A Generic Equation for Nitrogen-limited Leaf Area Index and its Application in Crop Growth Models for Predicting Leaf Senescence

Annals of Botany 85: 579±585, 2000 doi:10.1006/anbo.1999.1104, available online at http://www.idealibrary.com on

A Generic Equation for Nitrogen-limited Leaf Area Index and its Application in Crop Growth Models for Predicting Leaf Senescence X I N YO U Y I N * {, A D H . C . M . S C H A P E ND O N K{, M A R T I N J . KRO P F F}, M A R C E L VA N O I J E N{ { and P R E M S . B I N D R A B A N{ {Plant Research International, P.O. Box 14, 6700 AA Wageningen, The Netherlands and }Laboratory of Theoretical Production Ecology, Wageningen University, P.O. Box 430, 6700 AK Wageningen, The Netherlands Received: 4 October 1999 Returned for revision: 19 November 1999 Accepted: 17 December 1999 Appropriate quanti®cation of leaf area index (LAI) is important for accurate prediction of photosynthetic productivity by crop growth models. Estimation of LAI requires accurate modelling of leaf senescence. Many models use empirical turnover coecients, the relative leaf-death rate determined from frequent ®eld samplings, to describe senescence during growth. In this paper, we ®rst derive a generic equation for nitrogen-determined photosynthetically active LAI (LAIN), and then describe a method of using this equation in crop growth models to predict leaf senescence. Based on the theory that leaf-nitrogen at di€erent horizons of a canopy declines exponentially, LAIN , which is counted from the top of the canopy to the depth at which leaf-nitrogen equals the minimum value for leaf photosynthesis, is calculated analytically as a function of canopy leaf-nitrogen content. At each time-step of crop growth modelling, LAIN is compared to an independent calculation of the non-nitrogen-limited LAI assuming no leaf death during that time-step (LAINLD). In early stages, LAIN is higher than LAINLD ; but with the advancement of crop growth, LAIN will become smaller than LAINLD . The di€erence between LAINLD and LAIN , whenever LAIN is smaller than LAINLD , gives the estimate of leaf area senesced at the time-step; the senesced leaf area divided by speci®c leaf area (SLA) gives the estimate of senesced leaf mass. The method was incorporated into two crop models and the models adequately accounted for the LAI observed in ®eld experiments for rice and barley. The novel features of the approach are that: (1) it suggests a coherent, biologically reasonable picture of leaf senescence based on the link with photosynthesis and leaf nitrogen content; (2) it avoids the use of empirical leaf-turnover coecients; (3) it avoids over-sensitivity of LAI prediction to SLA; and (4) it is presumably of sucient generality as to be applicable to plant types other than crops. The method can be applied to models where leaf-nitrogen is used as an input variable or is # 2000 Annals of Botany Company simulated explicitly. Key words: Leaf area index, leaf senescence, canopy nitrogen, modelling.

I N T RO D U C T I O N Accurate estimation of the dynamic pattern of leaf area index (LAI) (see Appendix for list of abbreviations) is important for process-based crop growth models to predict biomass production and seed yields, since LAI determines light interception which is central to growth. Despite continued e€orts, accurate prediction of LAI under a wide range of environments remains a problem for crop modellers (Krop€, van Laar and Matthews, 1994). In many early crop models (e.g. Charles-Edwards and Fisher, 1980; Penning de Vries et al., 1989), LAI was simply calculated from leaf biomass using the parameter speci®c leaf area (SLA). Simulation of LAI in such a way, however, often appears to be over-sensitive to a small measurement error in SLA because of the positive feedback loop: leaf weight4leaf area4canopy photosynthesis4leaf growth4leaf weight (Penning de Vries et al., 1989). The value of SLA depends both on environmental variables (Penning de Vries et al., 1989; Tardieu, Granier and Muller, * For correspondence. Fax 31-317-423110, e-mail x.yin@plant. wag-ur.nl { Present address: Institute for Terrestrial Ecology, Edinburgh Research Station, Bush Estate, Penicuik, EH26 0QB, UK

0305-7364/00/050579+07 $35.00/00

1999) and development stages (Krop€ et al., 1994; Yin, Krop€ and Stam, 1999), and is thus dicult to determine accurately. In some recent models (e.g. Goudriaan and van Laar, 1994; Krop€ et al., 1994; Yin et al., 2000), a di€erent method was used, partly to reduce sensitivity to SLA, in which LAI is described as an exponential function of temperature sum between emergence and canopy closure; only afterwards is LAI estimated from SLA and the change in leaf mass. There are also many approaches for LAI prediction without invoking the use of SLA (e.g. Muchow and Carberry, 1989; Goudriaan, 1995; Battaglia et al., 1998). It appears that most existing approaches do not consider directly any response of canopy development to nitrogen, an important factor known to a€ect LAI in any environment, especially at later growth stages involving leaf senescence (Sinclair and de Wit, 1976). One important aspect of LAI modelling is the handling of leaf senescence. In ®eld-grown crops, leaf senescence occurs some days before ¯owering, and continues until maturity. Although leaf senescence is complex (Thomas and Stoddart, 1980), most modellers describe this process simply (CharlesEdwards and Fisher, 1980). A widely used approach is to de®ne an empirical leaf turnover coecient Ð relative death rate (RDR) Ð as a function of crop development # 2000 Annals of Botany Company

580

Yin et al.ÐModelling Leaf Senescence

stage. For example, in the model for rice (Oryza sativa L.) ORYZA-1 (Krop€ et al., 1994), and the model for barley (Hordeum vulgare L.) SYP-BL (Yin et al., 2000), RDR is applied to calculate the weight loss of leaves. The reduction in LAI is calculated as the loss of leaf weight multiplied by SLA. To determine stage-dependent RDR, frequent ®eld samplings are needed. The di€erence in green leaf weight between two samplings is used to estimate RDR at the average stage between samplings (Krop€ et al., 1994). This method is not only laborious in ®eld measurements, but also incorrect in the physiological sense because weight loss between sampling times can be due to other physiological processes (e.g. leaf respiration). Moreover, the method may underestimate leaf senescence during the period in which new leaf growth and leaf senescence occur simultaneously, which is common at stages prior to ¯owering. In this paper, we ®rst derive an equation for LAI based on canopy leaf-nitrogen distribution and the minimum leaf nitrogen required to support photosynthesis. Using this equation, we then describe a new method for simulating leaf death, which overcomes the above two problems of stagedependent RDR and over-sensitivity to SLA. The method can be feasibly used in most crop or plant growth models. Here, we incorporated it into both ORYZA-1 and SYP-BL models and compared it with the original RDR-based method of these two models in simulating leaf area development observed in ®eld experiments. THE MODEL Equation for nitrogen-determined leaf area index In a crop canopy, leaf nitrogen content per unit leaf area (leaf-nitrogen hereafter) is higher in the upper than in the lower leaves. It has been observed experimentally in various species that the pro®le of leaf-nitrogen in the canopy follows an exponential function with LAI counted from the top of the canopy (e.g. Field, 1983; Hirose and Werger, 1987; Krop€ et al., 1994; Anten, Schieving and Werger, 1995), that is: ni ˆ no e ÿkLi

…1†

where ni is the leaf-nitrogen of the ith layer of the canopy where the LAI counted from the top is Li , no is the leafnitrogen at the top of the canopy (i.e. Li ˆ 0) and k is the nitrogen extinction coecient. Often, the total canopy leaf-nitrogen content or the average leaf-nitrogen of the canopy is measured instead of no , because the value of no is not amenable to experimental collection. The total amount of canopy leaf-nitrogen (N) can be solved analytically from eqn (1): Z Nˆ

L 0

ni dLi ˆ no …1 ÿ e ÿkL †=k

…2†

where L is the total canopy LAI. Solving eqn (2) for no and substituting it into eqn (1) gives: ni ˆ kN e ÿkLi =…1 ÿ e ÿkL †

…3†

Many studies on leaf photosynthesis have shown a base leaf-nitrogen (nb) at which leaf photosynthesis is zero (Sinclair and Horie, 1989; Grindlay, 1997; Dreccer, Slafer and Rabbinge, 1998). Based on eqn (3), we can formulate the relation between nb and photosynthetically active LAI (Lphoto) as: nb ˆ kN e ÿkLphoto =…1 ÿ e ÿkL †

…4†

After leaf-nitrogen at the bottom of the canopy reaches nb (i.e. the onset of leaf senescence), crop LAI should be equal to Lphoto . Letting Lphoto equal L and then solving eqn (4) for L gives the LAI as determined by the nitrogen pro®le with a given amount of canopy leaf-nitrogen: LN ˆ …1=k†ln…1 ‡ kN=nb †

…5†

where LN is the nitrogen-determined LAI for a given value of N. Equation (5) is valid for estimating canopy LAI of the growth period after the onset of leaf senescence. In this period, a canopy with a LAI greater than LN contains photosynthetically-inactive leaves at the bottom which we assume then die. Before this period, leaf-nitrogen at the bottom of the canopy is higher than nb ; eqn (5) will therefore overestimate the real LAI, if applied to stages prior to this period. This forms the basis for our approach to estimating leaf senescence in crop growth models as described below. Method for predicting leaf senescence in crop growth models Our senescence model is based on the idea that leaves die when their nitrogen content falls below the minimum amount required for leaf photosynthesis. Because nitrogen is not the only factor determining LAI, we propose the existence of non-nitrogen limited LAI with no leaf death involved (denoted as LNLD), which can be calculated independently of LN at each time-step by crop growth models. In early stages, when leaf-nitrogen at the bottom of a crop canopy has not yet reached nb , LN given by eqn (5) is higher than LNLD ; but later, LN declines below the value of LNLD (see Results). Crop LAI can be chosen as the minimum value of LNLD and LN . The moment at which LN is just less than LNLD is the predicted time for the onset of leaf senescence. Since LNLD is computed assuming no leaf death at a time-step, the di€erence between LNLD and LN is attributed to leaf death at that time-step. The rate of daily leaf-mass loss due to senescence (r) can be calculated for that time-step by: r ˆ ‰LNLD ÿ min…LNLD ; LN †Š=…sDt†

…6†

where s is SLA and Dt is the time step for dynamic calculation. The total dead-leaf weight can be calculated as the accumulation of daily loss rate of leaf mass over the entire period until maturity. At early stages when LN is higher than LNLD , eqn (6) predicts no leaf death. At later stages when LN is lower than LNLD , eqn (6) predicts leaf death rate as (LNLD ÿ LN)/(sDt).

Yin et al.ÐModelling Leaf Senescence Method evaluation Two data sets were used to evaluate our method. The ®rst data set was published as the default data to parameterize the ORYZA-1 model (Krop€ et al., 1994). The data set was collected from a ®eld experiment using rice `IR72' conducted in the 1992 dry season at the International Rice Research Institute, the Philippines. The second data set came from a ®eld experiment for a set of recombinant inbred lines in barley, conducted in 1997 at Wageningen, The Netherlands (Yin et al., 2000). Data for the two parents of these lines, `Apex' and `Prisma', were complete, and used to test our model. In this second data set, not only LAI but also dead-leaf weight was measured. Equations (5) and (6) are incorporated into the ORYZA-1 rice model (Krop€ et al., 1994), and into the SYP-BL barley model (Yin et al., 2000), to replace the original modelling routine for senescence. The original routine of both ORYZA-1 and SYP-BL models uses the concept of stage-dependent RDR. As indicated earlier, the two models use the two-phase approach to simulate LAI: LAI is determined by temperature before canopy closure, after which LAI is calculated from SLA and the daily change in leaf mass. The LAI estimated using this approach is considered as the non-nitrogen limited LAI, i.e. LNLD in eqn (6). Both ORYZA-1 and SYP-BL models run on the timestep of 1 d. They do not simulate crop nitrogen budget, but use measured average leaf-nitrogen, nav , as inputs to calculate leaf and canopy photosynthesis (assuming an exponential nitrogen pro®le); so, canopy leaf-nitrogen content should be determined as: N ˆ navL, where nav is the average value of leaf nitrogen per unit leaf area in the canopy. However, this would create a loop in the calculation of L, N and LN , if L is calculated as min(LN , LNLD) as indicated earlier. To avoid this, we did not calculate L in this way but simply assumed L ˆ LNLD . This is exact for stages prior to the onset of senescence, but only approximate afterwards with a delay of 1 d in predicting LAI. This approximation can be explained as follows: although LNLD is computed assuming no leaf death at a given time-step, it does involve leaf death in the longer term because its value adjusts from one time-step to another, from the comparison between LNLD and LN made at each time-step. The di€erence between LNLD and LN is translated into senesced leaf mass by eqn (6); so LNLD will adjust in the next time-step to a new value based on the change in green-leaf mass between the two consecutive time-steps. To implement our method, the values of nb and k in eqn (5) have to be pre-determined. The value of 0.4 m2 m ÿ2 for k, as used in the ORYZA-1 and SYP-BL models, was used here. This value was estimated by Krop€ et al. (1994), using eqn (1), from experimental data for rice. The value of nb is dicult to measure directly, and is therefore obtained by extrapolating the relation between n and light-saturated leaf photosynthesis. The value of this parameter varies with species (Sinclair and Horie, 1989; Grindlay, 1997; Dreccer et al., 1998). Even within the same species, nb has been reported to vary. For example, data of Evans (1983)

581

indicated that nb for wheat (Triticum aesticum L.) was about 0.3 g m ÿ2, whereas Lawlor (1995) reported a value of about 0.5 g m ÿ2. To determine an appropriate nb , sensitivity analysis was conducted in which nb was varied between 0.1 and 0.5, with intervals of 0.1 g m ÿ2. No adjustment was made on parameters in other modelling components of either the ORYZA-1 or the SYP-BL model. Because our method in fact assumes that crop LAI after the onset of leaf senescence is determined by total canopy leaf-nitrogen, we tested whether our method would reduce the sensitivity of simulated LAI to the value of SLA, a common problem involved in many existing crop models. This was done by running the updated and the original routines for leaf senescence for a 20% increase or decrease in the values of SLA. R E S ULT S Not surprisingly, simulation of LAI with the new method is sensitive to the value of nb (Fig. 1). The simulated LAI that best agreed with the observed one was obtained by the use of 0.4 g m ÿ2 for nb in rice (Fig. 1A) and 0.3 g m ÿ2 for barley (Fig. 1B, C). These values for nb were used in the further analysis. The time course of calculated LNLD and LN is given in Fig. 2. For rice, LN was higher than LNLD until 45 d after transplanting, at which time LN became smaller than LNLD . Similarly for barley, LN was smaller than LNLD 53 and 50 d after emergence, for `Apex' and `Prisma', respectively. These days are the times for the onset of leaf senescence, predicted by our new method. The di€erence between LN and LNLD on each day after the onset of senescence whenever LN is smaller than LNLD gives the estimate of the area of leaves senesced on that day. Some days after the onset of the predicted senescence, LNLD also declines (Fig. 2). This happens when predicted leaf senescence outweighs new leaf growth at the preceding time-step, because LNLD is computed in the models from SLA and the daily change of leaf mass. The LAI simulated using the new method and the original routine of the two crop growth models is compared with the observed LAI in Fig. 3. In general, both methods adequately describe the dynamic pattern of the observed LAI. The trend for the weight of accumulated dead leaves simulated by the two methods is given in Fig. 4. The onset of leaf death predicted by the new method was 5, 14 and 12 d later than that predicted by the original routine, for `IR72', `Apex' and `Prisma', respectively. However, the new method predicted faster pre-¯owering senescence. For about 15 d after ¯owering, the new method predicted little or no leaf death, because the observed value of nav hardly varied over that period. In contrast, the original method predicted an increasing death rate over that time course. The 20% increase or decrease of the existing value of SLA used in the ORYZA-1 model had a strong impact on the simulated value of LAI of rice when the original routine for leaf senescence was used (Fig. 5A). However, this change in SLA only had a small impact on the simulated LAI when the new method was used in the ORYZA-1

Yin et al.ÐModelling Leaf Senescence 8

Leaf area index (m2 m−2)

Leaf area index (m2 m−2)

582 A IR72 6 4 2 0 0

20

40

60

80

100

8 A IR72 6 4 2 0 0

120

20

Leaf area index (m2 m−2)

Leaf area index (m2 m−2)

8 B Apex 6 4 2 0 20

40

60

80

100

2 0 60

80

100

120

100

120

B Apex

4 2 0 0

Leaf area index (m2 m−2)

Leaf area index (m2 m−2)

4

40

120

20

40

60

80

Days from emergence

6

20

100

6

120

C Prisma

0

80

8

Days from emergence

8

60

Days from transplanting

Days from transplanting

0

40

100

120

Days from emergence F I G . 1. Observed (d) and simulated (curves) LAI by the new method, using di€erent values of nb , for rice cultivar `IR72' (A), and barley cultivars `Apex' (B) and `Prisma' (C). Curves from the top to the bottom correspond to simulations using 0.1, 0.2, 0.3, 0.4 and 0.5 g m ÿ2 for nb , respectively. Simulations for `IR72' using 0.1 and 0.2 g m ÿ2 for nb yielded the identical result; thus, only four curves are present for `IR72'.

model (Fig. 5B). Similar results were obtained for barley by running the SYP-BL model using the original and new methods for calculating leaf senescence. DISCUSSION The new and the original routines used in the ORYZA-1 or SYP-BL model di€ered in their performance in simulating leaf senescence (Figs 3 and 4). The original routine in the two models uses an empirical stage-dependent leaf turnover coecient, RDR, which has to be determined from frequent destructive experimental samplings (Krop€ et al., 1994). The

8

C Prisma

6 4 2 0 0

20

40

60

80

Days from emergence F I G . 2. Leaf area index simulated assuming no leaf death at the timestep considered (ÐÐ) and nitrogen-determined leaf area index calculated by eqn (5) (± ± ±) for rice cultivar `IR72' (A), and barley cultivars `Apex' (B) and `Prisma' (C). The initial intersecting point of the two curves indicates the onset of leaf senescence.

value for RDR determined from weight loss between samplings is only approximate, because other processes (e.g. leaf respiration), besides leaf death, contribute to the loss of weight. Furthermore, the weight-loss approach may underestimate leaf senescence during the pre-¯owering period when new leaf growth and leaf senescence occur simultaneously. This may explain why the RDR method predicted slower pre-¯owering senescence than the new method (Fig. 4). The new method may not necessarily give a more accurate prediction of LAI than the RDR method for a particular case (Fig. 3); but it is physiologically correct and more robust. The new method, which relates leaf senescence to nitrogen reduction in the canopy, is supported by the generally held hypothesis that crop LAI declines in response

8

Dead leaf weight (g m−2)

Leaf area index (m2 m−2)

Yin et al.ÐModelling Leaf Senescence A IR72 6 4 2 0 0

20

40

60

80

100

200

A IR72

160 120 80 40 0 0

120

20

Dead leaf weight (g m−2)

Leaf area index (m2 m−2)

8 B Apex 6 4 2 0 20

40

60

80

100

200

2 0 60

80

120

100

120

100

120

80 40 0 0

Dead leaf weight (g m−2)

Leaf area index (m2 m−2)

4

40

100

20

40

60

80

Days from transplanting

6

20

80

120

120

C Prisma

0

60

B Apex

160

Days from emergence

8

40

Days from transplanting

Days from transplanting

0

583

100

120

Days from emergence F I G . 3. The time course of LAI observed experimentally (d) and simulated by the new method (thick curve) and by the original RDRbased method (thin curve), for rice cultivar `IR72' (A), and barley cultivars `Apex' (B) and `Prisma' (C).

to leaf-nitrogen remobilization such as to support seed®lling (Sinclair and de Wit, 1976). The method uses the parameter nb , which is physiologically meaningful and can be readily obtained from literature for various crops. Given that this parameter is already used for calculating leaf photosynthesis in many existing models, including the ORYZA-1 and SYP-BL models, in which, in particular, the exponential nitrogen pro®le was also assumed for estimating canopy photosynthesis, the use of RDR as an additional model-input parameter is super¯uous. Our method uses the exponential nitrogen pro®le, similar to the commonly modelled radiation pro®le in crop canopy, to describe leaf senescence. This is supported by the observation of Rousseaux, Hall and Sanchez (1996) that

200

C Prisma

160 120 80 40 0 0

20

40

60

80

Days from transplanting F I G . 4. Time course of the accumulated weight of senesced leaves observed in ®eld experiment (d) and simulated by the new method (thick curve) and by the RDR-based method (thin curve), for rice cultivar `IR72' (A), and barley cultivars `Apex' (B) and `Prisma' (C). No observations are available from the experiment for `IR72'. The arrow indicates the time of ¯owering.

leaf life span decreases strongly with decreasing radiation. Theoretical analyses by Anten et al. (1995), Goudriaan (1995) and Sands (1995) have shown that maximum canopy photosynthesis requires a distribution of nitrogen over the canopy such that the photosynthetic capacity of leaves is proportional to the mean absorbed photosynthetically active radiation. All these optimization analyses assumed a linear relationship between light-saturated leaf photosynthesis and leaf-nitrogen. The widely accepted biochemical leaf-photosynthesis model of Farquhar, von Caemmerer and Berry (1980), however, suggests a curvature of this relationship, as observed by many experimental

Leaf area index (m2 m−2)

Leaf area index (m2 m−2)

584

Yin et al.ÐModelling Leaf Senescence 8

A

6 4 2 0 8

B

6 4 2 0

0

20

40

60

80

100

120

Days from transplanting F I G . 5. Simulated LAI for rice cultivar `IR72' by the use of the RDR-based method (A) and the new method (B), with the 20% increase (continuous thin curve), the 20% decrease (broken thin curve), and no change (thick curve) in the value for SLA used in the ORYZA-1 model.

studies (Evans, 1983; Sinclair and Horie, 1989; Lawlor, 1995), when carboxylation is not Rubisco-limited but electron transport-limited at high leaf-nitrogen. Based on the model of Farquhar et al. (1980), Chen et al. (1993) indicated that a similar exponential but somewhat more uniform nitrogen pro®le than that given by the optimization theory is required in order to balance the Rubisco-limited carboxylation and the electron transport-limited carboxylation. This might explain why the nitrogen extinction coecient, k, in the majority of species reviewed by Dreccer et al. (1998), was smaller than their radiation extinction coecient. Therefore, use of a value for k di€erent to that of the radiation extinction coecient, as in ORYZA-1 (Krop€ et al., 1994) and SYP-BL (Yin et al., 2000) models, is justi®ed. However, even if the nitrogen distribution does not follow the exponential pro®le, e.g. the linear decline pro®le (Shiraiwa and Sinclair, 1993), our routine can still be used to model leaf senescence, provided some changes are made for describing the vertical nitrogen pro®le. Implementation of our approach in crop growth models requires a method for modelling non-nitrogen limited LAI of no leaf death (LAINLD). In our examples, we used the two-phase method present in ORYZA-1 (Krop€ et al., 1994) and SYP-BL (Yin et al., 2000) models to describe this LAINLD . There are many alternative models which may be used for describing LAINLD. Some of these are based on the premise that attained LAI is, in some sense, optimal for given environmental conditions (Goudriaan, 1995; Battaglia et al., 1998); some relate LAI to the number of leaves

(e.g. Muchow and Carberry, 1989); and others use source± sink relationships (Schapendonk et al., 1998). Future users of our senescence approach should use one of these or develop their own model to address LAINLD . Our method is suciently general as to be applicable to plant species other than crops. It is, however, suitable for models such as ORYZA-1 or SYP-BL where measured values of leaf-nitrogen are used as input, or for models where plant nitrogen budget is simulated explicitly. The method cannot be used in models such as MACROS (Penning de Vries et al., 1989) and SUCROS (Goudriaan and van Laar, 1994) where growth is assumed to depend on weather variables only. The examples we showed addressed crop growth that is limited by weather conditions and crop nitrogen status. It is not yet clear if our method can be used for situations where other senescence-inducing factors, e.g. water stress, heat stress and other nutrient de®ciency (Thomas and Stoddart, 1980), occurs as well. In the case of water stress, Wolfe et al. (1988) suggested that relative to irrigated plants, water stressed plants not only had a lower leaf nitrogen concentration but also a higher minimum leafnitrogen for photosynthesis. Therefore, a possible solution would be to adjust the value of nb . Further experimental and modelling work is needed to con®rm this. As indicated earlier, simulation of LAI by multiplying leaf weight with SLA is often over-sensitive to even a small error in SLA, due to the positive feedback loop between leaf area and leaf mass in the calculations (Penning de Vries et al., 1989). Because SLA is extremely variable, depending on environment (Penning de Vries et al., 1989) and development stage (Krop€ et al., 1994; Yin et al., 1999), many researchers (e.g. Tardieu et al., 1999) advise against using SLA as an input parameter in simulating LAI. The method described in this paper assumes that, starting from the onset of leaf senescence, LAI is mainly determined by canopy leaf-nitrogen. We demonstrated that this can e€ectively cut the positive feedback loop and greatly reduce the sensitivity of predicted LAI to the value of SLA (Fig. 5). Our study gives a generic equation, eqn (5), for determining nitrogen-limited LAI in terms of the exponential nitrogen pro®le. If the pro®le does not change with experimental treatment (e.g. planting density, nitrogen-supply etc.), the equation can be used to determine the value of the nitrogen extinction coecient, k, by curve ®tting, particularly when nb is known. Given that the value of k can di€er from the radiation extinction coecient (Dreccer et al., 1998), it should be determined separately. Equation (5) serves this purpose well, and is practically easier to use than eqn (1), for which both LAI and leaf-nitrogen at di€erent vertical layers of the canopy have to be measured. If it is necessary to relate LAI to canopy leaf-nitrogen for stages before the onset of leaf senescence, nb in eqn (5) has to be replaced by leaf-nitrogen at the bottom of the canopy. It is then possible to determine whether or not the value of k varies with developmental stageÐa poorly understood question to date (Krop€ et al., 1994). Elucidation of this would allow more accurate prediction of the nitrogen-limited LAI and canopy photosynthesis by crop models.

Yin et al.ÐModelling Leaf Senescence AC K N OW L E D GE M E N T S This work was supported by the European Commission Environment & Change Programme through the MAGEC (Modelling Agroecosystems under Global Environmental Change) project (European Union Contract ENV4CT97-0693). We thank Dr A. P. Whitmore for his contributions to this research and Dr R. C. Dewar and an anonymous reviewer for their constructive comments on the manuscript.

L I T E R AT U R E C I T E D Anten NPR, Schieving F, Werger MJA. 1995. Patterns of light and nitrogen distribution in relation to whole canopy carbon gain in C3 and C4 mono- and dicotyledonous species. Oecologia 101: 504±513. Battaglia M, Cherry ML, Beadle CL, Sands PJ, Hingston A. 1998. Prediction of leaf area index in eucalypt plantations: e€ects of water stress and temperature. Tree Physiology 18: 521±528. Charles-Edwards DA, Fisher MJ. 1980. A physiological approach to the analysis of crop growth data. I. Theoretical considerations. Annals of Botany 46: 413±423. Chen JL, Reynolds JF, Harley PC, Tenhunen JD. 1993. Coordination theory of leaf nitrogen distribution in a canopy. Oecologia 93: 63±69. Dreccer MF, Slafer GA, Rabbinge R. 1998. Optimization of vertical distribution of canopy nitrogen: an alternative trait to increase yield potential in winter cereals. Journal of Crop Production 1: 47±77. Evans JR. 1983. Nitrogen and photosynthesis in the ¯ag leaf of wheat (Triticum aestivum L.). Plant Physiology 72: 297±302. Farquhar GD, von Caemmerer S, Berry JA. 1980. A biochemical model of photosynthetic CO2 assimilation in leaves of C3 species. Planta 149: 78±90. Field C. 1983. Allocating leaf nitrogen for the maximization of carbon gain: leaf age as a control on the allocation program. Oecologia 56: 341±347. Goudriaan J. 1995. Optimization of nitrogen distribution and of leaf area index for maximum canopy assimilation rate. In: Thiyagarajan TM, ten Berge HFM, Wopereis MCS, eds. Nitrogen management studies in irrigated rice. Los BanÄos, Philippines: International Rice Research Institute, 85±97. Goudriaan J, van Laar HH. 1994. Modelling potential crop growth processes. Dordrecht, The Netherlands: Kluwer Academic Publishers. Grindlay DJC. 1997. Towards an explanation of crop nitrogen demand based on the optimization of leaf nitrogen per unit leaf area. Journal of Agricultural Science, Cambridge 128: 377±396. Hirose T, Werger MJA. 1987. Maximizing daily canopy photosynthesis with respect to the leaf nitrogen allocation pattern in the canopy. Oecologia 72: 520±526. Krop€ MJ, van Laar HH, Matthews RB. 1994. ORYZA-1: an ecophysiological model for irrigated rice production. Los BanÄos, Philippines: International Rice Research Institute. Lawlor DW. 1995. Photosynthesis, productivity and environment. Journal of Experimental Botany 46: 1449±1461. Muchow RC, Carberry PS. 1989. Environmental control of phenology and leaf growth in a tropically adapted maize. Field Crops Research 20: 221±236.

585

Penning de Vries FWT, Jansen DM, ten Berge HFM, Bakema A. 1989. Simulation of ecophysiological processes of growth in several annual crops. Wageningen, The Netherlands: Pudoc and Los BanÄos, Philippines: International Rice Research Institute. Rousseaux MC, Hall AJ, Sanchez RA. 1996. Far-red enrichment and photosynthetically active radiation level in¯uence leaf senescence in ®eld-grown sun¯ower. Physiologia Plantarum 96: 217±224. Sands PJ. 1995. Modelling canopy production. I. Optimal distribution of photosynthetic resources. Australian Journal of Plant Physiology 22: 593±601. Schapendonk AHCM, Stol W, Van Kraalingen DWG, Bouman BAM. 1998. LINGRA, a sink/source model to simulate grassland productivity in Europe. European Journal of Agronomy 9: 87±100. Shiraiwa T, Sinclair TR. 1993. Distribution of nitrogen among leaves in soybean canopies. Crop Science 33: 804±808. Sinclair TR, de Wit CT. 1976. Analysis of the carbon and nitrogen limitations to soybean yield. Agronomy Journal 68: 319±324. Sinclair TR, Horie T. 1989. Leaf nitrogen, photosynthesis, and crop radiation use eciency: a review. Crop Science 29: 90±98. Tardieu F, Granier C, Muller B. 1999. Modelling leaf expansion in a ¯uctuating environment: are changes in speci®c leaf area a consequence of changes in expansion rate?. New Phytologist 143: 33±43. Thomas H, Stoddart JL. 1980. Leaf senescence. Annual Review of Plant Physiology 31: 83±111. Wolfe DW, Henderson DW, Hsiao TC, Alvino A. 1988. Interactive water and nitrogen e€ects on senescence of maize. II. Photosynthetic decline and longevity of individual leaves. Agronomy Journal 80: 865±870. Yin X, Krop€ MJ, Stam P. 1999. The role of ecophysiological models in QTL analysis: the example of speci®c leaf area in barley. Heredity 82: 415±421. Yin X, Krop€ MJ, Goudriaan J, Stam P. 2000. A model analysis of yield di€erences among recombinant inbred lines in barley. Agronomy Journal 92 (in press).

APPENDIX List of main abbreviations and symbols used in equations Abbreviation Symbol De®nition k Nitrogen extinction coecient LAI L Leaf area index LN Nitrogen-determined LAIN LAI LAINLD LNLD LAI with no leaf death at a time-step n Leaf nitrogen per unit leaf area nav Average value of n in a canopy nb Base value of n for photosynthesis N Canopy leaf-nitrogen content r Rate of leaf-mass loss due to senescence RDR Relative leaf-death rate SLA s Speci®c leaf area Dt Time-step for dynamic calculation

Unit m2 ground m ÿ2 leaf m2 leaf m ÿ2 ground m2 leaf m ÿ2 ground m2 leaf m ÿ2 ground g N m ÿ2 leaf g N m ÿ2 leaf g N m ÿ2 leaf g N m ÿ2 ground g leaf m ÿ2 ground d ÿ1 d ÿ1 m ÿ2 leaf g ÿ1 leaf d