Leuning model
abridged from Leuning (1995). A critical appraisal of a combined stomatalphotosynthesis model for C3 plants. Plant, Cell and Environment, Vol. 18, No. 4. (1995), pp. 339355.
Contents
abstract
Gasexchange measurements on Eucalyptus grandisleaves and data extracted from the literature were used to test a semiempirical model of stomatal conductance for CO_{2},
where A is the assimilation rate;
 D_{s} is the humidity deficit at the leaf surface;
 C_{s} is the CO_{2} concentration at the leaf surface;
 g_{0} is the conductance as A → 0 (when leaf irradiance → 0);
 D_{0} and a_{1} are empirical coefficients.
This model is a modified version of
first proposed by Ball, Woodrow & Berry (1987), in which h_{s} is relative humidity. Inclusion of the CO_{2} compensation point Γ improved the behaviour of the model at low values of c_{s}. A hyperbolic function of D_{s} for humidity response correctly accounted for the observed hyperbolic and linear variation of g_{sc} and c_{i} / c_{s} as a function of D_{s}. Here c_{i}is the intercellular CO_{2} concentration.
In contrast, use of relative humidity as the humidity variable led to predictions of a linear decrease in g_{sc} and a hyperbolic variation in c_{i} / c_{s} as a function of D_{s}. This did not agree with the data from E. grandis leaves. The revised model also successfully described the response of stomata to vanations in A, D and c for published responses of the leaves of several other species. Coupling of the revised stomatal model with a biochemical model for photosynthesis of C3 plants synthesizes many of the observed responses of leaves to light, humidity deficit, leaf temperature and CO_{2} concentration. The best results are obtained for wellwatered plants.
Introduction
Stomata have evolved physiological control mechanisms to satisfy the conflicting demands of allowing a net carbon gain by leaves while restricting water loss to acceptable levels, under a range of environmental conditions.
Stomata respond to environmental stimuli in a complex way and the physiological mechanisms of stomatal behaviour are not yet fully understood. In the absence of a mechanistic description, semiempirical models are often used to describe stomatal response to environmental and physiological factors. Such models are required to predict the response of photosynthesis and plant water use under present and future CO_{2} concentrations and temperatures. Relatively simple phenomenological models of stomatal behaviour are also required to describe plant water use in response to diurnal and seasonal variations in environmental conditions.
Here we critically examine a semiempirical model for stomatal behaviour first proposed by Ball et al. (1987) and developed further by Leuning (1990). Stomatal conductance is assumed in the model to be a function of relative humidity at the leaf surface, assimilation rate, and surface CO_{2} concentration, but does not include the effects of plant water stress on conductance. The model is modified by replacing surface relative humidity with a hyperbolic function of humidity deficit, as this provides a superior description of the reponses of both conductance and intercellular CO_{2} concentration to varying humidity.
stomatal model
physical approach
Many modellers have followed Jarvis (1976) in assuming that environmental factors act independently in determining stomatal conductance, g_{sc}. Stomatal response is then described by
(1)
where f_{i} are nonlinear functions of
 Q: the absorbed shortwave irradiance;
 T_{l}: the leaf temperature;
 D_{s}: the water vapour deficit;
 C_{l}: the CO_{2} concentration, and
 Θ: the soil moisture content.
The conductance to water vapour g_{sw} = 1·56 g_{sc} where g_{sc} is the conductance to CO_{2}. The numerical factor 1.56 accounts for the ratio of diffusivities for water vapour and CO_{2} in air. Equation 1 describes a fivedimensional response surface. The possibility of other dimensions is indicated. The response surface will be complex because the responses of g_{sc} to individual variables are usually nonlinear. Stylized responses of stomata to individual factors were presented by Jarvis & Morison (1981). These functions have been determined by many workers using leaf gasexchange measurements under controlled conditions or by using boundaryline analyses of measurements under natural conditions. The results indicate that g_{sc}
 increases rapidly as irradiance increases from zero, and that
 saturates at relatively low values of Q.
Conductance shows a peaked response to temperature, although the optimum is usually fairly broad. Increasing atmospheric humidity deficits and CO_{2} concentrations both cause g_{sc} to decrease. Stomatal conductance is relatively insensitive to decreasing soil moisture availability until a threshold is reached, whereupon g_{sc} decreases approximately linearly.
introducing leaf physiology
An alternative approach to modelling stomatal behaviour only in terms of physical variables (as in Eqn 1) is to incorporate leaf physiology by combining models of stomatal behaviour and photosynthesis. This recognizes that stomatal control has evolved to maximize carbon gain while minimizing water loss. The apparent close link between stomatal conductance and assimilation led Farquhar & Wong (1984) to propose an empirical model relating stomatal conductance to the ATP content and photosynthetic capacity of leaves. Ball et al. (1987) presented an empirical relationship which incorporates the oftenobserved correlation between A and g_{sc} and includes the effects of humidity and ambient CO_{2} concentrations on conductance, namely
(2)
where
 g_{0}: the conductance as A → 0 (when leaf irradiance → 0);
 a_{1}: a coefficient, and
 h_{s}: the relative humidity at the leaf surface;
 c_{s}: the CO_{2} concentration at the leaf surface;
This equation predicts that stomatal conductance increases with assimilation rate and with relative humidity at the leaf surface when c_{s} is held constant. Conductance will decrease with rising ambient CO_{2} concentration, provided that A
increases more slowly than c_{s}.
Equation 2 satisfies the requirements of Eqn 1 in that the dependence of g_{sc} on Q, c and possibly T_{l} is incorporated in the dependence of A on these variables. The apparent simplicity of Eqn 2 has led to its adoption by modellers working at the scale of individual leaves, canopies, and landscape, and in some global climate models.
Equation 2 has no apparent mechanistic basis because the elegant experiments by Mott (1988) showed conclusively that stomata respond to substomatal CO_{2} concentration, c_{i}, rather than to the surfave concentration c_{s}. Despite this conclusion, formal justification for the term A/c_{s} in Eqn 2 can be obtained by examining the simplified equation for supply of CO_{2} by diffusion through the stomata,
(3a)
and rewriting it in the form
(3b)
This simple equation ignores interactions between molecules of water leaving and entering the stomata. Use of c_{s} rather than CO_{2} concentration outside the leaf boundary layer, c_{a} , eliminates complications of the equations arising from transfer through the boundary layer. We see that g_{sc} will be correlated to the ratio A/c_{s}, provided that the ratio c_{i}/c_{s} remains relatively constant when the leaf is subject to varying environmental conditions. While Eqn 3b defines g_{sc}, it does not mean that conductance is linked mechanistically to A/c_{s}.
a general humidity term
The BWB model has caused further concern because it is widely accepted that stomata respond to humidity deficit rather than to surface relative humidity. Stomata respond to atmospheric humidity through evaporation from the leaf, rather than to humidity deficit itself. Because there is a close link between transpiration, E_{t} and humidity deficit (E_{t} = g_{sw}.D_{s}) we shall continue to use D_{s}, rather than E_{t} as the humidity variable in subsequent developments. This avoids the need to include transpiration (and hence the full leaf energy balance) explicitly in the equations for g_{se} and A.
Succesful application of Eqn 2 suggests that this equation may still be useful if h_{s} is replaced by a more general humidity function f(D), i.e
(4)
(5)

BWB model  
(6)

Jarvis (1976)  describes f(D) at constant leaf temperature, but the slope 1/D_{0} depends on temperature. 
(7)

Lohammer et al. (1980)  developed from analysis of measurements of leaves in the field, where variation in D_{s} is largely caused by changes in leaf temperature rather than by changes in the absolute humidity of the air 
where
 e^{*}: the saturation water vapour pressure at leaf temperature.
 D_{0}: an empirically determined coefficient
In its present form, Eqn 4 is not capable of describing stomatal behaviour at low CO_{2} concentrations, since conductance increases to maximal values as c_{s} approaches the CO_{2} compensation point, Γ while A → 0. Equation 4 predicts that g_{sc} → 0 under these circumstances. Leuning (1990) accounted for these observations by replacing c_{s} with c_{s}  Γ, i.e.
(8)
This modification ensures that g_{sc} remains large as A → 0, while c_{s} → Γ. Note that
g_{sw} is not defined by this equation when c_{s} = Γ.
The implication of Eqn 8 for the expected value of c_{i} can be seen by combining it with Eqn 3 to give
(9)
This equation was called the supplyconstraint function by Leuning (1990). A plot of Eqn 9 shows that leaves that conform to the BWB model maintain c_{i} almost constant at the asymptotic value,
c_{s}  (c_{s}  Γ) / (a_{1}.f(D))
over a wide range of A, when Q varies while D_{s} and c_{s} are held constant (Fig.).
Note that g_{0} cannot be negative physically, and g_{0} ≠ O, otherwise c_{i}/c_{s} would be constant for all values of Q. The rate of convergence of c_{i} to c_{s} at low A (low Q) depends on the value of g_{0}: when g_{0} is large, c_{i} deviates from the asymptotic value at higher values of A than when g_{0} is small.
Figure 1. Hypothetical Ac_{i} curves showing the photosynthetic demand function (Eqn 10) at Q=1500 and 150 μmol quanta at T_{l} = 25 °C. Examples of the supplyconstraint function (Eqn 9) are also shown for c_{s}= 350 and 700 μmol/mol, and D_{s}= 1 and 3 kPa, with g_{0} = 0.01 mol m^{2}s^{1} and D_{s0}=0.35 kPa. The 'operating point' of the leaf is given by the intersection of the demand and supplyconstraint curves. A is more sensitive to changes in D_{s}, at lower values of c_{i} and when Q is high. The model predicts that c_{i} remains almost constant as A decreases with diminishing Q for any given values of D_{s} and c_{s}.