A test of the optimality approach to modelling canopy properties and CO2 uptake by natural vegetation



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    1. School of Environmental Systems Engineering, The University of Western Australia, Perth, Western Australia, Australia,
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      Present address: Max Planck Institute for Biogeochemistry, Postfach 10 01 64, D-07701 Jena, Germany.


    1. Environmental Biology Group, Research School of Biological Sciences, The Australian National University, Canberra, Australian Capital Territory, Australia,
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    1. Centre for Water Research, The University of Western Australia, Perth, Western Australia, Australia,
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    • Present address: Departments of Geography and Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana-Champaign, Illinois, USA.


    1. School of Science & Primary Industries, Charles Darwin University, Darwin, Northern Territory, Australia and
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    1. School of Geography and Environmental Science, Monash University, Melbourne, Victoria, Australia
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This article is corrected by:

  1. Errata: Corrigendum Volume 33, Issue 1, 130, Article first published online: 10 December 2009

S. J. Schymanski. Max Planck Institute for Biogeochemistry, Postfach 10 01 64, D-07701 Jena, Germany. Fax: +49 3641 577274; e-mail: sschym@bgc-jena.mpg.de


Photosynthesis provides plants with their main building material, carbohydrates, and with the energy necessary to thrive and prosper in their environment. We expect, therefore, that natural vegetation would evolve optimally to maximize its net carbon profit (NCP), the difference between carbon acquired by photosynthesis and carbon spent on maintenance of the organs involved in its uptake.

We modelled NCP for an optimal vegetation for a site in the wet-dry tropics of north Australia based on this hypothesis and on an ecophysiological gas exchange and photosynthesis model, and compared the modelled CO2 fluxes and canopy properties with observations from the site. The comparison gives insights into theoretical and real controls on gas exchange and canopy structure, and supports the optimality approach for the modelling of gas exchange of natural vegetation.

The main advantage of the optimality approach we adopt is that no assumptions about the particular vegetation of a site are required, making it a very powerful tool for predicting vegetation response to long-term climate or land use change.


Based on our understanding of the biochemical and physical processes controlling photosynthesis, sophisticated models have been developed to calculate the gas exchange between a canopy of leaves and the atmosphere if stomatal conductivity, canopy properties and atmospheric conditions are known. These models have proven relatively successful for crops, where canopy properties can be monitored with handheld devices and the response of stomata to environmental variables can be modelled empirically, based on prior observations of the given species. Natural vegetation, on the other hand, is very difficult to parameterize in these models, because it often consists of a range of species whose stomatal responses are largely unknown. In addition, both species composition and canopy structure of natural vegetation can vary widely in space and time. The application of gas exchange models that use prescribed vegetation properties becomes even more problematic if the aim is to make predictions into the future, particularly with respect to changes in climate or land use.

Cowan & Farquhar (1977) showed that it may be possible to get away from simple extrapolation of past observations if the problem is approached from a different perspective. They assumed a priori that plants would optimize stomatal conductivity dynamically in order to maximize total photosynthesis for a given amount of transpiration. This optimality assumption allowed them to formulate how stomatal conductivity should vary in response to the rate of photosynthesis and atmospheric water vapour deficit, given a fixed amount of water available for transpiration over a period of hours to days. However, the application of this approach at larger scales in time and space (e.g. canopies, days to years) requires detailed information about canopy properties and water availability.

Biochemical properties of foliage have been observed to adapt to environmental conditions as well, not only spatially within a canopy (Kull & Niinemets 1998; Niinemets et al. 1999; Niinemets, Kull & Tenhunen 2004), but also seasonally (Misson et al. 2006). Kull (2002) reviewed a range of models that derived theoretically optimal distributions of photosynthetic capacity in the canopy, and noted that all of these models over-predicted canopy photosynthesis and the slope of the nitrogen profile through the canopy, while under-predicting the leaf area index (LAI). The author blamed inappropriate merit functions, oversimplified photosynthesis models and the lack of consideration of whole plant processes in the acclimation of the canopy for the discrepancy between model results and reality. Kull (2002) also stated that

Poor results in optimum modelling are not proof that optimality fails; they merely imply that the function to be maximised in a natural community remains undiscovered.

If some of the self-organizing principles of vegetation could be summarized in an appropriate merit function (or ‘objective function’ in mathematical language), this would greatly improve our ability to describe how vegetation will change in a changing environment. Furthermore, the generality of such an objective function would facilitate the construction of global models and may reduce the uncertainty associated with the extrapolation of local observations, as is the case now. However, the appropriateness of the objective function and the associated constraints can only be tested by comparison of model predictions with observations in nature.

The aim of this paper was to formulate a quantitative, general concept of vegetation optimality, and to test whether it is consistent with observations. In order to test the generality of the concept, no attempt was made to ‘fit’ the parameters in this exercise.


Overall approach

The approach adopted in this study is based on the assumption that natural vegetation has co-evolved with its environment over a long period of time, and that natural selection has led to a species composition that is optimally adapted to the given conditions. Optimal adaptation was defined as the one that would maximize the net carbon profit (NCP, mol m−2) of the vegetation over the period of interest. NCP was defined as the difference between carbon acquired by photosynthesis and carbon spent on maintenance of the organs involved in its uptake. The organs ultimately involved in carbon uptake are not just leaves, but also roots and transport tissues, which supply the leaves with water and nutrients. For simplicity, the costs related to nutrient uptake were neglected in this model, so that water and light were the only resources considered. The chosen optimization problem was then to maximize NCP by adjusting foliage properties and stomatal conductivity dynamically, while adapting roots and transport tissues to meet the variable demand for water by the canopy (Fig. 1).

Figure 1.

Net carbon profit (NCP) as the difference between carbon acquired by photosynthesis and the carbon used for the construction and maintenance of organs necessary for its uptake. As CO2 uptake from the atmosphere is inevitably linked to the loss of water from the leaves, the root system as well as water transport and storage tissues are essential to support photosynthesis. Soil water supply, atmospheric water demand and daily radiation constitute the environmental forcing. Within those constraints, vegetation is assumed to optimize foliage, transport and storage tissues, roots and stomata dynamically to maximize its NCP.

The dynamics of the water supply from the soil, atmospheric water demand and daily radiation, together with the number of optimizable parameters, make the solution of the optimization problem mathematically and numerically challenging. Furthermore, modelling the processes of canopy photosynthesis, root water uptake and water transport is also a complex task. To overcome this, we have utilized an optimality approach where mechanistic details of plant functions are less important than processes associated with the costs and benefit terms of the NCP. Hence, we made simplifications to maintain the generality of the model while keeping the optimization problem tractable with a standard personal computer. For example, transpiration rates were prescribed using measurements on the site, so that the costs and benefits of roots could be neglected. By doing this, we were able to focus on formulating the problem in terms of canopy properties only.

Photosynthesis was modelled following Farquhar, von Caemmerer & Berry (1980), with the simplification that only electron transport-limited carboxylation was considered. This is deemed a good approximation for a canopy (Farquhar & von Caemmerer 1982), except that in bright sunlight, sun flecks can penetrate deep into the canopy at any instant in time and cause the sunlit fraction of foliage to be light-saturated (de Pury & Farquhar 1997).

Ideally, canopy photosynthesis would be computed as the sum of photosynthesis rates over all leaves, where each leaf would absorb a certain amount of light and would have a certain biochemical capacity and stomatal conductivity. However, such a level of detail was not feasible for the test of the optimality-based approach here. ‘Big leaf’ models on the other hand, which represent the canopy as a single horizontal leaf with homogenous light absorption, stomatal conductivity and biochemical properties, are computationally much more feasible, but do not permit the calculation of an optimal LAI or optimal biochemical capacities that could then be compared with observations. As a compromise, the current study assumed that the canopy is composed of leaves that are horizontal and randomly distributed in homogeneous layers of foliage for light processing purposes, but considered the canopy as a single big leaf for gas exchange purposes. This permitted us to include more realistic costs and benefits of maintaining a certain LAI than would be possible with a traditional ‘big leaf’ model, while holding the number of optimizable parameters in a computationally feasible range. Both LAI and photosynthetic capacity were implicitly constrained bythe carbon costs related to their maintenance. Giventhe observed meteorological data and transpiration, the number of foliage layers (resulting in the modelled LAI) and the photosynthetic capacity of each layer were optimized to achieve a maximum in NCP in each month.

To test the model, seasonally dynamic tall-grass savanna vegetation of the wet-dry tropics of north Australia was chosen. This vegetation is characterized by a C3 open-forest canopy (tree cover of less than 50%) that consists of evergreen and deciduous tree species above a dense, short-lived C4 grass layer, where the latter completes its life cycle during the 5 month wet season and then senesces (Beringer et al. 2007). In terms of canopy gas exchange, the vegetation changes intra-annually from a grass-dominated one in the wet season to a tree-dominated one in the dry season.

By comparing the model results with observations, we aimed at answering the following questions: (1) Can the observed magnitude of daily canopy CO2 uptake be reproduced by the calculated optimal vegetation, and (2) Are the modelled optimal LAI and distribution of photosynthetic capacity within the canopy consistent with observations?

It would be unlikely to achieve correspondence in both points if the vegetation optimality assumption was fundamentally flawed, or if factors other than the ones considered were limiting plant function. Of course, if parameter values were calibrated or the model structure modified with the sole aim to improve the match of model results and observations, affirmative results would be much more likely but a lot less meaningful. In the present study, all parameter values were taken from the literature, and the model structure was determined based on its capability to account for the most dominant costs and benefits of the vegetation traits under investigation.

Vegetation optimality model

In the following equations, all variables were calculated using International System of Units (SI) units (e.g. mol m−2 s−1 for transpiration), even if they will be expressed in more convenient units (e.g. µmol m−2 s−1) in later sections.

Canopy gas exchange

The aim of the canopy photosynthesis model was to keep it as simple as possible while including the key degrees of freedom by which a canopy can adapt to a certain habitat and their costs and benefits in terms of NCP. The process of photosynthesis was thus subdivided into two steps: light processing (generation of electron transport) and CO2 uptake.

We assumed that the electron transport rate (J, mol m−2 s−1) is a linear function of irradiance (Il, mol quanta m−2 s−1) with slope α (commonly set to 0.3) and a maximum determined by the electron transport capacity. We assumed further that the electron transport capacity is exponentially distributed over the leaf area with a mean value of Jmax (in mol m−2 s−1). This led to the formulation of Eqn 1 (Schymanski 2007).


The canopy was modelled as a series of layers with horizontal leaves that were randomly distributed in each layer, with a leaf area (LA) of 0.1 per layer (Fig. 2). Prescription of lower values of LA in each layer would lead to longer computation times but not to substantially different results (results not shown).

Figure 2.

Model canopy as an array of Nl layers with randomly distributed horizontal leaves with leaf area LA = 0.1 in each layer. Nl, number of foliage layers.

Each layer was further subdivided into a sunlit and shaded fraction, where the sunlit fraction receives direct and diffuse light, while the shaded fraction only receives diffuse light. The intensity of diffuse light in each layer (Id,i, where i denotes the layer) was calculated using Eqn 2, while the intensity of direct light was assumed to be the same throughout the canopy. The sunlit leaf area in each layer was calculated using Eqn 3.


The electron transport rate in each layer of foliage was calculated as the sum of the electron transport rates of the sunlit and shaded leaf area fractions, by multiplying Eqn 1 with the respective leaf area while accounting for the direct and diffuse irradiance of each leaf area fraction. To account for clumped vegetation interspersed with bare soil patches, the electron transport rate of the whole canopy per unit ground area (JA, in mol m−2 ground area s−1) was obtained by adding up the electron transport rates of all layers of foliage, and multiplying the sum by the ground fraction covered by vegetation (MA, ranging from 0 to 1). This is given in Eqn 4, where Nl denotes the total number of layers, and Ji denotes the electron transport rate in each layer of foliage.


CO2 uptake by the canopy was modelled following an established model of C3 photosynthesis (Farquhar et al. 1980; von Caemmerer 2000). The C4 pathway of photosynthesis could be added in the future. For the purpose of CO2 uptake, the canopy was treated as a single big leaf so that the equations and variables, originally defined for the leaf scale, were now formulated at the canopy scale. For simplicity, only the electron-transport-limited carboxylation was considered. This is shown in Eqn 5, where Ac denotes the carboxylation rate (mol m−2 ground area s−1), Cl denotes the mole fraction of CO2 inside the big leaf (mol CO2 mol−1 air) and Γ* denotes the CO2 compensation point in the absence of dark respiration (mol CO2 mol−1 air). The rate of CO2 exchange between the inside of the leaf and the atmosphere (Ag, mol CO2 m−2 ground area s−1) is formulated in Eqn 6, where Gsis the stomatal conductivity (mol m−2 ground area s−1), while Cl and Ca (both in mol CO2 mol−1 air) are the mole fractions of CO2 inside the leaf and in the atmosphere, respectively (Cowan & Farquhar 1977). Inside the leaf, leaf respiration (Rl, mol m−2 ground area s−1) replenishes some of the CO2 consumed by Ac, so that at steady state, only the difference between Ac and Rl has to be balanced by Ag. This is expressed in Eqn 7.


Inserting Eqns 5 and 6 into Eqn 7 gives Cl as a function of Gs, JA and Rl, which can then be inserted back into Eqn 6 to obtain Ag as a function of Gs, JA and Rl, as shown in Eqn 8.


Leaf respiration (Rl) was modelled as a linear function of photosynthetic capacity (Amax) using Eqn 9, where cRl = 0.07 following Givnish (1988), who showed for a wide range of species that Rl is approximately 7% of Amax.


Amax was modelled by substituting the total electron transport capacity of the canopy (Jmaxtot, given in Eqn 11) for JA, and Ca for Cl in Eqn 5. The result is shown inEqn 10.


The temperature dependencies of Γ* and Jmax were modelled empirically (Medlyn et al. 2002), where we assumed that leaf temperature is the same as air temperature, irrespective of the position in the canopy.

Equation 12 of Medlyn et al. (2002) was used for the temperature dependence of Γ* after converting it to SI units. Parameter values given for Eucalyptus pauciflora were used for the dependence of Jmax on temperature and on its reference value at 25 °C (Jmax25). Of these parameter values, only the ‘optimal temperature’ of the response function was modified and set to the mean monthly daytime temperature on the study site.

Transpiration (Et) was modelled as a diffusive process, similar to the CO2 uptake through stomata. The different diffusivities of CO2 and H2O vapour were taken into account by multiplying the stomatal conductance for CO2 (Gs) by a constant factor a, which was set to 1.6 (Cowan & Farquhar 1977). This is shown in Eqn 12, where Wl and Wa are the mole fractions of water vapour inside and outside the leaf, respectively. Assuming that the air inside the leaf is at 100% relative humidity, we approximated the term (Wl – Wa) by dividing the measured atmospheric vapour pressure deficit by the air pressure.


Foliage turnover costs (Rft, in mol s−1 m−2 ground area) were derived from Global Plant Trait Network (GLOPNET), a global database of leaf properties (Wright et al. 2004). This was achieved by taking the median of the ratios of ‘leaf mass per area’ to ‘leaf life span’ using all data sets for which such information was available.

The value of the median was 12.8 g dry mass m−2 leaf area month−1 leaf life span. Assuming a construction cost equivalent to 2 g CO2 per g leaf dry matter, we obtained Eqn 13, where LAI = 0.1 NlMA.



The NCP (mol m−2 ground area) of foliage was formulated as the difference between CO2 assimilation (Ag) and foliage turnover costs (Rft), integrated over a month. Ag is positive if there is sufficient electron transport and stomatal conductivity, and negative if there is no electron transport, for example, at night. Hence, it accounts for leaf respiration as well as CO2 assimilation. The total electron transport rate (JA) depends on the light and canopy properties, that is, the number of foliage layers (Nl), the vegetated fraction of the ground area (MA) and the electron transport capacity in each layer i (Jmax25,i). The adjustable parameters were thus MA, Nl and Jmax25,i for i = 1 to Nl. The costs for Jmax25,i are factored into leaf respiration (Rl), while the costs for Nl and MA are the foliage turnover costs (Rft).

Stomatal conductivity (Gs) was inferred from measured water vapour fluxes by inversion of Eqn 12. MA, Nl and the vertical distribution of Jmax25,i were optimized according to the following procedure:

  • 1Extract meteorological data and inferred Gs for 1 month.
  • 2Prescribe a low value of MA and set i = 0.
  • 3Add a layer of foliage with LA = 0.1 (thereby increasing i and Nl by 1).
  • 4Optimize Jmax25,i to maximize NCP: prescribe a positive value for Jmax25,i in this layer, calculate JA and compute NCP, using inferred Gs and meteorological data for 1 month. Iterate Jmax25,i to maximize NCP and save.
  • 5Return to step 3 and repeat until NCP peaks.
  • 6Reset, increase MA and go to step 3.
  • 7Keep increasing MA until NCP peaks or MA reaches 1.0.

From the optimized values of MA and Nl, we calculated the modelled ‘clumped leaf area index’ (LAIc = LA Nl) and the modelled average LAI of the site (LAI = LAIc MA), which were compared with observations. We also compared the resulting time series of foliage CO2 uptake (Ag) with the measured CO2 fluxes on the site. This procedure was repeated for each month of the investigated period to assess the temporal variability in canopy properties.

Study site

Empirical data to test the above optimization routines were taken from the Howard Springs eddy covariance site, which is located in the Northern Territory (Australia), 35 km south-east of Darwin, near Howard Springs in the Howard River catchment (12°29′39.30″S, 131°09′8.58″E). The climate is sub-humid, with approximately 1750 mm mean annual rainfall and 2300 mm mean annual class A pan evaporation. Around 95% of the annual precipitation falls during the wet season (December to March), when atmospheric water demand is low (daytime relative humidity >60%), while the dry season (May to September) is characterized by virtually no rainfall and high atmospheric water demand (daytime relative humidity 10–40%). Air temperatures range between roughly 25 and 35 °C in the wet season, and between 15 and 30 °C in the dry season. The vegetation has been classified as Eucalypt open-forest (Specht 1981), with a mean canopy height of 15 m and an overstorey cover of 50% (Hutley, O'Grady & Eamus 2000). Visual estimates of projected tree cover by analysis of cast shadows in June 2005 suggested values closer to 30 than 50%. The overstorey is dominated by the evergreen Eucalyptus miniata (Cunn. Ex Schauer) and Eucalyptus tetrodonta (F. Muell.), accompanied by a range of other tree species with varying degrees of deciduousness. A detailed account of the seasonal variation of canopy fullness of the different species was given elsewhere (Williams et al. 1997). According to these observations, the canopy fullness of the evergreen species in the tree layer varies by approximately 10% over the year, while the variation increases with increasing deciduousness for the other species. The understorey on the site comprises small individuals of the tree species, some fully or partly deciduous shrubs and some perennial grasses during the dry season. During the wet season, it is dominated by a thick layer of annual C4 grasses of the genus Sarga spp. (formerly known as Sorghum spp.). Recent estimates give a total LAI on the site of 0.8 in the dry season and 2.5 in the wet season (Hutley and Williams, unpublished data) when grass growth is significant.


The measurement techniques used on the site are described in detail elsewhere (Beringer et al. 2003; Hutley et al. 2005; Beringer et al. 2007) and will be only summarized here. Flux measurements were conducted at the top of a 23 m tall tower over the 15 m tall canopy, in flat terrain (slopes < 1°) with a near homogeneous fetch of more than 1 km in all directions. The eddy covariance technique was used to infer vertical fluxes of latent heat and CO2 from three-dimensional wind velocities and turbulent fluctuations of CO2 and water vapour in the air. Incoming shortwave radiation, air pressure and air temperature were also measured at the top of the tower. Soil moisture was measured using time domain reflectometry (TDR) probes (CS615 probes, Campbell Scientific, Logan, UT, USA) at 10 cm depth, and soil temperature was obtained from an averaging soil thermocouple with sensors at 2 and 6 cm depth. All flux variables were sampled at 10 Hz and were averaged over 30 min. Ongoing measurements have been logged since 2001. To create a continuous data set, small gaps (less than 2 h) were filled using linear interpolation, while larger gaps were filled using a neural network model, fitted to the whole data set. Periods with gap-filled data were flagged for later recognition. As the gap filling procedure is a major contributor of uncertainty in the results (Oren et al. 2006) and nocturnal fluxes in particular can be systematically underestimated by the eddy covariance method (Hutley et al. 2005), we chose a period in the data set with only a low proportion of gap filled values (July 2004 to June 2005), and we only considered daytime fluxes for the comparison between model results and observations. Daytime was hereby defined as the time intervals with a mean photosynthetically active photon flux (Ia) greater than 100 µmol s−1 m−2.

For compatibility with our model, a number of conversions of the measured data had to be made. The molar fraction of CO2 in the air was determined from the measured CO2 concentration in the air (mg m−3), the molar weight of CO2 (44 g mol−1), measured air temperature and the ideal gas law. Measured shortwave irradiance in the wavelengths 0.2–4.0 µm (Kshort, in W m−2) was converted to photosynthetically active irradiance (IaW, in W m−2) in the wavelengths 0.4–0.7 µm using a constant conversion coefficient of 0.45 (IaW = 0.45 Kshort), which was derived from an online database (Pinker & Laszlo 1997). To convert from energetic (W m−2) to molar units (Ia, mol quanta s−1 m−2), we used a conversion coefficient of 4.57 × 10−6 mol J−1 (Thimijan & Heins 1983). The ratio between diffuse and direct sunlight was estimated from the relationship between global irradiance and top-of-the-atmosphere irradiance (Spitters, Toussaint & Goudriaan 1986; Roderick 1999; Schymanski 2007).

In order to obtain an estimate of the transpiration part of the measured latent heat flux, soil evaporation had to be subtracted from the measurements, which is especially significant in the wet season when soil moisture is high. Soil evaporation was typically between 0.1 to 0.5 mm d−1 in the dry season and up to 0.65 mm d−1 in the wet season, that is, around 10% of total evapotranspiration (Hutley, unpublished data). The diurnal and day-to-day variation of soil evaporation (Es, in mol H2O m−2 s−1) was estimated using a flux-gradient approach as described in Eqn 14, where Ws denotes the mole fraction of water in the laminar layer immediately above the soil, and Wa the mole fraction of water in the atmosphere, while Gsoil is the conductivity of the soil to water vapour fluxes. Ws was calculated as the vapour pressure in the laminar layer immediately above the soil (pvs) divided by air pressure. pvs was modelled as a function of the atmospheric vapour pressure, the saturation vapour pressure at the measured soil temperature, measured volumetric soil moisture and volumetric soil moisture at field capacity on the site (Lee & Pielke 1992).


The value for soil moisture at field capacity for the site was set to 0.156, equivalent to the soil moisture at a matrix pressure head of −0.01 MPa (Kelley 2002).

The parameter Gsoil in Eqn 14 was set to the value 0.03 mol m−2 s−1, which resulted in a time series of soil evaporation equivalent to around 0.1–0.2 mm d−1 in the dry and 0.4–1.0 mm d−1 in the wet season. Transpiration (Et) was then calculated as the difference between the measured latent heat flux divided by the latent heat of evaporation and the estimated soil evaporation. Stomatal conductivity (Gs) was then inferred by inversion of Eqn 12.

The measured net CO2 uptake by the soil–vegetation system (FnC, mol m−2 s−1) was subdivided conceptually into net CO2 uptake by foliage (Ag), CO2 release by soil respiration (Rs) and CO2 release by heterotrophic tissue respiration (Rw), for example, sapwood. Soil respiration rates were estimated using a model formulated for an African savanna (Hanan et al. 1998), after setting the ‘critical temperature’ to the maximum soil temperature recorded in our data set (44.95 °C) and the intrinsic soil respiration rate at 20 °C to 1.86 µmol m−2 s−1. The estimated soil respiration rates corresponded well with point measurements on our study site (Chen, Eamus & Hutley 2002; Schymanski 2007). We assumed that respiration of woody tissues (sapwood) dominates aboveground heterotrophic respiration. Total aboveground woody-tissue respiration (Rw) has been estimated based on measurements at the study site to be around 297 g C m−2 year−1 (Cernusak et al. 2006), which is equivalent to 0.78 µmol m−2 s−1 averaged over the whole year. To estimate the diurnal variation in Rw, we followed Eqn 2 in Cernusak et al. (2006), where stem temperature (Tw) was modelled as a linear function of air temperature (Tw = 1.11 Ta – 1.36), while the stem CO2 flux at the reference temperature was fitted to match the observed total aboveground woody-tissue respiration of 297 g C m−2 year−1.


Wet season

As an example for wet season conditions, the optimal canopy properties were modelled using inferred stomatal conductivity and meteorological data for January 2005. The optimal canopy properties are summarized in Table 1. The model predicted that a closed canopy (MA = 1.0) with a LAI of 2.5 would be optimal for maximizing the NCP. This was consistent with reported values of wet season LAI of about 2.5 (Hutley and Beringer, unpublished data).

Table 1.  Optimal canopy properties for January 2005
  1. MA, ground fraction covered by vegetation; Nl, number of foliage layers; Jmax25,1, electron transport capacity in the top layer; Jmax25,25, electron transport capacity in the bottom layer; LAI, leaf area index; NCP, net carbon profit.

MA (optimized)1.0
Nl (optimized)25
Jmax25,1 (optimized)435.8 µmol s−1 m−2 leaf area
Jmax25,25 (optimized)14.0 µmol s−1 m−2 leaf area
LAI (computed)2.5
NCP (computed)14.22 mol m−2 ground area in 31 d

Optimal electron transport capacity (Jmax25,i) decreased from the top to the bottom of the canopy, but it decreased faster from layer to layer than the sunlit leaf area fraction (Fig. 3). The overall range of modelled Jmax25,i was between 435.8 µmol s−1 m−2 in the top layer of foliage and 14 µmol s−1 m−2 in the lowest layer. Initially, NCP increased steeply with the addition of each layer, before peaking in a flat plateau at an LAI of 2.5 (Fig. 4).

Figure 3.

Decrease in optimal electron transport capacity (Jmax25) and sunlit leaf area (LAsun) from the top to the bottom of the canopy. The fraction of the top-of-the-canopy irradiance calculated in each layer can be obtained by multiplying LAsun by 10. Canopy optimized for January 2005. The optimal canopy had 25 layers, and all subsequent layers would incur losses in terms of net carbon profit (NCP) (Fig. 4). The continuation of the plot beyond 25 layers shows the values of Jmax that would minimize the losses associated with the maintenance of these layers.

Figure 4.

Optimal electron transport capacity (Jmax25) in each foliage layer and net carbon profit (NCP) achieved with different numbers of layers. NCP peaked when the canopy had 25 layers, and with the addition of any more layers, NCP subsequently decreased. The leaf area of each foliage layer was prescribed as 0.1 m2 m−2, so that the number of layers can be translated directly into leaf area index by multiplying with 0.1.

Ensemble averages of the modelled diurnal cycle were consistent with the measurements (Fig. 5). Modelled and observed daily fluxes resembled each other, both in magnitude and day-to-day dynamics (Fig. 6).

Figure 5.

Ensemble means of measured and modelled diurnal canopy CO2 uptake rates. Ensemble means and SDs (error bars) were computed for 31 d in January 2005. Error bars equate to 2 × SD.

Figure 6.

Comparison of observed (thick, grey line) and modelled (thin, black line) daily canopy CO2 uptake rates during daylight hours in January 2005.

Dry season

As an example for dry season conditions, canopy properties were modelled using inferred stomatal conductivity and meteorological data for October 2004. The optimal canopy properties are summarized in Table 2. The model predicted that a continuous canopy (MA = 1.0) with an LAI of 2.0 would be optimal for maximizing the NCP. This was not consistent with reported values for LAI, which are around 0.8 in the dry season at this site (Hutley and Williams, unpublished data) when the annual grasses have senesced, and the deciduous trees and shrubs of the understorey have lost leaf area. As a result, the optimization also led to a large overestimation of canopy CO2 uptake rates.

Table 2.  Optimal canopy properties for October 2004, if the vegetated fraction of surface (MA) was optimized
  1. Nl, number of foliage layers; Jmax25,1, electron transport capacity in the top layer; Jmax25,25, electron transport capacity in the bottom layer; LAI, leaf area index; NCP, net carbon profit.

MA (optimized)1.0
Nl (optimized)20
Jmax25,1 (optimized)459.5 µmol s−1 m−2 leaf area
Jmax25,25 (optimized)21.0 µmol s−1 m−2 leaf area
LAI (computed)2.0
NCP (computed)11.66 mol m−2 ground area in 31 d

However, if MA was fixed at a value of 0.3, while allowing the number of foliage layers and Jmax25,i in each layer to be optimized, the resulting canopy had an LAI of 0.78 and realistic CO2 uptake rates (Table 3 and Fig. 7). The optimized values of Jmax25,i per unit leaf area were very similar to values for the wet season (Fig. 8 cf. Fig. 3).

Table 3.  Optimal canopy properties for October 2004, if the vegetated fraction of surface (MA) was prescribed as 0.3
  1. Nl, number of foliage layers; Jmax25,1, electron transport capacity in the top layer; Jmax25,25, electron transport capacity in the bottom layer; LAI, leaf area index; NCP, net carbon profit.

MA (prescribed)0.3
Nl (optimized)26
Jmax25,1 (optimized)480.0 µmol s−1 m−2 leaf area
Jmax25,25 (optimized)13.6 µmol s−1 m−2 leaf area
LAI (computed)0.78
NCP (computed)5.29 mol m−2 ground area in 31 d
Figure 7.

Comparison of observed (thick, grey line) and modelled (thin, black line) daily canopy CO2 uptake rates during daylight hours in October 2004. Modelled values were obtained by prescribing ground fraction covered by vegetation (MA) = 0.3, and optimizing canopy leaf area index (LAI) and electron transport capacity (Jmax25) in each layer.

Figure 8.

Decrease in optimal electron transport capacity (Jmax25, black triangles) and sunlit leaf area (LAsun, grey squares) from the top to the bottom of the canopy. The fraction of the top-of-the-canopy irradiance calculated in each layer can be obtained by multiplying LAsun by 10. Canopy optimized for October 2004, with prescribed ground fraction covered by vegetation (MA) = 0.3.

Monthly variation

The calculations discussed previously revealed that optimization of vegetation cover (MA) did not lead to realistic results during the dry season, and a prescribed value of MA for each month was required, based on the knowledge of the seasonality of phenology of this savanna vegetation (Williams et al. 1996; Williams et al. 1997) and site-based measurements of the annual LAI cycle. The savanna at the study site oscillates between a grass-dominated wet season state with the maximum vegetation cover (MA = 1.0) and a dry season state, where only evergreen and partially deciduous trees and shrubs prevail (MA = 0.3). Optimizations were run using values of MA that were prescribed for each month, and were allowed to take values of 1.0 or 0.3, whichever led to more realistic results for the month. The model was then run for each month between July 2004 and June 2005, optimizing the number of foliage layers and electron transport capacity (Jmax25,i) in each layer, in order to maximize NCP of the foliage in the given month. Modelled and observed canopy CO2 uptake rates were summed over the sunlight time of each day and plotted together for each month (Fig. 9).

Figure 9.

Daytime canopy CO2 uptake rates for each month between July 2004 and June 2005. Modelled (black, thin lines) and observed (grey, thick lines) canopy CO2 uptake rates were summed over the daylight hours of each day (defined as time intervals with mean Ia > 100 µmol s−1 m−2) and plotted together for each month. Modelled values based on prescribed values of ground fraction covered by vegetation (MA) and optimized number of foliage layers and electron transport capacity (Jmax25) in each layer. Values of MA were set to either 1.0 or 0.3, whichever led to a better fit with observed CO2 uptake rates. Leaf area index (LAI) in each month is calculated from the prescribed value of MA and the resulting optimal number of foliage layers (Nl).

Modelled CO2 uptake rates matched the observations best if MA was set to 0.3 between June and October, and to 1.0 between December and March. In August 2004, a bush fire led to a sudden decrease in the observed uptake rates, but they gradually recovered and reached the modelled values again after approximately 6 weeks. The periods between November and December, and March and May are transitional periods between the dry and wet season vegetation states, and model performance using a constant value of MA was reduced during these non-steady-state periods (Fig. 9).


Modelled and realistic canopy properties

The optimal distribution of photosynthetic capacity (expressed as Jmax25,i) follows the vertical distribution of sunlit leaf area relatively closely in the modelled canopy, with a slightly steeper decline (Fig. 3). This is in line with observations in other vegetation types, where electron transport capacity per unit leaf area has been shown to be closely related to the integrated light availability (Niinemets et al. 1999; Misson et al. 2006). The results are also consistent with the finding that if the light environment is dominated by beam irradiance, the optimal distribution of photosynthetic capacity (often expressed as the slope of leaf nitrogen versus canopy depth) is steeper than the slope of light versus canopy depth (de Pury & Farquhar 1997). The range of absolute values of optimal Jmax25,i within the modelled canopy was also similar to the values found in the literature. Optimal Jmax25,i given by the model had values between 13.6 µmol s−1 m−2 at the bottom of the canopy and 480 µmol s−1 m−2 at the top of the canopy, while the values observed in a deciduous forest in Estonia ranged from 14 to 335 µmol s−1 m−2 (Niinemets et al. 1999). A compilation of results from studies covering 109 C3 plant species contained average values for Jmax in the range between 17 and 372 µmol s−1 m−2 (Wullschleger 1993). This correspondence is particularly interesting, as we did not impose any upper or lower limits for Jmax25,i. Hence, the predicted range is solely a result of the trade-off between the prescribed maintenance costs of Jmax25,i and leaf area, and the achieved carbon uptake given the stomatal conductivity as calculated from the canopy flux data. While it should be expected that there is a physiological upper limit to photosynthetic capacity per unit leaf area because of, for example, space requirements for the photosynthetic apparatus, the lower limit should indeed be determined mainly by the associated costs and benefits. Below a certain light limit, the construction and maintenance costs even for the thinnest leaves would exceed their carbon uptake. Hence, if our model predicts the same lowest feasible value for Jmax25,i during periods with ample water supply as the lowest values observed in nature, we can be confident that the parameterization of the costs related to the maintenance of leaf area and photosynthetic capacity in our model are realistic. This also seems to be supported by the reasonable match between modelled and observed LAI on the site.

Many models of optimal canopy properties consider total nitrogen as a limit for photosynthetic capacity or leaf area (e.g. Friend 1991; Evans 1993; Badeck 1995; Dewar 1996; de Pury & Farquhar 1997; Farquhar, Buckley & Miller 2002; Hikosaka 2003; Buckley & Roberts 2006), for a review, see also Kull (2002). The nitrogen demand per unit of Jmax25 has been shown to increase with decreasing light (Evans 1987), so consideration of an additional nitrogen limitation in the present study would lead to a reduction of the number of foliage layers at the bottom of the canopy. The consequence would be that the modelled leaf area would be reduced and hence deviate from the observed values. Haxeltine & Prentice (1996), who modelled the optimal photosynthetic properties of canopies for unlimited nitrogen and water, found that the optimal light use efficiencies obtained were generally higher than the ones observed in natural vegetation, while the predicted optimal leaf nitrogen contents did not deviate as much from the observed values. They concluded that nutrient limitation might have led to a reduction in leaf area rather than a reduction in the nitrogen content per leaf area. In fact, if nitrogen and water limitation are considered, the optimal leaf area increases with increasing total nitrogen at a given irradiance, with the result that the changes in nitrogen content per leaf area remain relatively small (Farquhar et al. 2002). However, the latter results were obtained without considering the maintenance costs of leaf area or photosynthetic capacity. If these were included in the model, it would be possible to calculate the optimal total nitrogen content for a given irradiance and water availability. The present study included light and water limitation as well as the maintenance costs for leaf area and photosynthetic capacity.

The results of this study showed that nitrogen limitation did not need to be considered to reproduce both the observed fluxes and LAI in the wet season, despite the nutrient-poor soil on the study site (Russell-Smith, Needham & Brock 1995). The reason for this could be that optimal water and nutrient acquisition in natural vegetation lead to a co-limitation of both, such that the consideration of one or the other in a model suffices to obtain a correspondence between the modelled optimal and the observed canopy properties. In fact, prescribing a limited canopy nitrogen content in the present model would be very close to prescribing the total photosynthetic capacity, and would substantially decrease the generality of the model because the amount of nitrogen available for the canopy is generally hard to estimate, and it also depends on soil properties and the vegetation itself (e.g. on the presence of nitrogen-fixing plants).

The model presented here is based on a very simplistic parameterization of the canopy. The assumption that all leaves are horizontal and randomly distributed in homogeneous layers of foliage does not apply to real canopies (especially the Eucalypt and tall grass canopies of the savanna site used here). However, while non-horizontal leaf angles would increase optimal LAI (results not shown), non-random distributions of leaves would probably decrease the optimal LAI, so the effects of relaxing these two assumptions could (and appear to) partly negate each other.

The prescribed costs and benefits of leaf area and electron transport capacity (Jmax25) led to realistic predictions during the wet season, but were not sufficient to explain the low vegetation cover (MA) observed during the dry season. In the dry season, the optimization only led to realistic results if the vegetation cover was prescribed. In fact, we could have spared ourselves the optimization of MA, as there were no additional costs related to maintaining a certain value of MA, other than the costs related to the associated leaf area. As vegetation gets the largest returns for its leaf area in the top layer of foliage (no shading by other leaves), optimal MA was always 1.0, and leaf area was reduced at the bottom of the canopy if water supply was low (compare the values of Nl in Tables 2 and 3). With the same amount of water use, a closed canopy could have achieved a much higher NCP in the dry season than the actual open canopy (NCP = 11.66 mol m−2 in Table 2 compared with NCP = 5.29 mol m−2 in Table 3). This leads to the conclusion that factors other than the amount of water must be limiting MA. A possible explanation would be the location of available water in the soil. While water is available at the surface during the wet season, it needs to be taken up from deeper soil layers during the dry season, and redistributed at the surface. Observations show that the top 1–2 m of soil are severely depleted of water during the dry season in this region (Kelley 2002). The transport and redistribution of water requires the construction and maintenance of vascular infrastructure, which is obvious in tap roots, trunks and branches of the trees prevailing during the dry season. Tree species have been shown to use water from 2–5 m by the late dry season as surface water storage of the sandy upper soil profile is rapidly depleted at the end of the wet season (Kelley 2002). These maintenance costs were not considered in the present study, but will be included in a follow-up study that deals with the dynamics of soil water as well as vegetation dynamics. In the present study, the value of MA had to be prescribed for periods when water supply was limiting.

Canopy CO2 uptake rates

The relatively close match of the magnitudes of modelled and observed canopy CO2 uptake rates is, in our opinion, remarkable, particularly because no site-specific vegetation parameters were prescribed. In fact, the model did not even consider that the dominant CO2 uptake process during the wet season on this site is C4 photosynthesis (dominated by grasses of the genus Sarga spp.), and hence predicted the optimal properties of a canopy of C3 species. Our results suggest that the inherent ratio between costs and benefits related to the maintenance of leaf area and the photosynthetic apparatus may not change significantly between the different photosynthetic pathways. One interpretation is that increased benefits because of special adaptations are likely to bring along increased metabolic costs as well. The net benefit of employing a specialist photosynthetic pathway, although significant enough to favour certain species over others in a given environment, may be smaller at canopy scale.

During the wet season, there was also a good correspondence between modelled and observed day-to-dayvariation in canopy CO2 uptake (Fig. 6). Such a correspondence could not be seen during the dry season (Fig. 7). However, it has to be noted that the fluctuations in daily canopy CO2 uptake are naturally low during the dry season, so that failure to capture these fluctuations may be a result of the weakness of the signal, relative to the noise in the measurements.

Two relatively static states of vegetation cover (MA) have been identified on the study site, the wet season state with MA = 1.0 and the dry season state with MA = 0.3. By prescribing MA = 0.3 for the period between April and November, and MA = 1.0 for the period between December and March, the model could be used to describe the duration of the transition periods between these two states. The data presented in Fig. 9 reveal that the transition from the dry season state to the wet season state happened much faster (∼6 weeks) than from the wet to the dry season state (∼10 weeks). Comparison of observed and modelled fluxes also reveals the impact a bush fire had on the fluxes, after its occurrence on 6 August 2004. The data suggest that the recovery to the pre-bush fire CO2 uptake rates took ∼8 weeks. The match between modelled and observed fluxes during these dynamic periods of post-fire canopy regrowth and seasonal leaf senescence or flushing was reduced when compared to periods of more stable canopy dynamics (Fig. 9).

Validity of the approach

The optimality approach to vegetation modelling appeals to the theory of evolution and natural selection as a justification to assume a general tendency towards optimality in plant functioning. However, it is a great leap of faith to infer optimality of a community of plants from this theory, as implicitly done in the present study. In fact, it has been shown elsewhere that optimal water use by competing plants could be quite different in theory to optimal water use by a community of plants acting as an entity (Cowan 1982).

The question of whether natural selection favours competitive individuals or well-adapted plant communities (‘group selection’) is still the subject of considerable debate (Wilson 1983; Wynne-Edwards 1993; Smillie 1995; Okasha 2003). Some of the controversy about ‘group selection’ can be related to the question of whether selection is based on the ‘fitness of the group’ or the ‘average individual fitness of its constituent organisms’ (Okasha 2003). This is not different from looking at natural selection as an optimization problem, where the fitness function constitutes the objective function for the optimization. Thus, the search for the appropriate objective function in optimality modelling, as outlined in the introduction of this paper, is akin to the search for the appropriate fitness function in evolutionary modelling. Traditionally, evolutionary models often define ‘fitness’ as the number of offspring, while vegetation optimality models define the objective function as some quantity related to the use of resources. It could be argued that NCP, be it for an individual or a group, is a more general formulation of fitness than the number of offspring, as persistence in a given environment can be achieved by means other than the generation of a large number of offspring (e.g. the occurrence of vegetative suckers evident in these savanna tree species). Whatever the means to persist in the environment, the energy stored in carbohydrates is a universal currency that can be invested in reproduction or any other means.

The maximization of the Net Carbon Profit (NCP) is in contrast to the commonly assumed maximization of ‘net primary production’ (NPP) in other optimality-based models (Raupach 2005). NPP refers to ‘the rate at which solar energy is stored by plants as organic matter’ (Roxburgh et al. 2005), irrespective of whether this energy is subsequently available to the plants or not. In fact, most of the ‘observable’ part of NPP corresponds to energy that is locked up in cellulose and lignin, and hence not retrievable by the plants. We believe that maximization of NCP is a more appropriate objective function than the maximization of NPP, because NCP only refers to the energy that is available to the plants for increasing their ‘biological fitness’ (e.g. production of seeds, maintenance of defence mechanisms against pests and herbivores or maintenance of symbiotic relationships to improve nutrient uptake). While the growth of new leaves itself could be interpreted as a strategy to increase a plant's ‘fitness’, here, it is merely considered a means to increase the carbon gain because of light capture, which is offset by the carbon loss as a result of the associated maintenance costs of leaf area and the biochemical apparatus. These costs constrain the optimal leaf area, which would not be constrained if NPP were maximized instead of NCP.

The presented model contains a range of crude simplifications, for example, the assumption that all leaves are horizontal, the subdivision of the canopy into layers with similar leaf area and randomly distributed leaves, the neglect of the limitation of photosynthesis by the biochemical carboxylation capacity and the treatment of the canopy as a single leaf for carboxylation purposes. Some of these assumptions could probably be relaxed using the same optimality considerations as were used for the existing model. For example, the co-optimization of electron transport capacity and carboxylation capacity, and the optimization of leaf angles in a canopy have been applied elsewhere separately (Friend 1991; Herbert 1991; Herbert & Nilson 1991; Herbert 1992; Niinemets et al. 2004) and could be included in the overall vegetation optimality model when the available computing power improves.


  • 1Maximization of NCP is a possible principle for self-organization of plant communities

The objective function of maximizing NCP of foliage led to the emergence of vegetation properties and CO2 uptake rates in the model, which were consistent with observations during the wet season.

  • 2Costs and benefits of foliage are sufficient to predict fluxes and canopy properties if rooting and water transport costs are small

Although the study site is generally regarded as a nutrient-poor site, the LAI, photosynthetic capacity and canopy photosynthesis during the wet season were as high as the optimality model predicted without considering nutrient limitation at all. Meteorological data, observed water use and the prescribed costs related to the maintenance of the photosynthetic apparatus and the maintenance of leaf area were the only constraints needed to reproduce the wet season fluxes. We conclude that other costs, such as the costs involved in water uptake and water transport, are likely to have been relatively small during the wet season, so that they did not need to be considered.

  • 3If rooting costs are large (deep roots needed), they have to be included in order to predict canopy structure and fluxes

We found that in the dry season, vegetated cover (MA) had to be prescribed in order to achieve realistic model results. As the value of MA was not constrained by the amount of available water alone, we conclude that costs associated with deep rooting and water transport must be included to make realistic predictions during the dry season. Inclusion of these costs during the wet season was not necessary, as soil moisture was sufficient in the upper soil layers and the plants that grew in the wet season could survive with much ‘cheaper’ vascular systems. In fact, we will show in a follow-up study that the low projected cover of tree crowns on the site could well be explained by the costs related to the transport of water from deeper soil layers via a root system that extends to 5 m in depth (Kelley 2002), and enables an optimal and a seasonal pattern of tree water use despite large changes in available soil moisture (O'Grady, Eamus & Hutley 1999).


We thank Nigel Tapper for his support in the data acquisition, Erik Veneklas and Sandra Berry for providing useful literature references, and John Evans and Susanne von Caemmerer for the valuable insights into the physiology of photosynthesis. We are also thankful for the useful comments from an anonymous reviewer. The research was funded by the Australian Department for Education, Science and Training, and the Cooperative Research Centre for Greenhouse Accounting.