## Introduction

Strategies for investigating the interplay of carbon (C) and nitrogen (N) in plant growth include the evolutionary optimization and game paradigms (e.g. Mäkelä*et 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; Mäkelä*et 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 Mäkelä*et 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 (Mäkelä*et al.*, 2008). The steady-state optimization model of Mäkelä*et 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 (Mäkelä*et 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?