m -2 leaf kg CO 2 m -2 ground a

kg CO2 kg- l CO2 s- 1 /zmol phot m -2 ground s- l /zmol phot m -2 ground sm 2 leaf m-2 ground kg CO2 m -2 ground s- 1 kg CO2 m -2 ground kg CO2 m -2 ground s -~ kg CO2 m -2 ground K

Operators d -

increment over one environmental cycle temporal mean

Subscripts A b C D E e g L l m t x

triple point bottom of pool carbon dark period energy exponential growth light period linear maintenance top of pool maximum value

"Non-dimensional.

Note: 44 kg CO2 is equivalent to 12 kg C or approximately 30 kg dry matter. Braces enclose arguments of functions.

S

kg CO2 m -2 ground kg CO2 m-2 ground S-I K-I a

kg CO2 m -2 ground kg CO 2 m -2 ground s - l kg CO 2 m -2 ground s - I kg CO2 m -2 ground S-I K - l S-I

kg CO2 m -2 ground s- t m 2 leafkg - l CO2

L Seginer et al./Scientia Horticulturae 60 (1994) 55-80

57

vated a considerable amount of research on the effects of day and night temperatures on the growth of greenhouse crops. Mean daily temperature was clearly found to affect the growth rate of greenhouse plants. Growth, expressed in terms of dry weight, leaf area or stem length, reaches a maximum between 21 and 23 °C for vegetative sweet pepper (Bakker and Van Uffelen, 1988) and around 25°C for vegetative tomato plants (Went, 1944; Hussey, 1965). Optimal temperature increases with light level and decreases with plant age (Went, 1945 ). It also increases with CO2 concentration (Acock et al., 1990). Day and night temperatures do not have exactly the same effect. For example, total dry weight of vegetative tomato plants, as well as leaf and truss number, are more affected by day than by night temperatures (Hussey, 1965; Heuvelink, 1989 ). It was also found that height and leaf area of sweet pepper increase with temperature amplitude (Bakker and Van Uffelen, 1988). In the reproductive phase, dry matter accumulation, stem elongation and fruit setting increase when day temperature is higher than night temperature (Went, 1944; Gent, 1984). The best difference between day and night temperatures may be as high as 6 K (Verkerk, 1955 ) and even 9 K (Gent, 1988 ), resulting in fast fruit growth and early ripening. Mean fruit weight decreases with increasing temperature amplitude, probably owing to better fruit setting and faster fruit ripening (De Koning, 1988; Gent, 1988). However, temperature amplitude did not significantly affect the fresh weight of vegetative sweet pepper (Bakker and Van Uffelen, 1988 ). Several experimental studies (Langhans et al., 1981; Hurd and Graves, 1984; Challa and Brouwer, 1985; Heuvelink, 1989; De Koning, 1990) indicated that plants can tolerate a certain variation about the optimal temperature without negative effects, provided that the deviations of the temperature and of the temperature integral are within bounds. Many growth and development variables have been shown to depend on the temperature regime. This paper will focus on the plant's carbon balance (expressed in CO2 equivalent; 44 kg CO2 is equivalent to 12 kg C or approximately 30 kg dry matter). Following the accepted current view, gross photosynthesis (here, including photorespiration ) produces carbohydrates, which are temporarily stored in a buffer (carbohydrate pool). These, in turn, feed structural growth and respiration. This system is controlled by external variables: light, CO2 concentration and temperature, the latter affecting all carbon flows. There are also internal controls: gross photosynthesis may be restricted by over-accumulation of carbohydrates, generally as starch in chloroplasts (Foyer, 1988; Acock et al., 1990), while growth respiration is restricted by low carbohydrate levels (Gent and Enoch, 1983; Gary, 1988). This view of the crop system, as including an operational carbohydrate pool, explains intuitively the effect of certain strategies of temperature control, such as split-night temperature (ChaUa, 1976, 1978; Gent et al., 1979 ) or light and CO2 dependent temperature control (Calvert and Slack, 1974; Gary, 1989a; Acock et al., 1990). In the present paper we attempt to show how a crop model which contains a carbohydrate pool, can be used to obtain optimal day and night tem-

58

L Seginer et aL /Scientia Horticulturae 60 (1994) 55-80

peratures for a greenhouse crop. As a first attempt, a periodic environment is considered. Under such conditions the state of the carbohydrate pool is nearly periodic, which enables a simplified analysis. In the present study operating costs are ignored and optimization implies maximization of production. The motivation for this study stems from the need to optimize greenhouse temperature control. There are certain periods of time (e.g. under bright sunlight and low wind, or at night under a thermal screen) when it is cheaper to maintain high temperatures than it is at other times. If the crop has the claimed temperature integration capability, this characteristic could be exploited to obtain higher yields at lower costs. At a later stage a greenhouse model may be added to the crop model of the present paper, to enable economic optimization of the combined system's environmental control. 2. Basic model

2.1. Overall

A schematic representation of the assumed chain of events is shown in Fig. 1. This model has two state variables, c and s. The carbohydrate pool, c, is an operational storage (buffer), while structural biomass, s, normally increases monotonically. The total biomass content of the crop is, hence, w = s + c . Structural biomass can be broken down further into different organs, such as leaves and fruit. Three main carbon fluxes are considered: gross photosynthesis, P, maintenance respiration, Rm, and growth respiration, Rg (all expressed here per unit ground area). For a given crop size and as an approximation, photosynthesis is gross pl]otosynthesis

carbohydrate pool

C

F~m

n~antenance respLratLon

Rg

growth resp(ratLon

dS

dt S

structural matter

Fig. 1. A simple crop model with two state variables, c and s, and three basic rates (fluxes) P, R m and

gg.

L Segineret al./Scientia Horticulturae60 (1994) 55-80

59

assumed to depend only on light level, ignoring temperature and C O 2 concentration effects. Maintenance and growth respiration are regarded as functions of temperature and crop size, and the flux of biomass into the structure of the crop is considered proportional to growth respiration. We assume further that the carbohydrate pool has a limited capacity and that photosynthesis and growth respiration are restricted only when the carbohydrate pool reaches its limits. There is no gradual feedback effect on either flux as long as these limits are not reached. The characteristic time of the pool (the integration time) represents the ratio between the size of the pool and the characteristic net flow of carbohydrates. It is, therefore, supposed to depend on the size of the pool relative to the size of the crop, as well as on the environmental conditions. It is reasonable to expect that at any stage of the plant's life the pool is large enough to balance day and night activities for at least one bright day. In other words, the pool is expected to have a characteristic time of at least 1 day. 2. 2. Elements The proposed crop model is probably the simplest which still retains the intrinsic characteristics of the problem. It contains, nevertheless, a sufficient number of parameters (coefficients) to enable a reasonable tuning. The reference parameters in this study were obtained from results of experiments with tomato. Our simplified model assumes that the various rates (fluxes) are independent of the state of the carbohydrate pool as long as 0 < c < Cx,where Cxis the maximum capacity of the pool. In this range ofc the rates are as follows. ( 1 ) Photosynthesis rate (adapted from Van Keulen et al., 1982) is given by P{I,L}-dp/dt=H[I/(I+K)]

(1 - e -aL)

(1)

where P is gross photosynthesis rate, p is its integral over time, I is flux of light ( P A R ) , / / a n d Kare constants and L is the leaf area index (LAI) of the crop. The term l - - e - a L may be viewed as the fraction of solar radiation intercepted by the crop. As L-~0, 1 --e-aL-~aL. For horizontal leaves one would expect a = 1. A more elaborate model might include the effects of temperature and CO2. (2) Maintenance respiration, which is a monotonically increasing function, f{ T}, of temperature, and proportional to structural biomass, is represented by R m { T,s} =f{ T}s

(2)

where Rm is the maintenance respiration rate, T is temperature, and f{ T} > 0. (3) Growth respiration, with a similar temperature response, and proportional to effective ground cover, is given by Rg { T,L } =Of{ T} (1 - e -aL)

(3)

where Rg is growth respiration rate and 0 is a constant coefficient. Growth respiration is the loss of carbon during the process of upgrading simple carbohydrates to structural biomass. It is considered to be proportional to the rate of growth ofs

L Seginer et al. / Scientia Horticulturae 60 (1994) 55-80

60

ds/dt=TRg

(4)

where 1' is a constant coefficient. Structural biomass, s, is related to the LAI, L, through the definition of leaf area ratio (LAR), to

(5) Observation and experimental evidence indicate that to decreases as the crop matures, mainly owing to increased fruit biomass. An example (Bertin and Gary, 1993 ) is presented in Fig. 2, where tomato LAR is shown to decrease with plant age, here expressed in terms of the LAI. Note that this relation depends on plant spacing, here 2.5 plants m -2 ground. From Eqs. ( 3 ) - ( 5 ) one finds that, provided the carbohydrate pool is never empty, ds/dt is proportional to (1 -e-a~°s). For small s (before any fruit develops and to is practically constant with time), this formulation makes ds/dt proportional to s, implying that the growth of small plants is exponential. As the plant grows and L increases, ds/dt approaches a constant. An exponential growth followed by a constant growth rate is exhibited by practically all plant communities, which seems to support the formulation of Eq. (3). There is also some experimental evidence that respiration rate decreases exponentially with depth in the canopy, in a similar manner to photosynthesis (Amthor, 1989). This may mean that older organs participate to a lesser extent in the growth process, just as in photosynthesis. This, in turn, seems to justify making P/R~ (Eq. ( 1 ) divided by Eq. (3) ) independent of crop size, L. Finally, from a philosophical point of 25,

f

/

LAR = 3 + 20 e x p l - 0 . 9 2 LAI I

vl0 r~ .<

0

o.o

o.'s

,.'o

1.'s 2.'o

s.'o

3.'s

4;o

4.5

LAI ( m Z / m z) Fig. 2. The decrease of leaf area ratio (LAR) with the increase of leaf area index (LAI) for a tomato crop (data of Bertin and Gary, 1993).

L Segineret al./Scientia Horticulturae 60 (1994) 55-80

61

view, one would expect the fluxes into and out of the carbohydrate pool to be balanced, leading, again, to R, being proportional to ( 1 --e-aL). The upper constraint, namely the size of the carbohydrate pool, Cx, is assumed to be of the form

Cx=~(1-e -aL)

(6)

which, for young plants, reduces to (7)

Cx= EaL

Note that Cxis made to be proportional to the effective ground cover ( 1-e-aL), just as P and Rg. For a vigorously growing crop, where maintenance respiration is small compared with the other fluxes, this assumption leads to a characteristic time of the pool which is invariant with age. 2.3. State equations and constraints

As w = s + c , the overall carbon balance of the crop becomes d w / d t = ds/dt + dc/dt = P - Rm - Rg

(8)

Substituting from Eq. (4) into Eq. (8), one obtains the carbohydrate-pool dynamics d c / d t = P - R m - (7+ 1 )Rg = P - Uf{ T}

(9)

where

U{L}-s+

1)0(1-e-aL)

(10)

U is a property of the crop, not of the environment, and increases as the crop ages. Eqs. (8) and (9) are the equations of state of this two-state-variable dynamic model. When c is at its upper constraint (c= Cxand dc/dt = 0), photosynthetic inflow to the carbohydrate pool becomes limited, since it cannot be higher than the growth and respiration outflow (sink). Hence P=Rm + (Y+ 1 )R, = Uf{TL}

( 11 )

When c reaches its lower constraint, c=0, again dc/dt=O and there are two situations. If Rm < P, the difference, P - R m , is devoted to growth. If Rm > P, Rm is supported, as much as possible, by photosynthesis, the rest coming from the degradation of structural material, s. The latter situation has been observed particularly under low light and high night temperature conditions (Challa, 1976; Gary, 1989b). We assume that the conversion of carbohydrates to structural matter and back again does not involve any energy loss. Note that nowhere in the formulation of the model is an optimal temperature explicitly specified.

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I. Seginer et al. / Scientia Horticulturae 60 (1994) 55-80

3. Stationary problem

3.1. Possible cases

We shall now assume that light and temperature are periodic step functions of time. At the beginning of each cycle light is turned on at a level I and temperature is set to TL. Following a period tL, light is turned off ( I = 0 ) and temperature is set to TD. Following an additional period tD, the cycle repeats itself. For notational convenience the rate d c / d t during the light period is denoted by (12)

~{I, TL,L} = P - Uf{ TL}

and during the dark period by (13)

--rI{ TD,L } = -- Uf{ TD }

If the environmental conditions are periodic (repeat themselves each cycle), a simulation based on Eqs. ( 8 ) and (9) would reach, after several cycles, a nearly periodic trajectory of c, such as that shown schematically in Fig. 3. A mature crop may reach a true periodic behavior for s as well, since, as is the practice with some greenhouse crops, mature fruit and old leaves are removed at the same rate as new fruit and leaves are produced. In this way s is reduced by As before the next environmental cycle starts. The R o m a n numerals in Fig. 3 denote four out o f the five possible behavior categories (associated with phases in the cycle), as enumerated in Appendix 1. The figure shows only one out o f four possible quasi-stationary solutions. All four possible cases are shown in Fig. 4 in terms of the carbohydrate pool, c.

3.1.1. Case 1 (tL > tltD

rltD < Cx

(14)

The trajectory o f c reaches only the upper constraint (during the light period).

II

I

IV

V

P

T

!

i

respLratLon Loss over cycle

w ]_

~ t t ~

t L-

tb

cycle

t o- -

Fig. 3. Schematic variation of state variables over an environmental cycle. The carbohydrate pool returns to its initial state, while w accumulates material, tt is time to hit top constraint and tb is time to hit bottom constraint of pool.

I. Seginer et al. /Scientia Horticulturae 60 (1994) 55-80

63

o~s~ I

l (->

Fig. 4. Four possible solutions to the stationary periodic case, in terms of carbohydrate content, c.

Production is sink limited (insufficient growth due to low temperature integral relative to available light). 3.1.2. Case 2

(tL > Cx r//D> Cx

(15)

The c trajectory touches both constraints (upper during the light period, and lower during the dark period). Production is storage limited (small Cx relative to temporary storage requirements). 3.1.3. Case 3 ~tL
~tL < r/to

(16)

The c trajectory touches only the lower constraint (during the dark period). Production is limited owing to a high respiration rate (high temperature integral relative to available light). This may also be viewed as source limited production (low light compared with prevailing temperature ). 3.1.4. Case 4

(=0

(17)

The c trajectory stays on c= 0. Photosynthesis cannot support the growth potential even during daytime. This is a special (extreme) subset of Case 3.

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L Seginer et al. / Scientia Horticulturae 60 (1994) 55-80

3.2. Regions in the TL vs. To plane Fig. 5 shows the location, in the TL vs. To plane, of temperature combinations leading to the four cases just mentioned. This particular figure assumes that f{ 7 3 is a linear function of T, but the derivations in this section are still general. The figure clearly shows a production ridge (hereafter termed 'the ridge' ) extending from Point A 'upwards', toward lower dark-period and higher light-period temperatures (hereafter termed "dark and light temperatures' ). It divides the regions representing Cases 1 and 3 (hereafter termed 'Regions 1 and Y ). Point A, which is common to Regions 1, 2 and 3, will be referred to as 'the triple point'. Region 4 has a finite border with Region 3. Inspection of Fig. 4 shows that the borders are given by Regions 1 and 2: r/tD=Cx, which leads to TD=COnSt. Regions 2 and 3: (tL=Cx, which leads to TL=COnSt. Regions 3 and 4: ( = 0 , which leads to TL=COnSt. Regions 1 and 3: (tL=r/tD, which leads in general to a non-linear relation between TL and To along the ridge. f{TD} = (P/U-f{TL}) (tL/tD)

(18)

From the preceding list of borders it follows that the triple point must satisfy the two conditions , RGI~.=0.561 • / d

0

10 20 dark temperature,

30 T D ~C)

40

Fig. 5. Relative growth rate ( R G R ) , assuming linear respiration response, over a range of light and dark temperatures. Contour lines of Lts/s in percent per environmental cycle (day) for a mature crop under winter conditions. LAI = 5 m 2 m - z tL = 9.9 h, tD = 14.1 h, I = 430/~mol photons m - 2 ground s -1. The triple point ( T r n = 11.6°C, T D , = 2 0 . 2 ° C ) is indicated by A. Four different regions are indicated by numbers: ( 1 ) sink limitation; (2) storage limitation; (3) excessive respiration or source limitation; (4) extreme source limitation. Distance between contour lines is 0.1% day -I. Dashed lines in Regions 2, 3 and 4 represent negative RGR. The last solid line is R G R = 0.

L Seginer et aL/Scientia Horticulturae 60 (1994) 55-80

qtD = OL ~" Cx

65 (19)

leading, together with Eqs. (12) and ( 13 ), to

(P-Cx/tL)/U f{ ToA}= (Cx/tD)/U

f{ TLA} =

(20) (21)

Eq. (20) insures that by the end of the light period the carbohydrate pool is exactly full and Eq. (21) insures that by the end of the dark period it is exactly empty. The triple point may be located either above or below the TL= To line. To see this, consider the ratio f{ TLA} P--Cx/tL f{TDA }-- cx/tD

(22)

As f{ T} is an increasing function of T, TEAis greater than TDAwheneverf{ TEA}/ f{ TDA}> 1, and vice versa.

3.3.Effectofdaylength The climatic cycle is defined in terms of two time parameters: the total length of the cycle, t t - t L + to, and the ratio, p-tL/tD, between the lengths of the light and dark periods. Expressed in these terms the coordinates of the triple point (Eqs. (20) and (21))become

f{TLA}=u[P-(l+p)cx]prt _1

(23)

f{TDA}=I--u~(17)Cx I

(24)

From these equations one can see that, as p increases (from winter to summer), both f{ TLA} and f{TDA} increase. Since we only considered monotonically increasing functions f, the increase ofp leads to an increase of both light and dark temperatures. This means that longer days (more specifically, larger light integral), require higher temperatures to prevent the pool from limiting the growth. Fortunately, it so happens that a positive correlation between day length and temperature is the rule in our world. The effect of the period, ~t, is only interesting from a theoretical (laboratory) point of view. According to Eqs. (23) and (24), an increased period results in an increase of TEA and a decrease in TOA.In fact, as rt increases, the triple point moves up along the ridge, the remaining ridge being a section of the original one (this can be shown by eliminating ~ between Eqs. ( 23 ) and (24 ), the result being identical to Eq. (18) ). Longer periods mean processing larger batches. This scaling up would require a larger temporary storage (carbohydrate pool), or, in practice, higher light temperatures. The latter solution results in a shorter ridge.

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66

3.4. Growth rate and efficiency Total production during an environmental cycle (day) for any temperature pair TL and TD is given by the change ofs (or w) over the cycle. For Case 2 (Fig. 3) it is

zls=TR,{TL}Cx/~+rR,{TL}(tL --Cx/~) + TR,{TD}Cx/rl --TRm{TD}(tD--Cx/rl)/(7+l)

(25)

This is a sum of products of rates and times over the four stages of that case. The rates are taken from Appendix 1 and the duration of each stage is given by the expressions in Appendix 2. Similar formulae apply to the other cases. It turns out that the production along the ridge is constant. All points along the ridge satisfy the condition ~tL=qtD (previous section), which implies that the carbohydrate pool is never completely full or empty for a finite period of time. Therefore, the pool does not restrict photosynthesis, nor growth, and the only loss is due to maintenance respiration, which, over one cycle, is SRmdt= If{ TL}tL +f{TD}tD]S

(26)

Substituting from Eq. ( 18 ), which is satisfied by points along the ridge, into Eq. (26), leads to

~R mdt = etLs/U

(27)

which SHOWSthe loss, and hence the production, to be constant along the ridge. This production, easily calculated for the triple point, is

As= TR~{ TLA }tL + TRg{ TDA}tD

(28)

We have seen (Eq. (22)) that the triple point may lie either above or below the TL= To line, depending on whether P is large or small, respectively, compared with Cx. In other words: the model indicates that when photosynthesis is large compared with the carbohydrate pool (summer), daytime temperature should be higher than nighttime temperature. Under low light conditions (winter), most of the ridge is still above the TL= TD line, but the part close to the triple point is below the line. Therefore, higher night than day temperatures, provided that they still lie on the ridge, are expected not to reduce production. For comparisons with available experimental data, it is convenient to express production in terms of relative growth rate (RGR), defined as RGR = As/s

(29)

If the ridge extends across the TL= TD line, then, for the point of constant temperature, production over the cycle is balanced by maintenance and growth

etL = [Rm + ()'+ 1 )Re] (tL + tD)

(30)

L Segineret aL/ScientiaHorticulturae60(1994)55-80

67

Substituting from Eq. (4) for Rg, replacing ds/dt by As/( tL+ tD ) and dividing by s, Eq. (30) leads to

AS tL-bt D 7 ( PtL S --

S

7+ l \ t ; + t o

) Rm.

(31)

For vigorously growing plants (small plants in high light)

PtL/(tL -I-tD) >> Rm

(32)

SOthat Z~S 7 PtL s 7+ 1 s

(33)

which shows that for this case RGR is, to a first approximation, a function only of photosynthesis-associated parameters. It follows that when data for vigorously growing plants are used to estimate model parameters, 7 and the parameters of P{/} may be estimated almost independently of the parameters off{ T}. This is a convenient property of the model. Overall growth efficiency, e (Y in Thornley and Johnson, 1990), is defined as the ratio between structural growth and gross photosynthesis. Two measures are possible. The first, in terms of carbon balance

ec=-As/Ap

(34)

and the second, in terms of energy balance. Since we assumed no energy loss in moving between c and s eE = (7+ 1)As/(yAp)

(35)

The ratio between the two is constant, namely eE/ec= (7+ 1 )/7. The energybased efficiency has, however, the advantage that its limits are 0 to 1, while ec is bounded from above by 7/(7 + 1 ). Photosynthesis is divided into structure and maintenance as

zip= (7+ 1)As/7+ f Rmdt

(36)

hence eE= 1/(1-t 7fR_mdt~ (7+ 1)As]

(37)

Substituting from Eqs. (26), (28) and (3), one obtains, for points on the ridge L

eE= 1/(1-I O90(7+ 1) (1--e

--aL))

(38)

which is a function of L alone. This is an interesting prediction of the model.

3.5. Linear and exponential respiration Two simple respiration functions may now be considered: ( 1 ) Linear

f { T } = u + I,T

(39)

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L Seginer et al. / Scientia Horticulturae 60 (1994) 55-80

(2) Exponential

f~{T}=ae ~

(40)

where/2, v, a and fl are constant coefficients. Figs. 5 and 6 may be used to compare the consequences of these two formulations. The figures were obtained with the reference parameters of Appendix 3. Since a and p, as well as # and v were obtained from the same experimental data at 20 oC, good agreement between Figs. 5 and 6 at about this temperature should be expected. The linear formulation (Fig. 5) produces a straight ridge between Regions 1 and 3. The ridge, sloping at - t o / h . , follows a constant mean temperature (same temperature integral) line, as can be shown by substituting Eq. (39) into Eq. ( 18 ). This mean temperature is

f._

PtL _It vU(tL +t~,) v

(41)

A similar ridge direction is suggested by the results of Seginer and Raviv (1984). As can be seen in Fig. 5, all production contour-lines in Regions l, 3 and 4 are parallel to the ridge, meaning that in these regions production is only a function of the temperature integral. Moving along a constant temperature integral line into Region 2 results in production loss. For crops with lower Cx, the triple point is located higher on the ridge, increasing Region 2 at the expense of Regions 1 and 3. The ridge shortens, which means that the choice of temperature combinations for maximum production decreases. All lines parallel to the line TL= TD F______/ RGR=0.564 %/d 40

-

~

-

~

/""

4

E- 30

,/ /f// '

0

l0 20 30 40 dark t e m p e r a t u r e , TI) (*C)

Fig. 6. Same as Fig. 5, except for an exponential respiration response. RGR contour-lines in the negative production region (dashed lines in Fig. 5 ) were not plotted. Last solid line is RGR=0.

L Seginer et al./Scientia Horticulturae 60 (1994) 55-80

69

have the same maximum deviation from the temperature integral of the corresponding (same mean temperature ) constant temperature regime. Fig. 6, which was obtained with the exponential respiration model, is topologically identical to Fig. 5. The main difference is that the ridge, and all production contour lines, are now curved. As a result, constant temperature integral lines are no longer production contour lines. Note that the production ridge extends to unrealistically low dark temperatures. This is a serious deficiency of the model, which lacks a production loss mechanism due directly to extreme temperatures.

3. Results and discussion

As the literature review in the Introduction showed, quite a few experiments with tomato plants (mostly seedlings), were carried out to study particular growth aspects of this crop. Data collected in some of these experiments were used to compile a list of reference parameter values (Appendix 3 ). Some of the parameters, such as 7 and fl, were quoted directly, while parameters which are peculiar to the present model, such as ¢ or 0, had to be evaluated indirectly. The process of evaluation of the latter parameters is briefly described in Appendix 3. Figs. 5 and 6, for a mature crop under winter conditions, were obtained with these reference parameters, producing what seems to be a reasonable description around the center of the figures. In the following sub-section we find that fitting the model to a particular data set, required adjustment of a few of the parameters. Since the effects of crop age and light regime are only lightly coupled in this model (Eq. (33) ), the adjustment of parameter values is rather simple.

3.1. Growth chamber experiments A few experimental data exist which may be compared with the results of the model. Tomato RGR data of Hussey ( 1965 ), Gent (1986) and Heuvelink (1989) were used to compile Figs. 7 and 8 (carbohydrate content is normally not available from studies on thermoperiodicity). In these experiments small plants were grown in growth chambers at low light integrals of about 4.5 mol photons m -2 ground day -~. According to our previous analysis (Eq. (22)), and in conjunction with the reference parameter values, one would expect the triple point to lie below the line of uniform temperature for this low light level. The data of Figs. 7 and 8 indicate, however, that maximum production was obtained at a constant temperature of about 21 °C. Since the reference parameters would not predict the observed data, some of them had to be adjusted. Following the selection of the exponential respiration formulation (to be justified in the next subsection), two parameters were changed: First,//was slightly increased to yield a proper RGR level. Next, ¢ was reduced considerably to bring the triple point to the TL= TD line (Appendix 3 ). It turns out that changing parameter values within the range of uncertainty, has a substantial effect on the

70

I. Seginer et al. /Scientia Horticulturae 60 (1994) 55-80 0.25

0.20 ÷El /

0.15

c3/. /

Region

3

0.10

J

J J

Region

1

0.05

0.00

0

5 o

10

15

mean H u s s e y 65

20 25 30 temperature (~C) + H e u v e l i n k 89 ~

35

40

G e n t 86

Fig. 7. RGR response of tomato seedlings to different levels of constant temperature. Experimental results from three sources and line describing fitted model. 0.25 ~ ridge 0.20

0.15

~" 0.10

0.05

0.00

0

5 o

lb

t~

dark

temperature,

H u s s e y 65

20 +

25

30

35

40

TD 02) H e u v e l i n k B9

Fig. 8. RGR response of tomato seedlings to different temperature combinations at the same temperature integral. Mean temperature is about 21 °C. Experimental results from two sources and line describing fitted model.

location of the triple point and on the production slopes in the various regions. Since extracting parameters from different experiments produces considerably diverse estimates, one should not expect perfect fit to independent (new) data. For the very crude model of the present study and the limited experimental data available, one should, perhaps, be pleased with model predictions which are the right order of magnitude and which mimic the correct trends (as Figs. 7 and 8 do).

L Seginer et al./Scientia Horticulturae 60 (1994) 55-80

71

Two cross-sections of the (light and dark) temperature plane are shown. Fig. 7 is for TL= TD, and Fig. 8 is for a constant mean temperature of about 21 °C. The RGR data from all sources seem to agree fairly well among themselves. They show considerable loss of production at low mean temperatures (Fig. 7), presumably because of insufficient withdrawal from the carbohydrate pool (Region 1 ). They also show reduced production when TD > TL (Fig. 8 ), which may be attributed to limiting storage capacity (Region 2). Fig. 8 also indicates some (less pronounced) loss of production at To << TL, which one would expect, but the model is unable to predict (since the ridge is unlimited). Note the mild slope in Region 3 of Fig. 7, compared with the corresponding slopes in Figs. 5 and 6. This striking difference is due to the considerably higher LAR of small plants compared with mature plants. Note that the experiments from which the data of Figs. 7 and 8 were taken, are integrated level experiments. Agreement of these data with the predictions of our model are not necessarily a proof of its validity. Direct studies of carbohydrate pool behavior are required to validate the model.

3.2. Typical greenhouse situations It should be clear by now that the triple point is the central pivot around which the different regions are arranged. Its location is given by Eqs. (20) and (21), and hence is affected in a rather simple way by the light regime and by the size (age) of the crop. Note that P{LL} ofEq. ( 1 ) may be written as P{I}cx{L}/e. As a result, light regime is represented by P{I}, tL and to, while the size of the crop is represented by U{L} and cx{L}. Let us explore, on this basis, what might be reasonable temperature policies for particular combinations of crop size and season. Mean monthly global radiation and daylength data for Israel are given in Table 1. It was assumed that 0.7 of this radiation is available at the top of the canopy inside the greenhouse. The available light was divided uniformly over the light hours, to fit model requirements. Triple point locations for two crop sizes, L = 0.5 m 2 m -2 and 5 m 2 m -2 (young and mature), were calculated with the exponential respiration model (with reference parameter values) and plotted in Fig. 9. The resulting two heavy-line loops are identical, except for a shift parallel to the uniform temperature line. The loops are very thin, owing to the high correlation between global radiation and length of day. Note that Fig. 6 describes the production map around the lower point of the mature-crop loop. Table 1 Global radiation in Israel Month

Jan.

Feb.

Mar.

Apr.

May

Jun.

Jul.

Aug.

Sep.

Oct.

Nov.

Dec.

Radiation (MJ m -2 day -1 ) Daylength (h)

12.6

15.7

19.8

24.7

28.6

31.3

30.7

28.3

24.6

19.2

t3.7

11.0

10.2

10.9

11.8

12.8

13.7

14.1

14.0

13.2

12.3

11.3

10.4

10.0

72

I. Seginer et al. / Scientia Horticulturae 60 (1994) 55-80 4.0 35

# 30 -

:""---.~

I young crop ,j winter

lila[u



1-e

crop

15 ! 10

10

, 15 dark

3'5

20 25 30 temperature TD ~C)

40

Fig. 9. Location o f triple point for young and mature crop, for an annual light cycle in Israel. Results obtained with the exponential respiration model and the reference parameters. The cross is referred to in the text.

Several features of Fig. 9 are of interest. ( 1 ) The loop for the young plants is shifted by about 11 K, in both TL and To, towards higher temperatures, compared with the mature plants. The shift may be explained by the larger LAR of the young plants. Much of the biomass of the mature plants (fruits and stems) does not contribute to photosynthesis, but at the same time requires maintenance energy. As a result, maximum production of the mature crop is obtained at lower temperatures. This is qualitatively in agreement with the results of Went ( 1945 ), who found that optimal temperatures decrease with plant age. The magnitude of the shift seems, however, too large. In terms of the present model this may be due to incorrect values of a and 0. More discussion of this point follows the introduction of Table 2 below. (2) Summer triple points are above the TL= To line, namely TEA> TDA, as expected (Eq. (22)). The opposite is true during winter. In mid-winter, triplepoint night temperature is about 8 K higher than day temperature (equivalent to temperature integral deviation of about 50 Kh). While quantitatively this difference may be unrealistically large, the trend is probably correct, and can be exploited for optimization of greenhouse heating during cold winters: it is considerably less expensive to increase greenhouse temperature at night under a thermal screen than during daytime, when the screen is open (to let in light). As a result, operating on ridge-points close to the triple point is less expensive and as productive as operating further away along the ridge. Optimization studies with constraints on the temperature integral (Gutman et al., 1993) point to the same solution. (3) Optimal mean temperatures are about 10 K higher under summer conditions than in mid-winter. This is in qualitative agreement with the findings of Went ( 1945 ), who showed that optimal temperature is higher for high light con-

L Seginer et al./Scientia Horticulturae 60 (1994) 55-80

73

Table 2 Four extreme combinations of light and crop size. Results obtained with standard parameters (Appendix 3). All data except triple point coordinates are the same for both exponential and linear respiration models LAI

Light regime

L

I (pmol m-2) m - 2 s - I )

Mature crop Winter Exponential 5.0 linear Summer Exponential 5.0 Linear

TDA (°C)

T AS dS/S (°C) (gm -2 (%day -1) day - l )

ec (%)

eE (%)

430

9.9 22.7 24.1

31.2 43.6

27.7 35.6

6.4

19.9

69

93

866

14.1 38.0 69.5

36.2 61.5

37.2 66.2

11.9

37.1

69

93

9.9 11.6 20.2 11.2 20.2

16.6 16.5

8.8

0.6

32

43

26.2 30.8

16.4

1.1

32

43

430

866

tL (h)

Production along ridge

TEA (°C)

(m 2

Young crop Winter Exponential 0.5 Linear Summer Exponential 0.5 Linear

Triple point

14.1 26.9 32.3

25.2 28.6

ditions. If the model had considered the effect of elevated C O 2 concentration, the resulting increased photosynthesis would require a higher temperature level to promote growth (Acock et al., 1990). This trend may be exploited by growers in warm climates: higher permissible temperatures allow longer enrichment intervals. (4) Within each loop, the range of TD between summer and winter (5 K) is considerably smaller than the range of TL ( 15 K). This may be explained by Eqs. (23) and (24), which show that the effect of photosynthesis compounds the effect of p on TEA, but not on Tog. At its face value this result seems similar to actual practice. It is important to remember the existence of the ridge, which permits efficient operation at higher day temperatures and lower night temperatures than indicated by the triple point loops. As an example, consider a tomato crop planted in winter and maturing in summer. Two limiting ridges are shown in Fig. 9, one for the young crop in winter and one for the mature crop in summer. If a policy represented by the cross in Fig. 9 (TL= 30 ° C, T o = 24 °C) is applied during the whole season, production may be almost as good as if the triple point, or a point on the ridge, were accurately followed. Table 2 summarizes the results for the four combinations of young and mature crop under extreme winter and summer conditions (four extreme points on loops of Fig. 9 ). In addition to the triple point information (Fig. 9 ), it shows produc-

74

L Seginer et al. / Scientia Horticulturae 60 (1994) 55-80

tion data. The latter data are identical for the linear and exponential respiration, and are shown only once. The table shows that the linear respiration (in conjunction with the reference parameters), produces an extremely large difference (about 35 K), between triple point temperatures of young and mature crop under summer conditions. The exponential respiration reduces this difference to a third (11 K), because the slope off,{T} (Eq. (40)) is steeper than that o f f ( T } (Eq. (39)) in this range. Smaller maintenance respiration (smaller or) could further reduce the difference between young and mature crops. Reducing t~ would, however, require increasing 0, to maintain sufficient growth at comfortable temperatures. As additional respiration data become available, it may well turn out that older tissue, as well as fruit, respire less than young leaves (Amthor, 1989; Jones et al., 1991, eq. 18 ). In that case, Eq. (2) could, perhaps, be replaced by

Rm{T,s) =f{T} ( 1 - e -L)/to0 + f ' {T) [S-- ( 1 --e -L)/to0 ]

(42)

where tOo is the LAR of young plants (23 m 2 leaf kg-1 CO2 in Fig. 2 ), and f ' { T} is the respiration function of mature tissue and/or fruit. The first term on the right represents the respiration by the active (young) parts of the plant and the second term represents old tissue and fruit. Note that as L ~ 0, Rm approaches Eq. (2). If one assumes f ' {T)
4. Summary and conclusion Our carbohydrate pool model is obviously very rudimentary. Nevertheless, it produced some reasonable qualitative and order of magnitude results. The following summarizes the impressions obtained in the preceding sections for the stationary problem. ( 1 ) The model, using experiment-based parameter values, produces reasonable order-of-magnitude relative growth rates and temperature regimes. One conspicuous exception seems to be the large difference between the optimal temper-

L Seginer et al./Scientia Horticulturae 60 (1994) 55-80

75

atures for young and mature plants (Fig. 9 and Table 2 ), which can be reduced by adjusting two of the model parameters. (2) When relative growth rate (RGR) is plotted over temperature space, TL vs. I'D, a ridge of maximum production is found. At lower light and dark temperatures (Region 1, Figs. 5 and 6) production is limited by insufficient removal from the carbohydrate pool (sink limitation), while at high light and dark temperatures (Regions 3 and 4) production (for a given light level) is curtailed by excessive maintenance respiration. The ridge has one end, the triple point. Moving beyond the triple point into lower light temperatures and higher dark temperatures (Region 2 ), results in reduced production due to insufficient carbohydrate buffer space. (3) The ridge permits a choice from a range of temperature combinations for any given light period. This range decreases as total daily production increases (summer), requiring higher temperatures to prevent the carbohydrate pool from becoming the limiting factor. (4) Optimal mean temperature increases with photosynthetic production, whether this is due to higher light integral or higher CO2 concentration. It also increases with leaf area ratio (LAR), leading to reduced optimal temperature for a mature crop. Longer environmental cycles (under artificial conditions) would also lead to higher optimal temperatures. The most obvious deficiency of the model is that the production ridge, which presumably indicates the best temperature regimes, covers some clearly unacceptable combinations. The production of real plants suffers from exposure to extreme temperatures for reasons that are not related to the carbohydrate pool, leading to chill and heat injury. As an example, Longuenesse (1982), working with young tomato plants, measured similar net photosynthesis rates after one night at 20, 14 or 10°C. Net photosynthesis dropped, however, by about 30% after one night at 7 ° C. These effects are not included in the present model. Restricting the acceptable recommendations to a 'comfort zone', such as 10 ° C < T< 35 ° C, is an incomplete solution to the problem. Adding chill and heat injury as an intrinsic element of the model may correct this deficiency. Adding CO2 concentration explicitly into the photosynthesis model could also increase the utility of the model. We see the main significance of this study in helping to bridge the gap between physiological laboratory research and horticultural engineering applications. Several qualitative operational guidelines, which are compatible with horticultural experience, have already emerged. For example, the advantage of higher summer temperatures or recommending higher temperatures for small plants and plants subject to CO2 enrichment. The observed equivalence of production under certain temperature combinations (often same mean temperature) has been explained by the ridge in the TL vs. TD plane. The study also explains why high day and low night temperatures usually produce better results than the inverse combination, and why, under winter conditions, higher night temperatures may not result in reduced production. The point of view of this study could also stimulate certain laboratory experi-

76

L Seginer et aL / Scientia Horticulturae 60 (1994) 55-80

ments. We have shown that two diagonal cross sections of the temperature plane (even if not through the triple point) could provide sufficient data for model calibration. In addition, growth chamber experiments under unusual day lengths could be devised to test certain predictions of the model. Unfortunately, detailed studies with mature plants are expensive. However, data extracted from young plants may not be representative of mature plants, and therefore an effort aimed at extracting parameter values from mature plants may be worthwhile, despite the required effort.

Acknowledgment This study was supported by INRA and Technion visit grants, as well as by the Fund for Promotion of Research at the Technion. We are grateful to Dr. James Jones and Dr. Martin Gent for their critical review of the paper.

Appendix 1: Rates for five possible behavior categories Five categories of behavior (rates of change) are possible in this problem (four of them are shown in Fig. 3 ). (I) Light period, c not constrained

dp/dt=P{I} ds / dt = TRg{ TL } dc/dt=P{I}-Rm {TL } - (7+ 1 )Rg{TL} dw/dt=P{I}-gm{ TL}-Rg{ TL } (II) Light period, c=c.

dp/dt=Rm{TL}+ (7+ 1 )Rg{TL} ds/dt=TRg{TL} dc/dt = 0 dw / dt = TRg{ TL } (III) Light period, c= 0

dp/dt=P{1} ds/dt=7(P{I}-Rm{TL} )/ (7+ 1) dc/dt = 0 dw/dt=7(P{I}-Rm{TL}

) / ( 7 + l)

L Seginer et al./Scientia Horticulturae 60 (1994) 55-80

77

(IV) Dark period, c not constrained

dp/dt=O ds / dt = TRg { TD } dc/dt= - R m { T o } - (7+ 1 )Rg {To} dw/dt= --Rm { TD }--Rg{ TD ) (V) Dark period, c=O

dp/dt=O ds/dt= -TRm{TD}/ (7+ 1 ) dc/dt=O dw/dt=-TRm{TD}/ (y+ l ) Note that ( 1 ) during the dark period, tD, there is no photosynthesis, (2) in Category II photosynthesis is limited by lack of storage space in c, and (3) in Category V, and sometimes in Category III, structural biomass is reduced to support maintenance respiration. Appendix 2: Times on and off constraints

Time to hit top constraint in Case 1

t, = ~tD/ ~ Time to hit top constraint in Case 2

t, =Cx/~ Time to hit bottom constraint in Case 2 tb =Cx/ll

Time to hit bottom constraint in Case 3

tb =(tL/rl Refer to Figs. 3 and 4. Appendix 3: Parameter values

Note: 44 kg CO2 is equivalent to 12 kg C or approximately 30 kg dry matter.

Reference values Model parameters were extracted from experimental data for tomato.

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L Seginer et al. / Scientia Horticulturae 60 (1994) 55-80

From Bertin and Gary ( 1993 ) ~,= 3.0 /7= 1.5× 10 -6 kg CO2 m -2 ground sK = 4 0 0 /zmol photons m -2 ground s0 9 = 3 + 2 0 e -°92L m 2 leaf kg -~ CO2

(our Fig. 2)

From Van Keulen et al. (1982) a=0.8

m E ground m -2 leaf

From Gent (1984) For young plants (Eq. (7) ) Cx/s = ~09a= O.3 With a = 0 . 8 m 2 m -2 and 09=20 m E kg -~, this leads to E ~ 0 . 0 2 kg CO2 m -2 ground

Accepted value (Q~0= 2) fl= 0.0693 K - l From Jones et al. ( 1991 ) At 20°C, f{T} has a value between 0.12× l 0 -6 and 0.18× 10 -6 kg CO2 kg -1 CO2 s-1. An intermediate value was selected. Assuming/z=0 (zero respiration at 0°C), this leads to linear respiration (Eq. (39)) of p=0.14X10-6/20=7×10

- 9 oC - l

S-1

and exponential respiration (Eq. (40)) of a=0.14X

1 0 - 6 / e ° ° 6 9 3 x 2 ° = 3 5 × 10 -9 s -1

From Gary (1988) R~/Rm = 3.6 o9=14.6 m 2 leaf kg -~ CO2 For small plants, from Eq. (3): R~{T,L}=OaLJ{T}. From this, and Eqs. (2) and (5): O=RJ(a09Rm). 0 = 3 . 6 / ( 0 . 8 ) < 1 4 . 6 ) = 0 . 3 kg CO2 m -2 ground From Jones et al. ( 1991 ), Dayan et al. (1993) and Bertin and Gary (1993) for mature tomato: L ranges from 5 to 8 m 2 leaf m -2 ground, depending on whether old leaves are removed; s is about 1.5 kg CO2 m -2 ground; As is about 0.15 kg CO2 m -2 ground day-1; ds/s is approximately 1% day-1.

L Seginer et al./Scientia Horticulturae 60 (1994) 55-80

79

Modified values to fit growth chamber data ~=0.008 kg C O 2 m - 2 ground /-/=l.8x10 L=0.15 I=100

- 6 kg CO2 m - 2 ground s - l

m E l e a f m - 2 ground

/zmol photons m -2 ground s -1

t L = t D = 12 h

References Acock, B., Acock, M.C. and Pasternak, D., t 990. Interactions of C O 2 enrichment and temperature on carbohydrate production and accumulation in muskmelon leaves. J. Am. Soc. Hortic. Sci., 115: 525-529. Amthor, J.S., 1989. Respiration and Crop Productivity. Springer, New York, 215 pp. Bakker, J.C. and van Uffelen, J.A.M., 1988. The effects of diurnal temperature regimes on growth and yield of glasshouse sweet pepper. Neth. J. Agric. Sci., 36:201-208. Bertin, N. and Gary, C., 1993. Evaluation d'un modele dynamique de croissance et de developpement de la tomate (Lycopersicon esculentum Mill), TOMGRO, pour differents niveaux d'offre el de demande en assimilats. Agronomie, 13: 395-405. Calvert, A. and Slack, G., 1974. Light-dependent control of day temperature for early tomato crops. Acta Hortic., 51: 163-168. Challa, H., 1976. Analysis of the diurnal course of growth, carbon dioxide exchange, and carbohydrate reserve content of cucumber. Agric. Res. Rep. 86 l, Pudoc, Wageningen. Challa, H., 1978. Programming of night temperature in relation to the diurnal pattern of the physiological status of the plant. Acta Hortic., 76:147-150. Challa, H. and Brouwer, P., 1985. Growth of young cucumber plants under different diurnal temperature patterns. Acta Hortic., 174:211-217. Dayan, E., van Keulen, H., Jones, J.W., Zipori, I., Shmuel, D. and Challa, H., 1993. Development, calibration and validation of a greenhouse tomato growth model: II. Field calibration and validation. Agric. Syst., 43: 165-183. De Koning, A.N.M., 1988. The effect of day/night temperature on growth, development and yield of glasshouse tomatoes. J. Hortic. Sci., 63:465-471. De Koning, A.N.M., 1990. Long-term temperature integration of tomato. Growth and development under alternating temperature regimes. Sci. Hortic., 45:117-127. Foyer, C.H., 1988. Feedback inhibition of photosynthesis through source-sink regulation in leaves. Plant Physiol. Biochem., 26(4): 483-492. Gary, C., 1988. Relation entre temperature, teneur en glucides et respiration de la plante entiere chez la tomate en phase vegetative. Agronomie, 8 ( 5 ): 419-424. Gary, C., 1989a. Interest of a carbon balance model for on-line growth control: the example of a daylight dependent night temperature control. Acta Hortic., 248: 265-268. Gary, C., 1989b. Temperature and the time-course of the carbon balance of vegetative tomato plants during prolonged darkness: examination of a method of estimating maintenance respiration. Ann. Bot., 63: 449-458. Gent, M.P.N., 1984. Carbohydrate level and growth of tomato plants. I. The effect of CO: enrichment and diurnally fluctuant temperatures. Plant Physiol., 76: 694-699. Gent, M.P.N., 1986. Carbohydrate level and growth of tomato plants. 2. The effect of irradiance and temperature. Plant Physiol., 81: 1075-1079. Gent, M.P.N., 1988. Effect of diurnal temperature variation on early yield and fruit size of greenhouse tomato. Appl. Agric. Res., 3: 257-263.

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Gent, M. and Enoch, H.Z., 1983. Temperature dependence of vegetative growth and dark respiration: a mathematical model. Plant Physiol., 7 l: 562-567. Gent, M.P.N., Thorne, J.H. and Aylor, D.E., 1979. Split-night temperatures in a greenhouse: the effects on the physiology and growth of plants. Bull. 78 l, Connecticut Agricultural Experiment Station, New Haven. Gutman, P-O., Lindberg, P-O., Ioslovich, I. and Seginer, I., 1993. A non-linear optimal greenhouse control problem solved by linear programming. Agric. Eng. Res., 55:335-351. Hagihara, A. and Hozumi, K., 199 I. Respiration. In: A.S. Raghavendra (Editor), Physiology of Trees. John Wiley, New York, pp. 87-110. Heuvelink, E., 1989. Influence of day and night temperature on the growth of young tomato plants. Sci. Hortic., 38:11-22. Hurd, R.G. and Graves, C.J., 1984. The influence of different temperature patterns having the same integral on the earliness and yield of tomatoes. Acta Hortic., 148: 547-554. Hussey, G., 1965. Growth and development in the young tomato: the effect of night and day temperatures on vegetative growth. J. Exp. Bot., 16: 373-385. Jones, J.W., Dayan, E., Allen, L.H., van Keulen, H. and Challa, H., 1991. A dynamic tomato growth and yield model (TOMGRO). Trans. ASAE, 34: 663-672. Langhans, R.W., Wolfe, M. and Albright, L.D., 1981. Use of average night temperatures for plant growth for potential energy savings. Acta Hortic., 115:31-37. Longuenesse, J.J., 1982. Temperature nocturne et photosynthese. III. Influence de la temperature appliquee pendant une nuit sur les echanges gazeux de la tomate. Agronomie, 2:805-811. Penning de Vries, F.W.T., 1975. Use of assimilates in higher plants. In: J.P. Cooper (Editor), Photosynthesis and Productivity in Different Environments. Cambridge University Press, Cambridge, pp. 459-480. Seginer, I. and Raviv, M., 1984. Optimal night temperatures for greenhouse seedlings. Sci. Hortic., 23: 203-216. Thornley, J.H.M. and Johnson, I.R., 1990. Plant and Crop Modelling. A Mathematical Approach to Plant and Crop Physiology. Oxford University Press, Oxford, 669 pp. Van Keulen, H., Penning de Vries, F.W.T. and Drees, E.M., 1982. A summary model for crop growth. In: F.W.T Penning de Vries and H.H. van Laar (Editors), Simulation of Plant Growth and Crop Production. Wageningen Centre for Agricultural Publishing and Documentation (Pudoc), the Netherlands, pp. 87-97. Verkerk, K., 1955. Temperature, light and the tomato. Meded. Landbouwhogesch. Wageningen, 55: 175-224. Went, F.W., 1944. Plant growth under controlled conditions. 2. Thermoperiodicity in growth and fruiting of the tomato. Am. J. Bot., 31: 135-149. Went, F.W., 1945. Plant growth under controlled conditions. 5. The relation between age, light, variety and thermoperiodicity of tomatoes. Am. J. Bot., 32: 469-479.