Models can have any of a variety of purposes:
I've done my share of these
models. One I described above, to
illustrate models that use differential equations, is the optimal distribution
of photosynthetic capacity with depth in a plant canopy (a topic others have
also honed further). Another is the optimal pigment concentration in plant leaves. It struck me, back before 1980, that leaves
are too dark. They absorb 85% of
incident PAR but their photosynthetic rate saturates typically at a small
fraction of full sunlight, as little as 1.4 of full sun even for sun-adapted
plants. If leaves had less chlorophyll,
they could let more light reach lower in the canopy. Because it is Rubisco enzyme and not
chlorophyll that commonly limits the photosynthetic rate, a reduced pigment
content would not lead to much reduction of photosynthetic rates of upper
leaves - in fact, at high light, they could do slightly better. I modelled (1984) how a
change in pigment concentration would affect the distribution of light, CO2,
and photosynthetic rate within single leaves and also in the whole canopy. I knew about mutant peas and soybeans that
had half-normal chlorophyll content, and I modelled their performance in a full
agricultural stand, as well as that of the normal group. I predicted an 8% gain
in yield over a growing season. In 1989,
John Hesketh from the
Another optimization model I developed was finding a new combination of two traits for alfalfa that would increase yield, or increase water-use efficiency, or increase some mixed measure chosen by the farmer/breeder. Very elegant experiments and theory have shown that there is genetic variation within species for stomatal control that affect the ratio of internal (Ci) to external (Ca) levels of CO2 in leaves. A reduction in this ratio decreases photosynthetic rate (esp. per unit mass or per unit N in leaves) but increase water-use efficiency (WUE) even more. Also, my earlier work (e.g., 1988, with Wiegel) and that of others examined the genetic variation in mass per leaf area (call it m; it is the inverse of specific leaf area). There is an optimal average value of m in a canopy (as well as an optimal depth profile) for canopy photosynthesis. Changes in m also have a modest effect on WUE; thick leaves transpire more and are a bit cooler. I modelled how the combination of the two heritable traits, Ci/Ca and m, affected both yield and WUE. I predicted that there are new values, offset from those in current alfalfa cultivars, that do better in either yield or WUE. Limited experimental tests were not conclusive; we had a lot to learn about the best ways to measure Ci/Ca. One further prediction is that no major gains in WUE are possible for crops; 15% gains might be the limit before yield penalties become severe. Some forage grasses and other crops have been improved by selecting for Ci/Ca. It is notable that drought tolerance is not well related to WUE; I have discussed this, and the latest gains in drought tolerance in wheat used very different traits, esp. coleoptile length. Going back to Ci/Ca: a likely reason that C3 plants have a remarkable convergence in value and a suboptimal WUE is that water saved is saved largely for competitors.
A truly applied optimization model is
that for deficit irrigation, for which I've made
first desperate cut.
"Desperate" means that irrigation for nut crops in
I mentioned the functional balance model earlier for explaining how plant acclimations to low-nutrient conditions help maintain good growth rates. The results of this model indicate how RGR might be aided by physiologically-possible changes in root uptake properties and root:shoot ratio or root allocation. The formulas seemed to indicate that there was no saturation in benefit from increasing N uptake capacity, until one considered rather high costs of N uptake and assimilation. More disturbing was the prediction that the optimal root:shoot ratio was 1, independent of nutrient levels, the diffusiblity of nutrients in soil, root ageing characteristics, etc., while we observe ratios far smaller in crops and larger in some arid-zone plants. After some years of thinking, I considered water acquisition. A simple model predicts optimal ratios in the normal range. It is overly simple, applying to constant conditions of water availability, but it's a start.
Another problem is accounting for the important role of diffuse light in plant canopies, driving photosynthesis. The first interceptions of the direct solar beam are pretty easy to model, using geometry and basic models of the probable spatial distribution of leaves. Diffuse light propagates in many different directions, and is absorbed and generated differently in all these directions. If we want to characterize it with 25 representative angular directions, do we need a model of light propagation that is now 26 times more complicated? No. I alluded above to a model that solved for the histogram of diffuse light (skylight) levels on leaves, solved with some nice transformations. The result is that the distribution of diffuse-light intensities is pretty narrow around the mean value, independent of leaf orientation, until one gets so deep in the canopy that the light levels are rather insignificant for photosynthesis. We can therefore use a single diffuse-light intensity at each depth in a (uniform) canopy. Actually, we can use two flux densities (to use the correct term), an upward one and a downward one. With proper formulation of the capacities for individual leaves to absorb light and to scatter it, we get a model that is very simple, while predicting very accurately the observed absorption of radiation in canopies, the reflection from them, and the penetration to the soil or understory. The predictions are good in the PAR< near infrared, and thermal infrared, which differ widely in absorption and scattering behavior. I have a version in Excel that one can play with. It's not fully documented and the thermal infrared calculations need some attention but I'll get there.
I have used these results in recent work (yet to be published) on improving the accuracy of so-called big-leaf models. Some practical models of water use and photosynthesis over vast areas, as needed in climate models and ecosystem models, have to use very simple representations of plant performance. One simplification is to regard the plant cover on the land as a uniform leaf (or fraction thereof), operating at a single level of PAR, NIR, and TIR fluxes (the latter are often even ignored, though they are the biggest parts of the leaf energy balance). This is inaccurate for a variety of reasons, including that NIR is absorbed much more weakly than PAR. It goes deeper into the canopy. It affects the depth distribution of leaf temperature, thus, of photosynthesis. My model indicates relatively simple patterns and corrections for NIR interception by leaves.
For more accurate computation of all the light fluxes in a plant canopy, Frits Wiegel and I (1984) developed an integral equation that can be solved for discrete layers in the canopy (and just earlier I said that I rarely use discrete layers, sigh). The model replaces the detailed angular distribution of diffuse fluxes with an equivalent distribution that is uniform in angle (isotropic), although with different total intensity in upward and downward directions.
Sometimes a complex model can be fitted to a much simpler empirical model for use in a big effort where full computation is not practical. There are many such examples. Here I offer the simplification of a model I made of pecan orchard water use. The full model considered light interception, stomatal responses to photosynthetic conditions and humidity, and the diurnal and daily variation of meteorological conditions. The results, composited as water-use efficiency, were fitted to a simple function of monthly mean temperature and relative humidity.
Another sensitivity analysis is in (we hope) final review. Junming Wang involved several of us in such an analysis of the SEBAL method (see the page about inverse models) for estimating land evapotranspiration (ET) from satellite measurements of radiation from the surface. There are many measurements involved at the satellite (the radiances in as many as 13 bands), as well as estimates of vegetation amount and height (the latter from ground-based land classifications) and choices of points on the landscape that are used for calibrating the endpoints of ET (max and min). Junming took observations and considered variations about their estimated values (as from instrumental errors at the satellite, land-classification errors, etc.). He found that the proper identification of calibration points was generally the most critical to results, while other factors could be important but varied with the degree of vegetative cover.
Another model of mine, unpublished, is for the pattern of soil temperature with depth and with time of day, as sunlight interception changes, or windspeed, soil wetness, etc. This pattern is important for plant stem and root survival of extremes, as well as for animals on the surface or in burrows. The partial differential equation merits some careful numerical solution to avoid instabilities (violation of the well-known Courant condition).When soil wetness varies sharply with depth, the solutions are very challenging to formulate. I have used rational polynomials (Padé approximants) to give robust estimates of such rapidly varying soil properties, enabling practical solution of the soil temperature behavior.
A currently usable example, correcting leaf performance from the conditions in the measurement chamber to the original free-air conditions of the leaf. When one does gas-exchange measurements on a leaf, it is put into new conditions. The light level changes. So does the windspeed (thus, the boundary-layer conductance), the air humidity, and the leaf temperature. One can use the measurement conditions in the chamber to get core physiological parameters, such as photosynthetic capacity. One can then put this into the combined models of photosynthesis, stomatal conductance, energy balance, and transport to estimate leaf photosynthesis and transpiration in the original conditions on the plant…provide that one has measured those original conditions. Such a projection to the original conditions is important if one is trying to estimate transpiration and water-use efficiency, both of which are much more sensitive to environmental conditions than is photosynthesis. My model is available in Fortran and in Excel, though not documented in great detail
Another teaching model was not original but presented a complementary viewpoint of isotope discrimination in leaf photosynthesis. The common presentation of the topic uses canned results of discrimination by stomatal diffusion and Rubisco enzymatic action, or it goes into great biochemical detail. I presented a view that explains the discrimination during diffusion from gas theory and the discrimination by the enzyme with an appeal to elementary quantum mechanics (it can be done), and then it composes the final discrimination ratio with the combined process model.
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V. P.Gutschick. 1984b. Photosynthesis model for C3 leaves incorporating CO2 transport, radiation propagation, and biochemistry.. 2. Ecological and agricultural utility. Photosynthetica 18: 569-595.
V. P.Gutschick. 1984c. Statistical penetration of diffuse light into vegetative canopies: effect on photosynthetic rate and utility for canopy measurement. Agric. Meteorol. 30: 327-341.
V. P.Gutschick. 1988. Optimization of speciﬁc leaf mass, internal CO2 concentration, and chlorophyll content in crop canopies. Plant Physiol. Biochem. 26: 525-537.
V. P. Gutschick
and J. C. Pushnik. 2005. Internal regulation of nutrient uptake by relative
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J. M. Wang, D. R. Miller, T. W. Sammis, V. P. Gutschick, L. J. Simmons, A. A. Andales. 2007. Energy balance measurements and a simple model for estimating pecan water use efficiency. Agric. Water Manage. 91: 92-101. (PDF here)