Prentice et al. (2014a) raised this question about the development of next-generation land surface models. My interpretation of their ‘three R’s’ criteria for model development is as follows:

- A
*reliable*model should give the right answer. Therefore, reliability is the ultimate target of the models developed for the purpose of projecting the impacts of environmental change. - A
*robust*model should not be sensitive to the specification of one or more uncertain parameter values. And the more uncertain parameters are included in a model, the greater the risk of losing robustness. - A
*realistic*model should represent the Earth system accurately. Since the Earth system is complex, it is not surprising that anything approaching the full behaviour of the complex system cannot be described with a very simple model.

Therefore, there is a connection between realism and complexity; however, it should be borne in mind that complexity does not guarantee realism, and that there may be a trade-off between realism and robustness in the search for reliability.

So how are the three R’s to be achieved for state-of-the-art land models? Despite the apparent successes of land models that have been developed during the past several decades (Ciais *et al.* 2014), the ‘three R’s status’ of current models is becoming more and more worrying. Inconsistencies among models have persisted stubbornly through successive model generations (Friedlingstein *et al.* 2014) and the terrestrial carbon cycle has become one of the largest sources of uncertainty in climate projections. One original cause of this awkward situation was the limited data availability in the early days when land models were first developed. Many processes related to plant ecophysiology and ecosystem function were not very well understood, and when incorporated into models they had to be represented by equations with large numbers of parameters.

Many model parameters in current models, such as photosynthetic capacity, are still assigned fixed values (usually per plant functional type, PFT) even though they should really be variables that adapt to the environment. The typical modeller’s response to such criticisms is to represent processes in an even more complex way and in doing so to still further increase the number of parameters. However, realism does not inevitably increase with burgeoning complexity, whereas the robustness of models is inevitably lost. No wonder we end up with less reliable models!

On the other hand, a vast expansion of observations has happened in past couple of decades, and this situation has provided an opportunity for modellers to pursue a quantitative explanation of what is observed, and predict the variations of those fixed parameters. This opportunity unfortunately has been largely overlooked during business-as-usual model development.

Learning from previous model development work, we can propose a new strategy to improve the reliability of the next generation of land models. The key to this new strategy is optimal allocation theory, which has its roots in the principle of optimization by natural selection. This principle is central to understanding how and why plants allocate resources to different compartments and functions, and how allocation ‘decisions’ by plants vary in time and space. If the right optimality hypothesis is posed, remarkable predictive power can be obtained from a simple model with very few parameters (Wang *et al.* 2014) – paving the way for simpler, but also more powerful and robust models. Notably, optimality hypotheses can be formulated to maximize different criteria, and not all of them will make correct predictions. Therefore, the basis of our modelling strategy is to make use of the extensive observed data to test those hypotheses generated from optimality, select the one that gives the right answer, and finally meet the requirement of reliability.

To illustrate explicitly how to apply optimal allocation theory in pursuit of the ‘three R’s’, here I provide an example from my current work on predicting the variation of the ratio of leaf-internal to ambient CO_{2} concentration (the so-called *c _{i}*:

*c*ratio, denoted here by

_{a}*χ*). This ratio is regulated by stomatal behaviour in response to environmental conditions, and plays a crucial role in determining the photosynthetic assimilation rate under CO

_{2}limitation. In current land models the ratio is either considered as a constant, or allowed to respond to vapour pressure deficit (vpd) following one or another empirical equation. My work has shown that the behaviour of this ratio, inferred from stable isotope (δ

^{13}C) data, is predictable from optimal allocation theory.

According to the ‘least-cost’ hypothesis (Wright et al. 2003; Prentice et al. 2014b), evolutionary optimality requires plants to adjust *χ* so as to minimize the total respiratory costs required to maintain the capacities for both carboxylation and transpiration that are needed to support a given assimilation rate. The value of *χ* that minimizes these costs can be mathematically expressed as a function of vpd, the effective Michaelis-Menten coefficient of Rubisco (*K*), and the ratio of two carbon cost parameters (*a* for transpiration, *b* for carboxylation). The parameter *b* is the ratio of leaf dark respiration to carboxylation capacity, assumed to be constant. The parameter *a* is related to properties of water transport through sapwood, including the viscosity of water, the permeability of sapwood, and difference in water potential maintained between soil and leaves (Prentice et al. 2014b). Based on the well-established equations describing the relationships of *K* to temperature and atmospheric pressure, the viscosity of water to temperature, and vpd to atmospheric pressure, it can be predicted that the derivatives of ln [*χ*/(1 – *χ*)] with respect to temperature, ln (vpd) and elevation are ^{ }respectively 0.0545 K^{–1}, –0.5 and –0.0815 km^{–1}.

Therefore we have a hypothesized model for *χ*; the next step is to test it. An extensive global dataset of 3549 δ^{13}C measurements on leaves (compiled by Will Cornwell, University of New South Wales) was used. The data are from all biomes, including low and high elevations. δ^{13}C provides a measure of the long-term response of *χ* to environment, which is what optimality theory predicts. The environmental predictors should also be long-term mean values, therefore I used the growing-season mean values of temperature and vpd. A standard equation was used to transform δ^{13}C to *χ*. Multiple linear regression was performed on logit-transformed *χ* values using temperature, ln (vpd) and elevation as predictors.

The fitted regression coefficients of temperature, VPD and air pressure were all highly significant, and quantitatively consistent with predictions. Predicted values of *χ* based on the regression model are consistent with this data across all biomes and plant functional types (see Figure, below).

These results show that optimal allocation theory can successfully predict stomatal regulation of the leaf-internal to ambient CO_{2} concentration ratio as function of environmental factors. The prediction is more realistic than that of current models that treat this ratio as a function of vpd alone. No new empirical parameters were introduced during this work, so the model is robust. Finally, the comparison with observed data tells us that it is a reliable model.

A quick test was performed in which the theoretically predicted *χ* values were inserted into a simple light-use efficiency model of gross primary production (GPP). The results were compared with annual GPP data derived from publicly available FLUXNET measurements by Tyler Davis, and contrasted with the same model using a fixed value of *χ*. The use of predicted values of* χ* accounted for 8% more variation in the observations than the use of a fixed value, showing the potential for this model for *χ* to improve land models.

**References:**

Ciais, P. *et al.* (2014) *Carbon and Other Biogeochemical Cycles*. In *Climate Change 2013: The Physical Science Basis. *Cambridge University Press, Cambridge.

Friedlingstein, P., Meinshausen, M., Arora, V.K., Jones, C.D., Anav, A., Liddicoat, S.K., and Knutti, R. (2014): Uncertainties in CMIP5 climate projections due to carbon cycle feedbacks, J. Clim. 27: 511-526.

Prentice, I.C., X. Liang, B. Medlyn and Y. Wang (2014a) Reliable, robust and realistic: the three R’s of next-generation land-surface modelling. *Atmospheric Chemistry and Physics Discussions* 14: 24811-24861.

Prentice, I.C., N. Dong, S.M. Gleason, V. Maire and I.J. Wright (2014b) Balancing the costs of carbon gain and water loss: testing a new quantitative framework for plant functional ecology. *Ecology Letters* 17: 82-91.

Wang, H., Prentice I.C. and Davis T.W. (2014) Biophysical constraints on gross primary production by the terrestrial biosphere. *Biogeosciences* 11: 5987-6001.

Wright, I.J., Reich, P.B. & Westoby, M. (2003). Least-cost input mixtures of water and nitrogen for photosynthesis. Am. Nat., 161: 98–111.