Elevated carbon dioxide (

Predicting how plants respond to atmospheric carbon dioxide (

Understanding the mechanisms underlying predictions of ecosystem carbon
cycle processes is fundamental for the validity of prediction across space
and time. Comins and McMurtrie (1993) developed an analytical
framework, the “quasi-equilibrium” approach, to make model predictions
traceable to their underlying mechanisms. The approach is based on the
two-timing approximation method (Ludwig et al., 1978) and makes
use of the fact that ecosystem models typically represent a series of pools
with different equilibration times. The method involves the following: (1) choosing a time
interval (

Graphic expression of the baseline quasi-equilibrium framework in
understanding plant production response to elevated

In a CN model, plant net primary production (NPP) can be estimated from two constraints based on equilibration of the C balance (the “photosynthetic constraint”) and the N balance (the “nitrogen recycling constraint”) (Comins and McMurtrie, 1993). Both constraints link NPP with leaf chemistry (i.e., N : C ratio) (derivation in Sect. 3.1). The simulated production occurs at the intersection of these two constraint curves (shown graphically in Fig. 1). To understand behavior on medium and long timescales (e.g., wood and slow and passive soil organic pools in Fig. 2; 20–200 years), one can assume that plant pools with shorter equilibration times in the model (e.g., foliage, fine-root, or active soil organic pools in Fig. 2) have reached quasi-equilibrium, and model dynamics are thus driven by the behavior of the longer-timescale pools.

Framework of the Generic Decomposition And Yield (G'DAY) model. Boxes represent pools; arrowed lines represent fluxes. Boxes with dotted boundaries are M-term recycling pools (wood and slow soil). Box filled with diamonds is the L-term recycling pool (passive soil).

The recent era of model development has seen some significant advances in
representing complex plant–soil interactions, but models still diverge in
future projections of

Here, by constructing a quasi-equilibrium framework based on the structure of
the Generic Decomposition And Yield
(G'DAY) model (Comins and McMurtrie, 1993), we evaluate the effects on plant
responses to

Many of the assumptions currently being incorporated into CN models have
previously been explored using the quasi-equilibrium framework; here we
provide a brief literature review describing the outcomes of this work
(Table 1). Firstly, the flexibility of plant and soil stoichiometry has
recently been highlighted as a key assumption (Stocker et al., 2016;
Zaehle et al., 2014). A key finding from early papers applying the
quasi-equilibrium framework was that model assumptions about the flexibility
of the plant wood N : C ratio (Comins, 1994; Comins and McMurtrie, 1993;
Dewar and McMurtrie, 1996; Kirschbaum et al., 1994, 1998;
McMurtrie and Comins, 1996; Medlyn and Dewar, 1996) and soil N : C ratio
(McMurtrie and Comins, 1996; McMurtrie et al., 2001; Medlyn et al., 2000)
were critical determinants of the magnitude of the transient (10 to

A brief summary of the processes and model assumptions evaluated based on the quasi-equilibrium analyses. SLA: specific leaf area; LUE: light-use efficiency.

Changes in plant allocation with

Another hypothesis currently being explored in models is the idea that
increased belowground allocation can enhance nutrient availability under
elevated

The interaction between rising

This section combines both methods and results together because equation
derivation is fundamental to the analytical and graphic interpretation of
model performance within the quasi-equilibrium framework. Below we first
describe the baseline simulation model and derivation of the
quasi-equilibrium constraints (Sect. 3.1); we then follow with analytical
evaluations of new model assumptions using the quasi-equilibrium framework
(Sect. 3.2). Within each subsection (Sect. 3.2.1 to 3.2.3), we first
provide key equations for each assumption and the derivation of the
quasi-equilibrium constraints with these new assumptions; we then provide our
graphic interpretations and analyses to understand the effect of the model
assumption on plant NPP responses to

More specifically, we tested alternative model assumptions for three processes that affect plant carbon–nitrogen cycling: (1) Sect. 3.2.1 evaluates different ways of representing plant N uptake, namely plant N uptake as a fixed fraction of mineral N pools, as a saturating function of the mineral N pool linearly depending on root biomass (Zaehle and Friend, 2010), or as a saturating function of root biomass linearly depending on the mineral N pool (McMurtrie et al., 2012); (2) Sect. 3.2.2 tests the effect the potential NPP approach that downregulates potential NPP to represent N limitation (Oleson et al., 2004); and (3) Sect. 3.2.3 evaluates root exudation and its effect on the soil organic matter decomposition rate (i.e., priming effect). The first two assumptions have been incorporated into some existing land surface model structures (e.g., CLM, CABLE, O-CN, LPJ), whereas the third is a framework proposed following the observation that models did not simulate some key characteristic observations of the DukeFACE experiment (Walker et al., 2015; Zaehle et al., 2014) and therefore could be of importance in addressing some model limitations in representing soil processes (van Groenigen et al., 2014; Zaehle et al., 2014). It is our purpose to demonstrate how one can use this analytical framework to provide an a priori and generalizable understanding of the likely impact of new model assumptions on model behavior without having to run a complex simulation model. Here we do not target specific ecosystems to parameterize the model but anticipate the analytical interpretation of the quasi-equilibrium framework to be of general applicability for woody-dominated ecosystems. One could potentially adopt the quasi-equilibrium approach to provide case-specific evaluations of model behavior against observations (e.g., constraining the likely range of wood N : C ratio flexibility).

Our baseline simulation model is similar in structure to G'DAY (Generic
Decomposition And Yield; Comins and
McMurtrie, 1993), a generic ecosystem model that simulates biogeochemical
processes (C, N, and

Definitions of key variables for the baseline equations.

The baseline simulation model further assumes the following: (1) gross primary
production (GPP) is a function of a light-use efficiency (LUE), which depends
on the foliar N : C ratio (

We now summarize the key derivation of the two quasi-equilibrium constraints,
the photosynthetic constraint, and the nutrient cycling constraint from our
baseline simulation model (details provided in Appendix A1 and A2). The
derivation follows Comins and McMurtrie (1993), which is further elaborated
in work by McMurtrie et al. (2000) and Medlyn and Dewar (1996) and
evaluated by Comins (1994). First, the
photosynthetic constraint is derived by assuming that the foliage C pool
(

Graphic and mathematical illustrations of the

Secondly, the nitrogen cycling constraint is derived by assuming that
nitrogen inputs to and outputs from the equilibrated pools are equal.
Based on the assumed residence times of the passive SOM (

We now move to considering new model assumptions. We first consider
different representations of plant N uptake. In the baseline model, the
mineral N pool (

A mineral N pool was made explicit by specifying a constant coefficient
(

The second function represents plant N uptake as a saturating function of
root biomass (

Magnitudes of the

Relationship between nitrogen uptake coefficient (

The impacts of these alternative representations of N uptake are shown in
Fig. 4. First, the explicit consideration of the mineral N pool with a
fixed uptake constant (

Graphic interpretation of the effect of different nutrient uptake
assumptions on plant response to

Moreover, the approach that assumes N uptake as a saturating function of
root biomass linearly depending on the mineral P pool (McMurtrie et al., 2012) has comparable

By comparison, representing N uptake as a saturating function of mineral N
linearly depending on root biomass (Ghimire et al., 2016; Zaehle and
Friend, 2010) no longer involves the VL-term nutrient recycling constraint
on production (Fig. 4c), which is predicted by Eq. (11). Actual VL-term NPP
is determined only by

In several vegetation models, including CLM-CN, CABLE, and JSBACH, potential
(non-nutrient-limited) NPP is calculated from light, temperature, and water
limitations. Actual NPP is then calculated by downregulating the potential
NPP to match nutrient supply. Here we term this the potential NPP
approach. We examine this assumption in the quasi-equilibrium framework
following the implementation of this approach adopted in CLM-CN (Bonan
and Levis, 2010; Thornton et al., 2007). The potential NPP is reduced if
mineral N availability cannot match the demand from plant growth:

Graphic interpretation of the effect on

As shown in Fig. 5a, the potential NPP approach results in relatively flat
nutrient recycling constraint curves, suggesting that the

The priming effect is described as the
stimulation of the decomposition of native soil organic matter caused by
larger soil carbon input under

To account for the effect of priming on decomposition of SOM, we first
introduce a coefficient to determine the fraction of root growth allocated
to exudates,

Graphic interpretation of the priming effect on plant net primary
production

Comparison of medium-term (M) and very-long-term (VL) net primary
production response to elevated

Root exudation and the associated priming effect result in a strong M-term
plant response to

Effect of priming on key soil process coefficients. Coefficient

The quasi-equilibrium analysis of the time-varying plant response to

C–N coupled simulation models generally predict that the

Processes regulating progressive nitrogen limitation under

A strong M term and persistent

We also showed that the magnitude of the

Model–data intercomparisons have been shown as a viable means to investigate
how and why models differ in their predicted response to

The quasi-equilibrium framework requires that additional model assumptions
be analytically solvable, which is increasingly not the case for complex
modeling structures. However, as we demonstrate here, studying the
behavior of a reduced-complexity model can nonetheless provide real insight
into model behavior. In some cases, the quasi-equilibrium framework can
highlight where additional complexity is not valuable. For example, here we
showed that adding complexity in the representation of plant N uptake did
not result in significantly different predictions of plant response to

The multiple-element limitation framework developed by Rastetter
and Shaver (1992) analytically evaluates the relationship between short-term
and long-term plant responses to

A related model assumption evaluation tool is the traceability framework, which decomposes complex models into various simplified component variables, such as ecosystem C storage capacity or residence time, and hence helps to identify structures and parameters that are uncertain among models (Z. Shi et al., 2015; Xia et al., 2013, 2012). Both the traceability and quasi-equilibrium frameworks provide analytical solutions to describe how and why model predictions diverge. The traceability framework decomposes complex simulations into a common set of component variables, explaining differences due to these variables. In contrast, quasi-equilibrium analysis investigates the impacts and behavior of a specific model assumption, which is more indicative of mechanisms and processes. Subsequently, one can relate the effect of a model assumption more mechanistically to the processes that govern the relationship between the plant N : C ratio and NPP, as depicted in Fig. 1, thereby facilitating efforts to reduce model uncertainties.

Models diverge in future projections of plant responses to increases in

The code repository is publicly available via DOI

Here we show how the baseline quasi-equilibrium framework is derived. Specifically, there are two analytical constraints that form the foundation of the quasi-equilibrium framework, namely the photosynthetic constraint and the nitrogen cycling constraint. The derivation follows Comins and McMurtrie (1993), which is further elaborated in work by McMurtrie et al. (2000) and Medlyn and Dewar (1996) and evaluated Comins (1994).

Firstly, gross primary production (GPP) in the simulation mode is calculated
using a light-use efficiency approach named MATE (Model Any Terrestrial
Ecosystem) (McMurtrie et al., 2008; Medlyn et al., 2011; Sands, 1995), in
which absorbed photosynthetically active radiation is estimated from leaf
area index (

In the quasi-equilibrium framework, the photosynthetic constraint is derived
by assuming that the foliage C pool (

The nitrogen cycling constraint is derived by assuming that nitrogen inputs
to and outputs from the equilibrated pools are equal. Based on the
assumed residence times of the passive SOM (

In the VL term, we have

In the L term, we now have to consider N flows leaving and entering the
passive SOM pool, which is no longer equilibrated:

Similarly, in the M term, we have

The supplement related to this article is available online at:

BEM and MJ designed the study; MJ, BEM, and SZ performed the analyses; APW, MGDK, and SZ designed the priming effect equations; all authors contributed to results interpretation and paper writing.

The authors declare that they have no conflict of interest.

This paper builds heavily on ideas originally developed by Ross McMurtrie and Hugh Comins (now deceased). We would like to acknowledge their intellectual leadership and inspiration.

Sönke Zaehle and Silvia Caldararu were supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (QUINCY; grant no. 647204) and the German Academic Exchange Service (DAAD; project ID 57318796). David S. Ellsworth and Mingkai Jiang were also supported by the DAAD.

This paper was edited by David Lawrence and reviewed by two anonymous referees.