Basically, the production of biomass by a canopy relies on the net assimilation of atmospheric CO2, that is the balance between gross photosynthesis and respiration. It depends on the amount of available energy (light) and carbon substrate (CO2), and on the ability of the canopy to intercept light and assimilate CO2. In greenhouses, the assimilation of CO2 is not only important for crop growth, it interacts strongly with the composition of the atmosphere. The daily consump tion of carbon by a tomato canopy can be up to 10 times the amount of carbon available in the greenhouse atmosphere.6 It must be balanced either by ventilation or by CO2 enrichment.
Longuenesse et al.s and Gijzen9 have extensively reviewed models of photosynthesis of horticultural species at leaf and canopy levels. The leaf gross photosynthesis responds to light by a saturation-type curve. Various mathematical formulations have been proposed and tested on tomato data, for example the rectangular hyperbola,10 the non-rectangular hyperbola11 and the negative exponential.12 Despite their slight difference in shape, all these functions include two important parameters: the maximum rate of leaf photosynthesis (P^) and the initial (close to darkness) light use efficiency (a). Pmax increases with CO2 concentration and with the conductance of CO2 transfer from the atmosphere to the chloroplasts. It is limited at low and high temperatures (see examples of para-meterisation for tomato in Bertin and Heuvelink13). Initial light use efficiency a is positively affected by CO2 concentration and negatively by temperature. The conductance to CO2 transfer gets lower at low light intensity, high CO2 concentration, high vapour pressure deficit (VPD) and under water stress.14
Gross photosynthesis has been integrated at canopy scale in different ways. The simplest approach is to multiply the unit leaf activity by the leaf area index or by the projected leaf area ('big leaf' approach). Other models take the transmission of light in the canopy into account using an exponential law of extinc-tion.15 When the leaf light response curve is a rectangular hyperbola, analytical integration at canopy scale is possible (e.g. in Jones et al.16 for tomato crops). More sophisticated models are based on a detailed description of light distribution and absorption in canopies (see later).
The respiratory efflux of CO2 is significant: on a daily basis, it can represent a quarter to a half of the gross photosynthesis of a developed greenhouse tomato crop.6,9 Respiration of plants has functionally been divided in two components: maintenance and growth respiration. Maintenance respiration corresponds to the energy needed to maintain the ionic gradients across biological membranes and pools of macromolecules such as proteins. Growth respiration corresponds to the energy involved in the synthesis of new biomass from assimilates and minerals. Maintenance respiration is calculated as the product of the plant or organ dry weight times a maintenance coefficient. Growth respiration is calculated as the product of the plant or organ growth rate times a CO2 production factor. In crop models, maintenance and growth respiration are summed to estimate total respiration, generally on a daily basis. Respiration rate increases exponentially with temperature. For tomato, Heuvelink17 has hypothesised that the maintenance coefficient decreases with ageing of organs. The CO2 production factor is proportional to the energy cost of biomass synthesis; it varies between organs and with ageing (see Gary et al.18 for tomato).
The crop carbon balance includes carbon exchanges between the atmosphere and the canopy (net photosynthesis), and the partitioning of carbon in the plant between one or several pools of photoassimilates and the growing organs. Gent and Enoch19 put together simple formulations for gross photosynthesis and res piration, and provided a relationship between the availability of photoassimilates and growth. With these simple formulations, the 24-hour dynamics of CO2 exchanges and of the variations in the assimilate pool of young tomato plants could be simulated.20,21 Such a simple carbon balance model was reshaped for control purposes by Seginer et al.22
The water balance in the crop is an important crop property in various respects. Water import contributes to the plant growth, as water status influences cell extension in growing organs and water flow conveys nutrients to growing or storage organs. Water status also partly controls the stomatal conductance and may therefore affect photosynthesis. And last, the evaporation of water during transpiration is connected to the absorption of latent heat: it strongly determines the temperature of the canopy and, therefore, of the air inside a greenhouse.3
The modelling of water relations of horticultural crops has been reviewed by Jones and Tardieu,23 van de Sanden24 and Jolliet.25 Research in this domain has been motivated by two main concerns: (1) simulating the water status and its relation with various physiological functions (organ extension, stomatal opening, water flux and so on) and (2) simulating the water flux through the canopy to estimate the water requirements of crops. The basic framework that has generally been adopted is an analogue of Ohm's law: the water volume flux along a certain path is proportional to the gradient of water potential and to the inverse of a flow resistance. For tomato, van Ieperen26 designed a model describing the pathway of water from the root environment to the atmosphere through one root compartment and three shoot layers within a vegetative plant, and the dynamics of water potential in roots, stems and leaves. Premises for modelling the water fluxes to the tomato fruit through the phloem and xylem vessels can be found in Guichard et al.21 These premises are based on Fishman and Genard's model.28 The dominating phloem flux depends on the concentration of carbohydrates in the phloem vessels and on the ability of the fruit to unload these carbohydrates.28 The xylem flux varies with the water potential in the stem, since the fruit water potential remains fairly stable in time and in different environmental conditions.21 Owing to a high resistance to water flux in its epidermis, the transpiration of the tomato fruit is limited; it was modelled as a function of irradiance and VPD by Leonardi et al.29
On the canopy scale, the transpiration of tomato crops has been modelled applying the classical Penman-Monteith approach30 as the sum of a radiative component, proportional to the global radiation absorbed by the canopy, and of a convective component, proportional to the VPD. The canopy resistance to transfer water vapour comprises the aerodynamic resistance that depends on wind speed and air and leaf temperatures, and the stomatal resistance that depends on radiation, leaf air saturation deficit and leaf temperature (e.g. Boulard et al.31 for tomato crops). For operational purposes, the complete analytical model has been simplified to a two-parameter formula, the parameters being either derived from the complex model or identified in situ?2
A crop canopy can be compared to a solar collector. The absorbed radiation is the balance between incident, reflected and transmitted global radiation. In their study of light interception by glasshouse crops, Warren Wilson et al.33 measured, for a tomato canopy, an average reflectance of 13% and an average transmittance of 23.5% of the incident light in the photosynthetic active radiation (PAR) waveband. Light absorption was improved by about 10% when the soil was covered with a white plastic sheet. It also increased with the foliage development to almost complete absorption with a leaf area index (LAI) of 4 or above. Light absorption is related to plant density and row spacing as it tends to increase when the plant distribution is more uniform.34 The distribution of light and its absorption by rows of canopies such as tomato crops have been modelled by using several approaches reviewed by Critten.15 Among these are the exponential extinction curve, and various models that take light scattering and the distribution of diffuse and direct light35 and leaf angle distribution into account.36
Part of the absorbed radiation is used by photosynthesis for carbon assimilation and biomass production. This proportion is estimated by the radiation use efficiency (RUE), that is the ratio between the energy equivalent of biomass and the absorbed (or incident) global (or PAR) radiation. For a tomato crop, Aikman37 estimated the absorbed radiation to be about 7% when based on the absorbed PAR or 1.6% when based on the global radiation outside the greenhouse.
A significant part of the absorbed energy is actually dissipated by the crop as latent heat by transpiration. As a consequence, the temperature of a transpiring canopy is lower than the air temperature. This difference generates a flux of sensible heat from the air to the canopy. In a greenhouse, depending on the LAI, 50-70% of the solar energy input is used for evapotranspiration.3 This justifies the fact that the crop water requirements are estimated from the absorbed or incident global radiation.
In the same way as for carbon and water, both mechanistic and black-box models have been designed (see the extensive review of Le Bot et al.).38 The mechanistic models describe specific processes like nutrient uptake, transport and assimilation. Even for nitrogen, the most studied element, the regulation and the integration of these processes on a whole-plant scale are still in discussion. For tomato, two main approaches of mechanistic modelling have been proposed. According to Le Bot et al.,38 the time-course of nitrate uptake is related to the translocation of carbohydrates to the roots to cover the energy cost of nutrient uptake. According to Cardenas-Navarro et al.,39 nitrate uptake is related to the maintenance of a steady internal ion concentration.
More general (black-box) models link the demand of nutrients directly to the growth rate. It has been established for several elements (nitrogen, potassium, phosphorus) that a critical concentration in plant tissues should be maintained to approach the potential growth based on total intercepted radiation. For nitrogen, this critical concentration gradually declines with the accumulation of biomass during the vegetative phase.40 Le Bot et al41 parameterised this relation for tomato plants. To explain this decline in nitrogen content, Caloin and Yu42 suggested two compartments in the biomass, one mostly active for growth and having a high nitrogen content and another dedicated to structures and storage having a lower nitrogen content. With crop development, the second compartment tends to dominate. This model was calibrated for a greenhouse tomato crop by Bellert et al.43 A comparable approach to the nitrogen demand by processing tomatoes has been implemented in the EPIC model to evaluate different fertilisation policies in terms of crop growth and nitrogen dynamics in the soil.44
Few models are available at the time of writing for other nutrients.45 A first model simulating the flux of calcium in pepper fruit and its relation to the occurrence of blossom-end-rot (a quality defect also observed on tomato) was reported.46
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