•We formulate a dynamic evolutionary optimization problem to predict the optimal pattern by which carbon (C) and nitrogen (N) are co-allocated to fine-root, leaf, and wood production, with the objective of maximizing height growth rate, year by year, in an even-aged stand.
•Height growth is maximized with respect to two adaptive traits, leaf N concentration and the ratio of fine-root mass to sapwood cross-sectional area. Constraints on the optimization include pipe-model structure, the C cost of N acquisition, and agreement between the C and N balances. The latter is determined by two models of height growth rate, one derived from the C balance and the other from the N balance; agreement is defined by identical growth rates.
•Predicted time-courses of maximized height growth rate accord with general observations. Across an N gradient, higher N availability leads to greater N utilization and net primary productivity, larger trees, and greater stocks of leaf and live wood biomass, with declining gains as a result of saturation effects at high N availability. Fine-root biomass is greatest at intermediate N availability.
•Predicted leaf and fine-root stocks agree with data from coniferous stands across Finland. Optimal C-allocation patterns agree with published observations and model analyses.
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Strategies for investigating the interplay of carbon (C) and nitrogen (N) in plant growth include the evolutionary optimization and game paradigms (e.g. Mäkeläet al., 2002). Optimization ordinarily involves the formulation of a teleonomic model to maximize an objective function with respect to one or more variable plant traits. The trait optimization is subject to physiological, structural, and environmental constraints on plant structure and function. In a game problem, the effect of competing individuals on the success of the alternative strategies is also considered (Maynard Smith, 1982; Mäkelä, 1985; King, 1993; Dybzinski et al., 2011).
Examples of objective functions include net growth rate (Franklin, 2007), gross primary productivity (McMurtrie et al., 2008), net energy for reproduction (Finnoff & Tschirhart, 2009), and total entropy production (Dewar, 2010). The optimization approach to modeling is efficient because it eliminates the need for explicit submodels of the variable traits and their attendant parameters (Dewar et al., 2009). The prime justification for this modeling approach, however, is based on Darwinian selection: plants that are best at optimizing functional traits are the likeliest to survive to produce offspring. In this regard, the objective function serves as a proxy of fitness.
To date, applications of the evolutionary paradigm have provided testable predictions about the co-allocation of C and N to shoots and roots for either of the two extremes of plant growth, namely, exponential growth (Hilbert, 1990; Ågren & Franklin, 2003) or steady state (Dewar, 1996; Franklin, 2007; McMurtrie et al., 2008; Mäkeläet al., 2008; Dybzinski et al., 2011). Models of exponential growth provide optimal patterns of co-allocation that maintain a balance between shoots and roots. The co-allocation satisfies demands for C and N that are defined by the fixed or plastic stoichiometries of the two tissues. The acquisition of C and N, respectively, is assumed to be proportional to shoot and root size (Johnson & Thornley, 1987; Mäkelä & Sievänen, 1987; Hilbert, 1990; Ågren & Franklin, 2003). Many of the optimal steady-state models focus on canopy properties when all available N is utilized, and therefore do not explicitly consider a C cost of N acquisition through the maintenance of a system of fine roots (e.g. Dewar, 1996; Franklin, 2007). However, a recent model by Mäkeläet al. (2008) generalized the idea of the exponential balanced-growth models to steady-state, resource-limited situations (see also Dybzinski et al., 2011). Given the availability of soil N, this steady-state model maximizes net primary productivity (NPP) by optimizing the co-allocation of C and N to leaf and fine-root production, and to leaf N concentration. The optimal co-allocation is constrained by: a structural model, that is, the pipe model (Shinozaki et al., 1964); saturation levels of leaf and fine-root biomass with regard to C and N acquisition; and agreement between C and N balances. The steady-state model accounts for the C costs for producing and maintaining leaves, wood, and fine roots, and, in so doing, accounts for the cost of acquiring N by the root system.
The results of the balanced-growth optimization models, both exponential and steady state, are in reasonable agreement with our general understanding of how N availability influences C allocation. They suggest, for example, that increasing N availability reduces the allocation of C to fine roots while simultaneously increasing the leaf N concentration (Hilbert, 1990; Ågren & Franklin, 2003), and that the optimum steady-state stocks of leaves and fine roots are determined by the relative availabilities of C and N (Mäkeläet al., 2008). The steady-state optimization model of Mäkeläet al. (2008) and the model of Dybzinski et al. (2011), which is based on evolutionarily stable strategies, indicate that the fraction of NPP allocated to leaf production in stands at steady state is largely unaffected by increasing N availability. The primary trade-off in allocation is between wood and fine-root production, the fraction of NPP allocated to wood increasing with N availability at the expense of fine roots. These model predictions are consistent with recent observations (Litton et al., 2007). However, models differ in which adaptive traits are considered and the degree of plasticity allowed, particularly with regard to tissue N concentration.
To our knowledge, a dynamic analysis of optimal C and N co-allocation, bridging the early and later phases of stand growth, has not been reported (but see Franklin, 2007). A dynamic analysis of stand growth seems to require an explicit treatment of height growth, which, so far, has been ignored in most of the evolutionary optimization studies. In closed stands, trees compete for light and other resources, so rapid height growth is a crucial means of survival and reproductive success for species of low or moderate shade tolerance (e.g. Horn, 1971; Mäkelä, 1985). Key to a dynamic analysis is an understanding of how different resources influence shoot production and elongation so we can relate height growth to the material balances of the model. The optimization must account for the fact that the growth of shoots, coarse roots, and supporting stems involves a trade-off with the growth of the leaves and fine roots that capture the resources.
In this paper, we formulate a dynamic evolutionary optimization problem to predict the optimal pattern of co-allocation of C and N to the fine roots, leaves, and wood that maximizes height growth rate over the course of development of an even-aged stand. The analysis draws from our previous work on optimal C and N co-allocation in an even-aged, closed-canopy stand of trees at quasi steady state (Mäkeläet al., 2008). Here we advance our previous approach with a height growth model derived from the requirements of C balance (Valentine & Mäkelä, 2005), the pipe model (Shinozaki et al., 1964), and optimal crown allometry (Mäkelä & Sievänen, 1992). We also derive a second height growth model from the N balance in analogy to height growth rate based on the C balance. The two height growth rates are the same if the C and N balances are in agreement. In addition, we define the mortality rate of trees on the basis of leaf and fine-root density, and assume that crown rise is driven by crowding in space (after Valentine et al., 1994). Finally, we maximize the height growth rate of the trees by optimizing two adaptive traits – leaf N concentration and fine-root biomass per unit sapwood area – on a yearly time step. This strategy leads to optimizing photosynthetic N concentration and the co-allocation of C and N to the production of fine-root, leaf, and wood biomass, while accounting for the C costs of respiration and N acquisition from the soil.
The model is applied with parameter values appropriate for Scots pine (Pinus sylvestris) in southern Finland. We consider the dynamics of the co-allocation of N and C to fine roots, leaves, and wood in stands with different maximum rates at which N can be taken up from the soil. Because the focus of this paper is on growth dynamics, we ignore the N dynamics and assume that the abundance of available N within the soil at a given site is constant over time.
In order to assess the significance of some of the postulated constraints, we consider the implications of the degree of plasticity of the adaptive traits by comparing stand development patterns that emerge from optimal C and N balances against patterns that emerge from a law-of-the-minimum (LM) approach, that is, stand growth based on the minimum of the C or N-based height growth rates.
We bound our analysis by focusing on three research questions: first, to what degree do the values of the two adaptive parameters change over the course of stand development? Second, do the results under the law-of-the-minimum approach differ markedly from those under the optimal-balance approach? Finally, does a steady-state response represent the essence of the closed-canopy dynamics?
The growth model is presented in sufficient detail for replication and advancement of our work.
Components of live biomass
We consider a pure stand of N (ha−1) identical trees. Each tree has three components of live biomass (kg): Wf, leaves; Wr, fine roots; and Ww, sapwood. The total sapwood comprises the transport roots and aboveground sapwood. Leaves die and turn over after Tf years, and fine roots after Tr years. Sapwood volume is defined in terms of three state variables: A (m2), the cross-sectional area of sapwood at the crown base (‘sapwood area’); H (m), the height of the crown tip; and Hc (m), the height of the crown base.
Using a simple pipe model, we assume that the length of a strand of sapwood from a fine root to a leaf – the average length of an active pipe – is Hc + βc(H−Hc) + βr(H−Hc), where βc(H−Hc) is the average length of pipe from the crown base to a leaf, and βr(H−Hc) is the average length of pipe below ground. Hence, average pipe length is β1H + β2Hc, where β1 = βc + βr and β2 = 1−β1 are dimensionless parameters. The three live components of biomass, expressed in terms of the pipe model, are
where ρf and ρr are ratios of biomass to sapwood area (kg m−2). The parameter ρw is the bulk density of sapwood (kg m−3). Hence the volume of sapwood is Ww/ρw.
Rates of change
Let dWi /dt (kg yr−1) be the rate of change of component i = f, r, w. Then, by definition,
where Gi is the rate of production of new tissue, and Si is the rate of loss of old tissue to senescence or deactivation. Similarly, the rate of change of sapwood area (m2 yr−1) is, by definition,
where GA is the rate of production of new sapwood area, and SA is the rate of loss of old sapwood area to deactivation. Based on the optimal crown model of Mäkelä & Sievänen (1992) and the pipe model, Valentine & Mäkelä (2005) assumed that A ∝ (H−Hc)z, and decomposed the time derivative
Note that subtraction of Eqn 8 from Eqn 7 gives Eqn 6. With the decomposition, the production rate of new sapwood is associated with the rate of height growth, and the deactivation rate of old sapwood is associated with the rate of crown rise, that is, the rate at which the crown base rises. This decomposition is key to the development of our model because with Eqns 7, 8, together with Eqns 1–3, we can express the rates of production of all three components of live biomass in terms of the dynamics of H and Hc, that is,
The production rates of leaf and fine-root biomass, respectively, include annual rates of replacement of senescent leaf and fine-root biomass, Wf/Tf and Wr/Tr. The rates of growth and loss for all three components of biomass are provided in Supporting Information Notes S1.
The net primary productivity of the model stand is
where P and Rm, respectively, are the stand-level rates of photosynthesis and maintenance respiration (kg C ha−1 yr−1) and Y (kg biomass (kg C)−1) is the conversion efficiency of C to biomass, accounting for construction respiration (Thornley, 1970). Substitution of Eqns 9–11 into Eqn 12 provides the C-based rate of height growth (dHCB/dt) as a function of P and Rm (Valentine & Mäkelä, 2005), that is,
where σfM (kg C (kg leaf)−1 yr−1) is the light-saturated specific rate of photosynthesis, and Kf (kg ha−1 yr−1) is the stand leaf biomass at which P equals half the saturated rate. We chose this model because its mathematical properties afford a closed-form solution at steady state. A formulation based on leaf area is an option, but would increase the number of parameters.
where σfM0 (kg C (kg leaf)−1 yr−1) is the N-saturated specific rate of photosynthesis, and [N]ref is the value of [N]p that provides half the N-saturated rate. [N]p is the difference between the total and structural leaf N concentration ([N]0),
The total leaf N concentration, [N]f, changes with time because it is optimized from year to year. We assume that fine-root N concentration is proportional to [N]f, that is, [N]r = rN[N]f (Helmisaari et al., 2007).
We model the rate of maintenance respiration in terms of the component N concentrations (e.g. Ryan, 1991):
where rm (kg C (kg N)−1 yr−1) is the N-specific respiration rate.
We assume that the rate at which N is concentrated in new biomass equals the rate of N uptake from the soil plus the rate of resorption of N from senescent biomass (kg N ha−1 yr−1), that is,
Substituting Eqns 9–11 into Eqn 18, and solving for the N-based height growth rate (dHNB/dt) yields
(i = f,r,w).
Analogous to our model of photosynthesis, Eqn 14, we model the rate of N uptake (kg N ha−1 yr−1) as a saturating function of Wr:
where σrM (kg N (kg fine root)−1 yr−1) is the maximum fine-root specific rate of N uptake, and Kr (kg ha−1) is the stand fine-root biomass at which the N uptake rate equals half the maximum rate.
As leaf biomass turns over, a fraction of its N content is resorbed for reuse in production. It seems plausible that resorption also may occur as fine roots turn over. Let ff and fr, respectively, be resorption fractions from senescent leaves and fine roots; then
Hydraulic resistance to sap flow reduces the specific rate of photosynthesis, σfM, in some species (Yoder et al., 1994). As the greatest resistance occurs in branches (Hellkvist et al., 1974; Tyree, 1988), we follow Mäkelä (1997), and account for the resistance effect by letting σfM decrease with crown length, that is,
We assume a proportional decrease in the maximum fine-root specific rate of N uptake:
Stand density and spacing
By definition, average tree spacing, X (m), depends on the stand density, N (ha−1), that is, . Consequently, the rate of increase in the average spacing (m yr−1) attributable to tree mortality (ha−1 yr−1) is:
We assume that tree mortality begins when either leaf or fine-root biomass per unit land area reaches a critical value. We define the rate of mortality with parameters of the saturating photosynthesis and N uptake functions. Hence, the mortality rate, Mi (ha−1 yr−1), triggered by leaf (i = f) or fine-root (i = r) abundance, is
where mi (yr−1) is a rate parameter. The actual mortality rate is the maximum of Mf and Mr, that is,
The rate of crown rise is based on the model of Valentine et al. (1994), which posits that average crown length in a closed stand is proportional to the average spacing among trees, that is, H−Hc = β3X. A model stand is deemed open if H < β3X + Hc. Hence, crown rise begins (or resumes) as H→β3X + Hc; therefore,
In concert with observations from fertilizer trials of loblolly pine (Pinus taeda) (Albaugh et al., 2006), we assume that longer crowns are supported by greater fertility. We achieve this effect by letting β3 vary with the leaf-specific rate of photosynthesis, that is,
The model is organized as an initial value problem and solved, year by year, by numerical integration. Starting values for the solutions presented here are A = 0.284 × 10−5m2,H = 0.1 m, and Hc = 0 m. The initial stand density is N = 1600 trees (ha−1) for most scenarios. Eqns 1–3 provide the initial values of the components of biomass. Parameter values are provided in Table 1. (Note: the resorption fraction, ff, is fixed at 0 because N cycling is ignored, and the availability of soil N is assumed constant.)
Table 1. Parameter symbols, definitions, values, and units
Reduction in leaf-specific rate of photosynthesis per unit crown length (hydraulic limitation)
kg C m−1
N resorption fraction from senescent leaves
N resorption fraction from senescent fine roots
Leaf biomass for half maximal photosynthesis
Fine-root biomass for half maximal N uptake
Mortality rate from aboveground forcing
Mortality rate from belowground forcing
Non photosynthetic N concentration in leaves
Total N concentration in leaves
N concentration in leaves for half maximal leaf-specific rate of photosynthesis
N concentration in sapwood
Ratio of fine-root to leaf N concentration
N specific rate of maintenance respriation
kg C kg−1 yr−1
Conversion ratio of C to biomass
kg (kg C)−1
Scaling exponent: A ∝ (H−Hc)z
Ratio of average pipe length above the crown base to crown length
Ratio of average pipe length below ground to crown length
Baseline value of β3, the ratio of crown length to tree spacing after closure
Slope of β3 vs leaf-specific rate of photosynthesis
Ratio of leaf biomass to sapwood area
Ratio of fine-root biomass to sapwood area
Ratio of sapwood biomass to sapwood volume
N-saturated leaf-specific rate of photosynthesis
kg C (kg leaf)−1 yr−1
We assume that each of the N identical ‘model trees’ in a stand is endowed with two adaptive traits: the ratio of fine-root biomass to sapwood area, ρr, and the leaf N concentration, [N]f. The values of these two free parameters must satisfy the balanced growth constraint, that is, dHCB/dt−dHNB/dt = 0. To achieve this, we express the right-hand sides of the two rate equations, Eqns 13, 19, in terms of the pipe model as factors of ρr, which provides
where a1, a2, and a3 are functions of the state variables and parameters, including [N]f (Notes S2). Thus, Eqn 29 provides the value(s) of ρr for which dHCB/dt = dHNB/dt given any proposed value of [N]f and, therefore, any set of trait values comprising the proposed value of [N]f and the calculated value of ρr ensures balanced growth. Of interest is the optimal set of values that maximizes dHCB/dt (or minimizes −dHCB/dt). We used numerical search or the Nelder–Mead simplex method (see, e.g. Press et al., 2003) to find the optimal sets.
In our solutions, we re-optimized the values of the free parameters at the start of each year, and used those values for the duration of that year. By re-optimizing these trait values, we effectively optimized the co-allocation of C and N to the production of the three components of biomass: Wf,Wr, and Ww.
The model also can be solved without year-to-year optimization by assigning fixed values to free parameters. Or, given the maximum rate of N uptake, we can search for the optimal fixed values of [N]f and ρr that maximize total tree height at a reference age of, say, 50 yr. As fixed parameters do not change year to year, equality of C-based and N-based height growth rates over any lengthy period of time is unlikely. Therefore, we solve the model using an LM approach, that is, height growth rate equals the slower of the C-based or N-based growth rates:
Because the model can be solved in two ways, we call solutions with adaptive traits and optimal C and N balances the optimal balance (OB) approach, and solutions with fixed traits the LM approach.
We solved the model for six different initial stand densities: 800, 1200,…, 2800 (trees ha−1), by six different maximum N uptake rates: Nup,max = 40,60,…,140 kg N ha−1yr−1. As it turned out, altering the initial stand density has only a minor effect on the behavior of the model so we concentrate on results for six different maximum uptake rates given an initial stand density of 1600 trees ha−1. We do, however, provide some results for the six initial stand densities, given a maximum N uptake rate of 100 kg N ha−1 yr−1 (Notes S3, Fig. S3).
Plasticity of traits and model stand behavior
With the OB approach, variation in the availability of N leads to wide variation in the two adaptive traits, leaf N concentration, [N]f, and the ratio of fine-root mass to sapwood area, ρr, the predicted optimal values of former increasing, and the latter decreasing, with increasing values of Nup,max (Fig. 1a,b). Because these traits are optimized from year to year, they also show plasticity over time (Fig. 1c,d). The temporal plasticity is more pronounced in the values of ρr, which are greatest at stand initiation, decreasing to approach their respective, nearly constant, final values at stand closure. The ratio of leaf mass to sapwood area, ρf, is fixed at 460 kg m−2, so the ratio ρr : ρf initially exceeds 1 for every value of Nup,max, meaning that fine-root mass initially exceeds leaf mass in every case.
The OB approach gives rise to a wide array of dynamics (see Notes S3, Figs S1–S3). The predicted time-courses of gross primary productivity (GPP), NPP, and the percentages of NPP allocated to the three components of biomass are depicted in Fig. 2. GPP increases rapidly after initiation and then tends to plateau or gradually decline by a small amount after closure (Fig. 2a). NPP, by contrast, tends to decline more rapidly after peaking, except at the lowest levels of N uptake. The ratio of net to gross primary productivity declines over the course of stand development from about 0.72, initially, to about 0.4 at 150 yr, with some variation, the N-poor stands having the larger final ratios (not shown).
The predicted fraction of NPP allocated to leaf (Fig. 2b) and fine-root production (Fig. 2c) decreases between stand initiation and closure. An initially large fraction of NPP allocated to leaf biomass decreases rapidly before the stand closes, and then the fraction tends to stabilize at about 0.2 (or 0.25 at the lowest levels of N uptake) with a slight increase as the stand ages. An obvious trade-off in C allocation between fine-root and sapwood production is evident in Fig. 2(c,d). At any given point in time, less C allocation to fine roots is matched by more C allocation to wood, and vice versa. Fine-root production decreases, and wood production increases, with increasing N availability. N saturation effects are apparent at the higher values of Nup,max with regard to GPP, NPP, and the allocation of NPP to fine roots and wood.
Predicted time-courses of the components of biomass and total N in biomass are provided in Fig. 3. Greater N content (Fig. 3a) is associated with greater sapwood mass (Fig. 3b), and leaf mass (Fig. 3c). After closure, fine-root mass is greatest for intermediate N uptake rates (Fig. 3f), which accords with the steady-state model of Mäkeläet al. (2008). N saturation effects at the higher rates of N uptake are evident for all the traits except fine-root mass.
Optimal vs limited growth
The optimum time-course of the objective function, height growth rate, behaves realistically, increasing from stand initiation and peaking before closure, and then declining as the model stand ages (Fig. 4a). The time integral of this rate provides the expected sigmoid time-course of average tree height (Fig. 4c).
For the LM approach (Fig 4b,c), fixed values of [N]f and ρr were optimized to maximize total tree height at age 50 yr, given the maximum rate of N uptake, Nup,max. Thus, the LM solutions for each value of Nup,max are based on a unique set of fixed trait values.
Compared with the LM approach, stands modeled under the OB approach achieve closure faster and yield taller average tree heights at any given age, and for any given value of Nup,max. Decreasing differences between tree heights for the faster maximum rates of N uptake indicate a tendency toward N saturation. Note that, even with this tendency, the height growth rates under the LM approach initially are limited by N uptake (Fig. 4b). N limitation persists after closure at the two slowest rates of N uptake, but C becomes limiting to growth after closure at the four faster rates. By comparison (Fig. 4c,d), the two approaches to optimization do not yield markedly different results, but it is clear that maintaining optimal C and N balances – which presumes plasticity in tree structure and function with regard to the acquisition of C and N – is the more beneficial life style if rapid height growth promotes survival.
Leaf N concentrations are similar under the two optimization approaches, and the post-closure values of ρr for the OB approach are very similar to the fixed LM values for corresponding N uptake rates (Fig. S4). This seems to suggest that OB trees gain much of their advantage over LM trees early in life. But that is because the fixed parameters of the LM trees maximize tree height at age 50 yr. Were we to choose fixed parameter values to maximize tree height at, say, age 10 yr, then the OB trees would seem to gain more of their advantage later in life (Notes S4, Fig. S5).
Dynamic vs steady state
To address the question of whether a steady-state model can capture the essence of certain traits after closure, we compared solutions of the OB model at ages 50, 100, and 150 yr, over a range of N availability. We quantified N availability as the asymptotic fine-root-specific rate of N uptake, σrM = Nup,max/Kr. This result is obtained from Eqn 20 with NWr→∞.
Fig. 5 depicts three predicted trend lines for particular traits vs N availability, one line for each age. Coincidence of the three trend lines would indicate a steady state over the 100-yr period. That trend lines are proximal, but not coincident, suggests that steady-state solutions do not apply exactly, but may apply approximately over one or more decades. The steady-state approximation applies better to leaf biomass and GPP (Fig. 5a,c) than to fine-root biomass, NPP, or leaf N concentration.
Some traits, such as height and stemwood biomass, increase continuously, and therefore are ignored in steady-state models. However, the fractions of NPP allocated to the three components of live biomass, including sapwood biomass, do tend to approach steady states (Fig. 5d). A hint of a trade-off in C allocation between sapwood and leaf production is seen at the lowest levels of N availability, but a pronounced trade-off is evident between fine-root and sapwood production, with increasing allocation to sapwood, and decreasing allocation to fine roots, with increasing N availability. These results are nearly identical to those reported by Mäkeläet al. (2008).
Field estimates of leaf and fine-root biomass, and leaf [N] were obtained for eight stands dominated by Scots pine across a climatic gradient in Finland (Helmisaari et al., 2007). The stands occur on different site types with different stages of development, and they were all situated in the Boreal zone, where the effective temperature sum varies between 660 and 1290 degree-days per year. A summary of the data can be found in the supporting information for Mäkeläet al. (2008). There was considerable scatter in the estimates for both leaf and fine-root biomass vs [N]f, but the present model suggests that some of this scatter could be attributable to factors associated with stand age and tree density – inasmuch as the elevations of the trend lines from the model solutions decrease with age and increase with density (Fig. 6a,b). Regardless of which factors are at play, the order of magnitude and general patterns of the model solutions over a range of N availability are consistent with the field estimates. Interestingly, all the trend lines from the present OB model for the leaf to fine-root ratio are nearly coincident, and show reasonable agreement with both the steady-state model of Mäkeläet al. (2008) and the field estimates (Fig. 6c).
Plasticity of traits and model stand behavior
A major rationale for applying evolutionary optimization to models of plant function is the need to understand the life strategies of plants in terms of the plasticity of environmentally adaptive or acclimative traits. In this respect, the present model is consistent with previous work (Hilbert, 1990; Dewar, 1996; Ågren & Franklin, 2003; Franklin, 2007; McMurtrie et al., 2008; Mäkeläet al., 2008; Dybzinski et al., 2011) and our general understanding of plant responses to variations in N availability. For example, the C fraction allocated to leaf production after closure is near the 20% value reported by Litton et al. (2007) for most forests (Fig. 2b). The remaining C allocation is, therefore, a trade-off between fine roots and sapwood (Figs 2c,d, 5d). The predicted total fine-root biomass in a mature stand is lowest where N is either least or most available. Fine-root biomass is highest where N availability is mid-range, in agreement with field data. This complex behavior of predicted fine-root biomass with regard to stand age and N availability is an emergent property of the optimization.
The predictions of the degree of plasticity in the adaptive traits, from stand initiation onwards, are new results. Few data are available for direct comparisons with these predictions, though other researchers have assumed that fine-root allocation is greater in the early phase of stand development (Running & Gower, 1991; Bossel, 1996).
To formulate the present model, we simplified features of stand development in many ways, focusing on what we considered essential. That we, like Dybzinski et al. (2011), ignored the dynamics of N cycling in the soil may be important concerning the temporal patterns of the adaptive traits. Explicit N cycling might change the solution to some extent, especially under nutrient-limited conditions, or where considerable N is bound within accumulations of litter and humus. However, we may have partly compensated for these conditions by our assumption of no N resorption from senescent leaves – at least that was our intent. Inasmuch as our optimization method would lend itself equally well to situations where the availability of nutrients varies with time, inclusion of an N cycling model is a logical next step for advancement of the model.
In this study, we limited our consideration of plasticity to two traits. We did not optimize plasticity of crown shape or foliar density in the crown, although we did include an impact of N availability on crown length. Plasticity of crown shape has implications regarding the acclimation of trees to different canopy layers and, hence, to competition between individuals in the stand (see Horn, 1971; Oliver & Larson, 1990). Considering competition from the evolutionary point of view would require that we define the problem as a dynamic game so that the competitive interactions could be considered explicitly (see, e.g. Mäkelä, 1985). Because we optimized height growth directly, allowance for plasticity in crown shape would have rendered the problem ill-defined, as a narrower or sparser crown would always lead to faster height growth in the short term. However, several studies suggest that the architecture of tree structure is an evolutionary optimization problem that is somewhat independent of growth rate (see Horn, 1971; Mäkelä & Sievänen, 1992; West et al., 1999; Duursma et al., 2010), supporting the idea of separating the treatment of the two problems.
Many studies have considered the plasticity of N content and the consequent variability of photosynthetic capacity inside the canopy. For example, Hollinger (1996) demonstrated that trees allocate N to leaves in a nearly optimal way, with greater N concentrations per unit leaf area where photons are abundant, and lower concentrations where photons are sparse. On the basis of leaf mass, however, the N concentrations did not vary significantly, that is, N mass per unit leaf mass was roughly uniform throughout the crown. The two observations conform with each other if we interpret the leaf area per unit mass as a simple physical response to the light environment. To wit, leaves stretch their masses over larger areas in the shaded parts of the canopy, decreasing the N mass to leaf area ratio, while keeping the N mass to leaf mass roughly constant. This means that, when considering both the rate of photosynthesis and the N concentration in leaves on a mass basis, as was done in this study, the within-canopy variation of photosynthesis can be regarded solely as a function of light availability.
Optimal vs limited growth
Several models have assumed that the dynamic co-allocation of N and C is based on a combination of balanced growth and N limitation (Running & Gower, 1991; Bossel, 1996; Landsberg & Waring, 1997). The balanced growth hypothesis implies that more fine roots are required relative to leaves in N-poor sites, while the N limitation leads to reduced growth when the available N is insufficient to support the amount of production implied by the availability of C. These approaches leave open questions concerning the transfer from the early balanced growth to presumed N limitation as the trees grow larger: How much growth is allocated to fine roots if N is limited, that is, what amount of fine-root mass is sufficient to harvest the available N? How does tissue N concentration and, hence, the N demand of production vary with N availability? How do variations in tissue N concentration affect the physiological rates related to C and N acquisition? Heuristic assumptions (Running & Gower, 1991), empirical observation (Landsberg & Waring, 1997), and detailed mechanistic models (Bossel, 1996) have been explored to suggest answers to these questions for purposes of predicting the dynamic response of growth to N availability.
By modeling stand growth with optimal balances, however, these questions become moot, as the overarching assumption is that, through plasticity of key traits, the trees adjust structure and function to optimize the use of available resources. Relaxing this overarching assumption, and taking an LM approach with the present model (fixing [N]f and ρr to maximize tree height at a reference age) produced an interesting result: N was limiting initially at all levels of N availability, but a switch to C limitation occurred at closure where N availability was moderate to high (Fig. 4b). That C may be limiting after closure may explain, to some degree, the popularity and success of C-balance models for modeling forest growth.
Modeling stand growth using the concept of limiting factors, rather than achieving agreement between the N and C balances, may yield good results and insights (e.g. Dybzinski et al., 2011). The time-courses of height for the OB and LM approaches from the present model are very similar (Fig. 4a,d), although greater tree heights, at a given point in time, are achieved with optimal balances of C and N. Nonetheless, if the traits crucial for achieving the optimal balances lack sufficient plasticity, then the LM solution may be closest to the truth. As a consequence, however, the excess element might be wasted, for example, through an increased rate of respiration (Dewar, 2001), or by a release of sugars as root exudates (Hari & Kulmala, 2008). In this regard, growth based on agreement between the C and N balances is more efficient, and, we conjecture, more likely from a Darwinian standpoint. We further conjecture that if a model indicates that it is less profitable to achieve agreement between the N and C balances than to waste resources, then it is likely that the model lacks the mechanism to achieve profitable agreement. However, if optimal results suggest wider plasticity than has been observed, then it is likely that additional constraints are at play, which have not been considered.
Dynamic vs steady state
Our primary objective for this study was to extend the steady-state model of Mäkeläet al. (2008) to a dynamic situation to provide insights into the dynamics of N and C co-allocation and its implications for whole-stand dynamics, and also test the generality of the steady-state model. In accordance with both goals, we formulated the dynamic model to be as consistent as possible with the steady-state model. The key differences were the inclusion of height growth, crown rise, and mortality. Importantly, the objective function was chosen to be the maximization of height growth rate instead of NPP.
As already noted, previous evolutionary optimization studies have used a range of objective functions, chosen appropriately for the problem at hand. In the steady-state model, our objective function was NPP. We could not use height growth as the objective function because, by definition, there is no growth at steady state, as production equals senescence. However, we allowed the steady-state height to increase with increasing N availability. For the dynamic model, we assumed that rapid height growth is a trait of most low to moderately shade-tolerant trees that survive self-thinning. It has been shown earlier that maximization of height growth is an evolutionarily stable strategy for trees competing for light (Mäkelä, 1985; King, 1993), and that this strategy does not necessarily maximize proxies of stand-level fitness, such as NPP (Maynard Smith, 1982).
Despite the differences, the results indicate qualitative agreement between the two models, where such agreement can be expected. This applies to the growth allocation patterns as well as the trends in biomass components with increasing N availability (Figs 2, 5) (see Mäkeläet al., 2008). We conclude that these results essentially rely on our assumptions concerning tree structure, that is, the pipe model, and our assumptions regarding N and C acquisition, which were shared by both the static and the dynamic approaches. In the dynamic approach, we explicitly connected height growth with the N and C balance of the tree, resulting in a dependence between maximum height growth and N availability. In the static approach, a similar pattern was obtained by maximizing NPP at steady state while demanding that the stable height increase with N availability. These results suggest that optimization problems may be simplified with appropriate choices of constraints and objectives, although it is difficult to provide any general rule for guiding such choices.
In conclusion, this study concerned the optimal co-allocation of C and N in pure, even-aged stands with the objective of maximizing height growth rate. This means that the primary field of application of the results is among species that naturally form single-cohort stands, for example, through fire-controlled regeneration patterns. Particularly, this would cover several species of pine, as well as other shade-intolerant conifers. However, as height growth is crucial for the survival of most species in approximately even-aged populations, it seems likely that similar selective pressures also have prevailed for a wide range of species. Moreover, the model and the related optimization method, which allows for temporally varying traits, also would be applicable to the analysis of the possible optimal responses of trees to silvicultural treatments, for example, harvests or fertilization. The results of this study can be regarded as hypotheses concerning growth patterns and traits that should be more thoroughly tested against empirical data in future research.
We thank H.-S. Helmisaari for the data used in this article. We thank the referees for their thoughtful and useful reviews.