## Introduction

Modelling of forest responses to elevated CO_{2} and environmental factors is a cornerstone of climate change research. In many models, light-use efficiency (ɛ_{GPP}) in combination with estimates of light absorption, often obtained from remote sensing methods, is used to estimate gross primary production (GPP). A maximum potential *ɛ*_{GPP} is reduced in response to environmental constraints and combined with a net primary production (NPP) : GPP ratio or a respiration function to obtain NPP. However, the assumptions and values used for these factors vary substantially. For the same forest type, maximum *ɛ*_{GPP} varies threefold between models (Xiao *et al*., 2005), while others argue that ɛ_{GPP} is conservative (Goetz & Prince, 1999). The common assumption that the NPP : GPP ratio is invariable (Waring *et al*., 1998; Medlyn & Dewar, 1999) has also been challenged (Goetz & Prince, 1999). In summary, a general consensus on the appropriate assumptions for forest NPP modelling is lacking, which also carries over to the modelling of elevated CO_{2} effects.

Experimental evidence of the effects of elevated CO_{2} from free air CO_{2} enrichment (FACE) and other investigations are accumulating. When all studies are compared, the range of observed responses in productivity and biomass is large. This large range is mostly the result of differences in the age of the studied stands; that is, if it is a young forest still increasing resource acquisition or if it has reached steady state in terms of fine roots and LAI (Korner, 2006). For closed-canopy forests, NPP responses are conservative across species and sites (on average 23% higher at 550 than at 376 ppm CO_{2}) (Norby *et al*., 2005). Nutrient additions strongly enhance the growth response to elevated CO_{2} (Oren *et al*., 2001; Reich *et al*., 2006a). Light-use efficiency (*ɛ*_{NPP}) is increased while leaf area index (LAI) is usually little affected (Norby *et al*., 2003), although at low LAI the CO_{2} effect on LAI can be as large or larger than the effect on *ɛ*_{NPP} (Norby *et al*., 2005). For the allocation to wood relative to fast turnover tissues (leaves and fine roots), both positive (Hamilton *et al*., 2002; DeLucia *et al*., 2005) and negative effects (Norby *et al*., 2004) have been observed. Explanations for the above responses have been suggested mostly in qualitative terms. Nutrient limitation constrains the potential growth response, particularly for woody biomass increment (Korner, 2006), LAI is conservative because the gain in absorbed PAR per additional unit of LAI is small at higher LAI. To my knowledge, an explanation for the conservative NPP responses among closed-canopy forests (Norby *et al*., 2005) is lacking. However, the question addressed here is as follows: can these different CO_{2} responses be integrated and explained in a common framework?

In contrast to the variable responses of whole-plant properties to CO_{2}, the primary functional responses of leaves appear to be limited to increased photosynthesis and reduced stomatal conductance (Gifford, 2004). Observed down-regulation of photosynthetic capacity at elevated CO_{2} can be attributed to reduced leaf nitrogen (N) (Ellsworth *et al*., 2004), while photosynthetic capacity per leaf N per area (*a*) and quantum efficiency (initial slope of leaf photosynthesis vs PAR, φ) are consistently increased by elevated CO_{2}. This consistency suggest changes in *a* and φ as primary effects for up-scaling of elevated CO_{2} responses. To scale up these leaf responses to the whole-plant level, they must be put into a framework that includes the effects of other limiting resources, such as nutrients and water. For water, however, the main response is stomatal regulation, which in this framework is included indirectly through *a*.

Nutrients, particularly N, commonly limit plant growth and its response to elevated CO_{2} (Reich *et al*., 2006a). The fact that N is very often a limiting and depletable resource for plant growth suggests that the plants should strive to optimize their use of N. Furthermore, for exponentially growing plants, growth rate is linearly related to plant N concentration and N supply (Ingestad, 1979), which is a strong indication of the close relation between plant N and plant growth. Because of self-shading, the linearity does not hold as the plants get larger, but this is no reason to question the link between N and plant functioning. On the contrary, because of its key role in the metabolic machinery, not only photosynthesis but also respiration scales with N (Vose & Ryan, 2002; Reich *et al*., 2006b). Based on this dual role of plant N, it has been hypothesized that plant canopy N content is determined by optimizing NPP through a tradeoff between N-induced photosynthesis and whole-plant respiration (Dewar, 1996).

Here I extend Dewar's (1996) theory of optimal canopy N by including effects of shifts in foliage : root : sapwood ratios, to capture effects of changes in fine-root allocation in response to nutrient availability as well as the effect of sapwood accumulation. In this framework, leaf-level effects of CO_{2} (on *a* and φ) are scaled up to the whole-tree level. The hypothesis is that this optimization at the whole-plant level can elucidate less well understood forest responses to elevated CO_{2}, such as the variation in NPP response among sites, the small responses in LAI (and absorbed PAR) relative to ɛ in most but not all cases, and the effects on wood : litter production ratio. The model should also provide insights into why and under what circumstances the NPP : GPP ratio is conserved in response to CO_{2} and nutrient availability. Observations from four forest FACE experiments are used to evaluate the hypothesized model.

### Theory and model description

Evolutionary principles state that optimization of fitness, determined by reproductive success, ultimately controls plant behaviour. But since fitness is difficult to measure and model, for tree modelling some measure of production is usually used as a substitute, for example NPP (Dewar, 1996) or canopy carbon export (Dewar *et al*., 1998). However, here I assume that net biomass increment + reproductive production is more closely linked to fitness than either NPP or canopy carbon export. Looking at any instant in time, it seems logical that a plant maximizes its NPP, which is allocated according to the current demands of different organs. But in optimizing instantaneous NPP we are not accounting for the development over time, that is, that the amount and allocation of current NPP will affect NPP and survival the next year. Over its lifetime, growth and survival of a tree are determined by competitiveness and ability to acquire resources, for example, avoiding being overtopped by neighbouring trees, which is directly related to size (and often height). As size is equal to integrated biomass increment over time, size and hence fitness are likely to be maximized if biomass increment (net growth) is maximized at each moment in time. Furthermore, maximizing biomass increment at each instant in time can be seen as an approximate way of maximizing NPP over the lifetime, as NPP (per plant) generally increases with size. In addition to biomass increment, by definition, reproductive production should contribute to fitness. Therefore, net growth + reproductive production, that is, NPP less annual turnover (and any other carbon expenses) of leaves and fine roots, is the chosen target for optimization and is hereafter denoted as *G*. Turnover of woody structures, such as branches, are not deducted from *G* because of their long life span relative to leaves and fine roots. To obtain a framework for optimization of *G*, *G* must be formulated in terms of its components: canopy photosynthesis (gross primary production, GPP), respiration (*R*) and litter turnover (*T*).

### Upscaling from leaf to canopy photosynthesis

Leaf photosynthesis is described by the nonrectangular hyperbola model, which predicts leaf responses much more accurately than the rectangular hyperbola model (Thornley, 2002). In this model, light-saturated photosynthesis (*A*_{max}) is a linear function of N content per unit area (*N*_{A}) and minimum *N*_{A} per leaf area (*N*_{min}), *A*_{max} = *a*(*N*–*N*_{min}). The slope of the photosynthetic capacity vs leaf N (*a*) is central to the optimal plant behaviour and is related to the allocation of N to structural and photosynthetic uses in the leaf and to stomatal conductance (Hikosaka, 2004). *a* is increased by elevated CO_{2} and reduced by water deficit (through stomatal regulation).

Using Beer's law of light extinction and optimal distribution of canopy N (*N*_{c}), as described in Franklin & Agren (2002), the total daily canopy photosynthesis (GPP, see Eqn 1a) can be derived as a function of *N*_{c} and absorbed PAR (*I*_{a}) through integration of leaf photosynthesis over the canopy (Supplementary Material, Eqns S1, S2). More complex canopy models, for example, differentiating sun and shade leaves, would probably be more accurate in absolute terms (dePury & Farquhar, 1997; Thornley, 2002). However, for the purpose of this paper, that is, to elucidate relative differences between CO_{2} treatments, the increased complexity would mainly serve to obscure the results.

(*h*, day length; φ, quantum efficiency; θ, a curvature parameter of leaf light response (parameter values and units are given in supplementary material Table S1)). *I*_{a} is related to radiation above the canopy (*I*_{0}), LAI (*L*) and the light extinction coefficient (*k*) according to:

*I*

_{a}=

*I*

_{0}(1 −

*e*

^{−kL}) (Eqn 1b)

LAI (*L*) is determined by maximizing GPP (Eqn 1a) for a fixed *N*_{c}. Increasing LAI at small LAI increases GPP through increased light absorption. As LAI gets larger, light absorption saturates while the proportion of N that is nonphotosynthetic increases linearly with LAI (through the term *N*_{min}* L* in Eqn 1a), reducing GPP. Thus, for a fixed *N*_{c}, GPP has a maximum with respect to LAI (Fig. 1). This optimal LAI, for simplicity hereafter denoted just LAI or *L*, is approximately linearly related to *N*_{c} (Fig. 2), which means that mean canopy leaf N per area (*N*_{A} = *N*_{c}/LAI) is conservative during canopy development and among mature stands with differing *N*_{c}. However, *N*_{A} increases with photosynthetic capacity per N (*a*) and decreases with quantum efficiency (φ) (Fig. 1). These shifts in *N*_{A} happen because, for a fixed *N*_{c}, φ increases the initial slope of GPP vs LAI, while *a* increases GPP only at higher LAI (Eqn 1, Fig. 1). A combined increase in *a* and φ, where the increase in φ is one-third of the increase in *a*, as expected at elevated CO_{2} (Cannell & Thornley, 1998), decreases *N*_{A} slightly for all *N*_{c} (Fig. 2). As GPP and LAI have been derived as functions of *N*_{c}, it is necessary to evaluate optimal *N*_{c} before they are fully defined.

### Respiration and turnover

Respiration is modelled using the maintenance + growth respiration approach. Growth respiration (*R*_{g}) is a fraction of net assimilation, *R*_{g} = (1 – *y*)(GPP – *R*_{m}), where *y* is the biosynthetic conversion efficiency, which is conservative; *y* = 0.7 for whole-plant, woody species (Choudhury, 2001). Maintenance respiration (*R*_{m}) is proportional to N content across all living tissues, that is, canopy, sapwood and roots (Vose & Ryan, 2002), which can be explained through respiratory costs of protein turnover (Dewar *et al*., 1998). A factor *q*_{r} > 1 accounts for the fact that fine roots have a higher respiration per N than other tissues (Ryan *et al*., 1996):

*R*

_{m}=

*r*(1 +

*q*

_{r}

*f*

_{r}+

*f*

_{s})

*N*

_{c}=

*r*

_{w}

*N*

_{c}( Eqn 2a)

(*r*, basic respiration rate per unit nitrogen; *f*_{s}, *f*_{r}, ratios of N in sapwood and fine roots to N in canopy; *r*_{w}, whole-plant respiration per canopy nitrogen). Subscripts *c*, *r* and *s*, represent canopy, fine roots, and sapwood, respectively. The ratios *f*_{s}, *f*_{r} are restricted by the need for root and stem tissue to maintain the canopy (pipe theory; Shinozaki *et al*., 1964), but changes in response to environmental and ontogenetic factors, such as soil nutrient availability (changes *f*_{r}) and tree height (changes *f*_{s}).

Turnover (*T*) is expressed as a function of *N*_{c}, mean residence times of tissues (*t*), and N : C ratios (*n*) of canopy and roots:

(*l*_{w}, whole-plant litter production (turnover) per *N*_{c}).

To simplify the following expressions, an aggregated variable (*w*) of respiration and turnover per *N*_{c} is defined to represent total carbon costs per canopy nitrogen:

*w*=

*yr*

_{w}+

*l*

_{w}( Eqn 2c)

*f*_{r} and *f*_{s} are the main controls of *w* and are ultimately increased by root allocation in response to reduced soil N availability and increased mass of living wood, respectively. Compared to *f*_{r} and *f*_{s}, changes in the N : C ratios (*n*) and turnover times (*t*) have a smaller impact since they only enter the turnover function (Eqn 2b) and not the respiration function (Eqn 2a). Furthermore, they tend to be inversely correlated, for example, increased leaf N : C ratio (*n*_{c}) is correlated with shorter life span (*t*_{c}) (Wright *et al*., 2004). Nevertheless, no such relation is imposed here and effects of changes in *n*_{c} at elevated CO_{2} were evaluated.

It should be noted that, although *N*_{c} and *n*_{c} both occur in the expression for turnover, because leaf mass per area is not fixed, they are mathematically independent. Thus the derivation of optimal *N*_{c} (see following section) would not be invalidated by a changing *n*_{c}.

### Production and efficiency of expanding and steady-state canopies

For expanding canopies, the development of GPP, *R*_{m} and *T* as functions of canopy N (*N*_{c}) can be derived directly from Eqns 1a and 2a,b. Here LAI is optimized for each *N*_{c}, as described above. NPP and *G* are then calculated according to Eqn 3a. As the canopy expands, *N*_{c} eventually reaches an optimal value where *G* is maximized and where no further expansion occurs (Fig. 3), unless there is a change in parameters. Optimal *N*_{c} is thus determined by the optimal tradeoff between the N-based carbon gain (GPP) and carbon losses (*R*_{m} + *R*_{g} + *T*). The state reached after full canopy expansion is hereafter referred to as steady state, which also includes any subsequent shifts in canopy optimal size resulting from changes in parameters, such as increasing sapwood (*f*_{s}) with height. Because all the terms of *G* are functions of *N*_{c}, the steady-state *N*_{c} (Eqn 3b) is readily derived by maximizing *G* (Eqn 3a) with respect to *N*_{c} (Supplementary Material). This optimal state of the system represents acclimation over timescales not shorter than the response time of the slowest relevant response mechanism. For example, in response to a change in CO_{2} concentration, it may take a growing season for the root : foliage ratio and *N*_{c} to reach a new equilibrium; therefore, modelled optimal growth rate represents a growing season average and not shorter-term fluctuations.

By inserting in the expressions for GPP and *R*_{m}, the properties GPP*, NPP*, and *G** of the steady-state canopy are obtained (Eqns 3b–f, where * denotes the optimized steady-state canopy). NPP and *G*, but not GPP, follow different paths in relation to *N*_{c} during canopy expansion and at steady state (Fig. 3).

*G*= NPP −

*T*=

*y*(GPP −

*R*

_{m}) −

*T*=

*y*GPP −

*w*

*N*

_{c}(Eqn 3a)

(*ɛ*_{sat}=*a*(–*N*_{min}*L*)/*I*_{a}, light-saturated photosynthetic light-use efficiency, that is, if all leaves operate at *A*_{max}).

*hI*

_{a}ɛ

_{GPP}( Eqn 3c)

where

The light-use efficiency of GPP (ɛ_{GPP}) has an upper theoretical limit of φ (2.73 µg C J^{−1}, Wong *et al.*, 1979).

*G** and NPP* are given by

(ɛ_{G}, slope of the light-use efficiency of *G*).

GPP*, NPP*, *G** and are all approximately linear functions of absorbed PAR where the light-use efficiencies (the slopes) are controlled by the leaf quantum efficiency (φ) and the ratio *ahy/w*. The factors in this ratio thus strongly regulate the system; the numerator reflects responses in photosynthetic capacity (*a*) and day length (*h*) and the denominator, total carbon costs per canopy N (*w*), responds to the relative allocation to nonphotosynthetic parts, that is, roots (*f*_{r}) and sapwood (*f*_{s}). While GPP*, NPP*, *G** all monotonically increase with the ratio *ahy/w*, the effect on is more complex. is increased by the ratio *ahy/w* but at the same time it decreases with *a* (denominator in the first factor in Eqn 3b), which makes the response to *a* small. An important consequence of this is that is much less sensitive to changes in photosynthetic capacity (*a*) than to changes in root and sapwood allocation (through *w*). The response to *a* is also variable, that is, positive at small and negative at larger . As occurs where the slopes of GPP and *yR*_{m} +*T* vs *N*_{c} are parallel (cf. Fig. 4), the response of to *a* is determined by the change in slope of GPP vs *N*_{c} in response to *a *[= (∂/∂*a*) (∂GPP/∂*N*_{c})]. This change of slope is positive at small and negative at larger .

Effects of elevated CO_{2} enter the system through a combined increase in *a* and φ (Cannell & Thornley, 1998). These parameters directly affect GPP and, as shown in Eqn 1a, elevated CO_{2} raises both the initial slope (through *a*) and the maximum (through φ) of GPP as a function of *N*_{c}. The increase in φ has a positive effect on , while, as discussed above, the effect of *a* on is negative unless is very small. The combined increases in φ and *a* cause a small positive net effect of elevated CO_{2} on , although the effect increases at decreasing (Fig. 4). However, if at the same time carbon costs per *N*_{c} (*w*) increases, for example because of increased fine-root allocation, the total effect may be reduced at elevated CO_{2} (Fig. 4).

The carbon use efficiency = NPP/GPP ratio can be expressed and, for steady-state canopies, approximated by:

where the function *Q* is insensitive to variation of its parameters (*Q* ≈ 1.3 for all realistic parameter values; Supplementary Material, Eqn S7). The NPP/GPP* ratio has an upper limit of *y* (≈ 0.7) and is rather insensitive to changes in relative root and stem allocation (*f*_{r} and *f*_{s}) and associated changes in . This invariance is not surprising since both GPP and *R*_{m} are increasing functions of *N*_{c} (Eqns 1a, 2a). The NPP/GPP* ratio is increased by elevated CO_{2} through the effect on *a* (Eqn 3g) and decreases during canopy expansion (Fig. 5). The effect of canopy expansion is the result of the saturating response of GPP combined with the linear increase in *R*_{m} (Fig. 3).

Optimal LAI of the expanding canopy (*L*) and the steady-state canopy (*L**) are identical for a fixed *N*_{c} (Fig. 1). However, for the steady-state canopy an analytical expression for *L**, including the effect of optimal *N*_{c}, can be obtained by inserting *I*_{a} (Eqn 1b) in *G** (Eqn 3e) and maximizing with respect to *L*, which gives:

The PAR absorption of the optimized canopy (*I*_{a}) is then obtained through its direct link to *L* (Eqn 1b) as

PAR absorption saturates earlier with increasing canopy size than light-use efficiency because of its exponential nature (Fig. 2).

As shown above, LAI vs *N*_{c} (*N*_{A}) increases slightly with CO_{2} for a fixed *N*_{c}. Here Eqn 3h shows that also when including the effect of CO_{2} on *N*_{c}, for steady-state canopies, *L** (at constant *w*) is slightly increased by elevated CO_{2} through the increased light-use efficiency (*ɛ*_{G}). However, the *L** response to CO_{2} is always smaller than the response of ɛ_{G}, since *L** is a logarithmic function of *ɛ*_{G}.

### Modelling the CO_{2} effects in FACE experiments

To evaluate the hypothesized model, published observations from four forest FACE sites (whole stand elevated CO_{2} experiments) representing closed-canopy stands (meaning canopies horizontally filling the growing space) were used. The sites are: Oak Ridge (sweet gum, steady-state canopies), Duke forest (Loblolly pines, steady-state canopies), POPFACE (poplars, expanding canopies) and Aspen FACE (mixed aspen dominated, expanding canopies). The observations used here all represent elevated and ambient CO_{2} treatments where no other treatments were applied. Further information about the sites and relevant references are given in Table 1.

Site name | Location | Mean temp (°C) | Species | Planting year | Canopy state^{a} | n^{b} | Δa (%)^{c} | Δφ (= Δa/3) | Δf_{r} (%)^{d} |
---|---|---|---|---|---|---|---|---|---|

- *
Further details of the sites available in King *et al*. (2004). Superscripted numbers represent the following: 1, Calfapietra*et al*. (2005); 2, Gielen*et al*. (2005); 3, Lukac*et al*. (2003); 4, Takeuchi*et al*. (2001); 5, King*et al*. (2005); 6, Springer*et al*. (2005); 7, Finzi*et al*. (2002); 8, Finzi*et al*. (2006); 9, Norby & Iversen (2006). - a
Expanding canopies have not yet reached their optimal state (steady state). - b
Number of pairs of plots (elevated and ambient CO _{2}) used for each site (only observations from years when the canopy was closed were included). - c
Observed CO _{2}effect on leaf photosynthetic capacity per N (*a*). - d
Observed CO _{2}effect on fine-root : leaf N ratio (*f*_{r}).
| |||||||||

POPFACE | Italy: 42°22′-N, 11°48′-E | 14.1 | Three poplar species | 1999 | Expanding | 3 | 53^{1} | 17.7 | 49.8^{2,3} |

Aspen FACE | USA: 45°36′-N, 89°42′-W | 4.9 | Aspen, birch | 1997 | Expanding | 8 | 21^{4} | 7 | 4.1^{5} |

Duke forest | USA: 35°59′-N, 79°6′-W | 15.5 | Loblolly pine | 1983 | Steady state | 1 | 38^{6} | 12.7 | 52.2^{7,8} |

Oak Ridge | USA: 35°54′-N, 84°20′-W | 14.2 | Sweetgum | 1988 | Steady state | 1 | 58^{6} | 19.3 | 105.3^{9} |

Parameters for the control plots of the FACE stands were collected from publications, except for *a*, *f*_{r} and *r*, which were determined by fitting of observed data (GPP, *G*, NPP, LAI) (supplementary material, Tables S1 and S2), using length of growing season to derive annual numbers from the daily values given by the equations. The modelling of the elevated CO_{2} stands was then done with the same parameter values as for the control stands, except for the leaf photosynthetic capacity per N (*a*) and the root N : canopy N ratio (*f*_{r}), which were multiplied by the observed relative changes (in percentage) in these parameters caused by the elevated CO_{2} treatment for each site (Table 1). In addition, an *a*-dependent increase in quantum efficiency (φ) was applied, equal to one-third of the increase in *a* (Cannell & Thornley, 1998; Long *et al*., 2004). By fitting the parameters *a* and *f*_{r} for control stands and then applying observed relative changes to model the changes caused by elevated CO_{2}, the influence of potential errors in the measured absolute values in these highly spatially and temporally variable parameters is removed. In this way focus is kept on the changes caused by elevated CO_{2} while minimizing potential effects of problems in baseline predictions or observations.

The observed CO_{2}-induced increase in *a* (Δ*a*) was obtained by fitting observed leaf *A*_{max}–*N*_{A} data for ambient and elevated CO_{2} treatments to separate slopes and a common *N*_{min}, since estimated *N*_{min} was not significantly different between treatments. The effects on *f*_{r} (Δ*f*_{r}) were calculated from observed data on differences between elevated and ambient CO_{2} treatments in amounts and nitrogen concentrations of fine roots and leaves. For Duke forest, Δ*f*_{r} was partly estimated by extrapolation in time because of missing data for the main part of the investigated period. For the poplar sites (POPFACE and Aspen FACE), despite slight differences among the species within each site, one value of Δ*f*_{r} per site was used because Δ*a* was available for one species per site only.

Small changes in physiological parameters resulting from CO_{2} treatment were also observed for leaf and fine-root N : C ratios and turnover rates. These effects only marginally affected the results (slightly improving the fit in Fig. 7) and were excluded from the further modelling in order to keep focus on the more important factors. The reasons for the small effects of changes in N : C ratios in this framework are that photosynthesis (within a species) is much more strongly linked to *N*_{A}, which is used in this model, than to N : C ratio (Meir *et al*., 2002), while for respiration, the effects of N : C ratio are indirectly included through changes in total N.

Because the poplar stands were in a phase of expanding canopy, the preoptimal production equations as functions of canopy N were used, while for Oak Ridge and Duke forest the optimized equations were used since these stands had reached steady-state canopies.