Models can have any of a variety of purposes:

  • optimization.  It is often the case that the model has one or a few final results, such as growth rate (plant, population, etc.), and we want to find out how to get the best result, such as maximal growth rate, minimal resource use, or the like.  This result is the objective function that we are optimizing, possibly subject to constraints on some other variables such as total nutrient pool available.  These models have an inherent appeal in applied science, such as agriculture, where one can anticipate manipulating plants, animals, even microbes by management techniques or genetics.  The models are also good benchmarks for testing some combination of organismal evolution and our understanding of constraints.  If we were fully (and wrongly) adaptationist, we would expect that organisms do a certain function or set of functions optimally.  We can formulate the possibilities, such as the way that plant leaves are displayed, their pigment content, their root allocation…or the biomechanics of animals for swimming, flying, running, etc.  Given the objective function and the potential variations in organism structure or physiology, we can estimate what in the latter gives the optimum.  If it's not what's observed, we can 1) expect continuing selection pressure (but ask, why has it been insufficient to fix the optimal traits after all these eons) or 2) look for constraints and compromises that we have overlooked: Is nutrient use a constraint on efficient water use that we forgot about?  Is there phylogenetic constraint? …

            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 University of Illinoise notified me that his group tested this in the field with soybeans….and got an 8% increase in yield!  Do seed companies and farmers want to adopt this?  No.  First, they think that pale mutants look sickly, even if they are superior.   Also, most pale mutants are heterozygous, so they don't breed true, although the pea mutants are homozygotes and would work in a seed-production program.  Finally, breeders spend over 90% of their effort on pest and disease resistance and little on yield.  One of the few yield-improving programs not involving fertilizer responsiveness but light interception instead was the adoption of erect leaves in crops such as maize.  This accomplishes some of the same thing as pale mutants.  Of course, wild plants don't adopt this optimum because light shared is shared with competitors.  Shade 'em and accept a somewhat lower growth rate.  Schieving and Poorter (1999) followed up on my speculation and modelled this sharing vs. shading.  Their results supported the idea of the compromise.

     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 California may be as low as 15% of normal, potentially lethal.  In 3 important nut-growing states (CA, TX, NM) with something on the order of a billion-dollar crop, irrigation water is under long-term threats from climate change and city growth.  I am collaborating with a 13-member team to estimate the best management.  I developed a model of pecan responses to reduced water availability in two growth stages - leafout and nutfill.  Using physiological models and cruder developmental models, I estimated how reduced water in leafout decreases final leaf area, only modestly, and how reduced water in nutfill compounds the saving from reduced area with an increase in water-use efficiency.  I solved for the optimal reduction fractions in both stages - 36% of normal in leafout, 52% of normal in nutfill - that meet the constraint of 50% of normal whole-season water use while maximizing yield at 60% of normal.  Some standard techniques of deficit irrigation (full water in leafout, bigger reduction in nutfill) lead to disastrously lower yields (21%).  I have communicated the results to the California water managers to see if we can apply the ideas in this crisis year, not waiting for the 3-years-hence conclusion of the project!

       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.

  •  process simplification.  Let me start with a "soft target," the role of mass flow in plants' acquisition of nutrients from soil.  Nutrients reach plant roots in soil solution.  The nutrients diffuse in the solution (aided by the gradient formed by uptake itself, one might say), and they are carried passively as the soil water moves to the roots that take it up to support shoot transpiration.  Does transpiration improve nutrient delivery and uptake, over and above what diffusion can accomplish?  Is this the reason that a number of plants are observed to transpire at night, when this seems otherwise wasteful of water, no photosynthesis being possible.  Short, robust answer: no, no, no.  I modelled both processes.  Transpiration make a negligible difference overall.  While one can compute a mass-flow contribution to uptake, it turns out that mass flow flattens the concentration profile from bulk soil to the root surface,  This reduces diffusive uptake in almost exactly the same amount as mass flow contributes.  This has been hard for a number of authors to accept.  Their problem.

           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.

  • exploring phenomena, to develop testable hypotheses.  Earlier on this page, I described the model for pale mutant plants, which generated the hypothesis that they do better photosynthesis in a whole stand than normal plants.  This hypothesis was tested and supported.
  • process synthesis.  I described earlier the model that combined our understanding of how leaf stomatal conductance (and transpiration and photosynthesis) is controlled by the aerial environment (light, air temperature, humidity, windspeed, CO2 content, other radiant fluxes) with our understanding of the control contributed by soil water status.  The model also resolved the role of plant physiology (photosynthetic capacity, three parameters of stomatal control) and structure (leaf dimension, for determining the leaf boundary layer).
  • sensitivity analysis - finding out how important are various factors in determining the performance of an organism or ecosystem.  An example is a model that Jim Pushnik and I developed for the growth rate (RGR) of plants at various combinations of soil nitrogen content, diffusibility of nitrate in soil, root:shoot ratio, and root uptake capacity for nitrate.  One can use the results as guidelines for experimental investigations.  One can test if, for example, uptake capacity  matters much only at low root:shoot ratio.  If the model is found to be accurate, then one can use the results another way - to determine which organism traits or environmental conditions affect the results most, and which therefore merit the greatest effort to measure accurately in order to get accurate predictions.

             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.

  • expanding our view of possible phenomena. I described on another page my model of how diverse are the expected responses to elevated CO2 among different plant species.
  • developing practical computational solutions to important submodels used in big models.  Climate and ecosystem models demand accurate estimates of water use and photosynthesis by vegetated areas.  Climate models, in particular, have to be applied over many geographic locations, with many process equations to solve, and with many time steps to perform.  Computations must be done as efficiently as possible.  One submodel used in such climate models (GCMs, general circulation models) is the simple biosphere model (SiB, and its improved "sibs', SiB2 and SiB3).  It uses the combined equations for photosynthesis, CO2 and water vapor transport, stomatal control, and energy balance that I noted using in my own models (Wang et al., 2007, p. 258; also, a report) for related purposes.  Other research groups developed efficient solutions, somewhat different from mine.  I use my combined model often to estimate effects of, say, nitrogen nutrition or water stress on water fluxes.

             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.

  • Assessing the adaptive value of organismal traits, and identifying places to look for selection pressure, possibly designing population genetic experiments.  Earlier, in the section on different mathematical types of models, I described our functional balance model.  This allowed some attribution of the value, for growth rate, of various plant acclimations to low nutrient conditions.
  • Correcting measurement errors.  First, an example that's passé now.  I developed a complete model of how the prior generation of leaf gas-exchange systems (LI-COR LI-6200) recorded data.  The correct estimates of leaf photosynthesis, stomatal conductance, etc. depended upon entering the correct values of such data as leaf area or air pressure.  The LI-6200 software allowed correction of wrong leaf areas, recomputing the results.  However, entering the wrong air pressure was not correctible.  I delved into the guts of the LI-COR, using copious information from LI-COR, to reconstruct the very voltage signals of the sensors.  I then used these to recompute the whole schmeer, successfully.

              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

  • verifying our understanding of processes that control the performance of organisms or ecosystems.  In 1999-2000, my research group participated in measurements of evapotranspiration in a massive riparian zone on the Rio Grande, one that had been invaded in large areas by non-native tamarisk trees.  Our role was to examine physiological control of water use (by leaf physiology and leaf area development) and verify our understanding of such control against 1) our own measurements of sapflow in single trees, and 2) on a large, stand scale, the flux-tower measurements using eddy covariance.  The results are presented in a Ph. D. thesis by Ernesto Catalan.   We do see fine agreement with both measurements. We also support the finding of other groups doing such studies on other reaches of the Rio Grande, that tamarisk and native cottonwood use very similar amounts of water, when both canopies are fully developed.  Demonizing tamarisk for its water use is bas science and has led to bad policy and environmental damage (brute-force bulldozing of tamarisk polluted the Rio Pecos with vast amounts of silt).
  • teaching.  Virtually all the models I've developed have been used in teaching concepts of physiology, ecology, agricultural management, remote sensing, mathematical biology, and other areas.   That said, I had to run most of them, rather than having students run them and comprehend them, or, even better, develop the models themselves.  The simpler models, however, are readily used in the latter mode.  As an example, I developed a model in Excel that simulated the growth of plants.  It used a limited set of parameters - initial biomass, leaf photosynthetic rate, the biosynthetic efficiency of using photosynthate to make biomass, the light interception rate of leaves (extinction coefficient), the leaf mass per area, the fraction of shoot mass as leaf mass, the root:shoot ratio, and the hours of daylight.  Students could vary any of these and see the results, which were resolved for each day.. 

              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.

  • parameter estimation.  Image analysis to estimate the leaf area on individual plants has been described in detail, under the topic of inverse models.


Some references

G. J. Collatz, J. T. Ball, et al. (1991). "Physiological and environmental regulation of stomatal conductance, photosynthesis and transpiration: A model that includes a laminar boundary layer." Agric. For. Meteorol. 54: 107-136.

E. Finkel. 2009. Profile: Richard Richards: Making every drop count in the buildup to a Blue Revolution. Science 323: 1004-1005.

V. P.Gutschick. 1984a. Photosynthesis model for C3 leaves incorporating CO2 transport, radiation propagation, and biochemistry. 1. Kinetics and their parametrization. Photosynthetica. 18: 549-568.

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 L. E. Kay.1995. Nutrient-limited growth rates: Quantitative benefits of stress responses and some aspects of regulation. J. Exp. Bot. 46: 995-1009. (PDF here)

V. P. Gutschick and J. C. Pushnik. 2005. Internal regulation of nutrient uptake by relative growth rate and nutrient-use efficiency. In Ecological Studies: Nutrient Acquisition by Plants: An Ecological Perspective, ed. H. BassiriRad. Springer, Heidelberg, pp. 63-88. (PDF here)

V. P.Gutschick and F.W.Wiegel. 1984. Radiative transfer in plant canopies and other layered media: rapidly solvable exact integral equation not requiring Fourier resolution. J. Quant. Spectrosc. Radiat. Transfer 31: 71-82.

V. P.Gutschick and F.W.Wiegel. 1988. Optimizing the canopy photosynthetic rate by patterns of investment in specific leaf mass. Am. Nat. 132: 67-86.

D. A. Johnson, K. H. Asay, et al. (1993). Genotypic and environmental variation for carbon isotope discrimination in crested wheatgrass, a perennial forage grass. In: Stable Isotopes and Plant Carbon-Water Relations. Eds. B. Saugier, G. D. Farquhar, and A. E. Hall, Academic Press: 269-280.

W. T. Pettigrew, J. T. Hesketh, D. B. Peters, and J. T. Woolley. 1989. Characterization of canopy photosynthesis of chlorophyll-deficient soybean isolines. Crop Sci. 29: 1025-1029.

F. Schieving and H. Poorter. 1999. Carbon gain in a multispecies canopy: the role of specific leaf area and photosynthetic nitrogen-use efficiency in the tragedy of the commons. New Phytol. 143: 201-211.

P. J. Sellers, D. A. Randall, et al. 1996. A revised land surface parameterization (SiB2) for atmospheric GCMs. Part I: Model formulation. J. Climate 9: 676-705.

B. R. Trenbath and J. F. Angus. 1975. Leaf inclination and crop production. Field Crop Abstr., 28: 231-244.

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)

J. M. Wang, T. W. Sammis, A. A. Andales, L. J. Simmons, V. P. Gutschick, and D. R. Miller. 2007.Crop coefficients of open-canopy pecan orchards. Agricultural Water Management 88: 253-262. (PDF here)