Types of models, according to the kind of math they embody:

  • algebraic.  No explicit time- or space-dependence.  Some relatively simple ones are fairly powerful. 

I offer my functional balance (FB) model for nutrient (nitrogen) uptake and use by plants.  My wife and I (Gutschick and Kay, 1995) measured a suite of plant physiological and developmental responses to low nutrient levels - root:shoot ratio, maximal uptake rate per unit mass of roots, Michaelis constant for uptake , fractional allocation of shoot mass to leaves, photosynthetic nitrogen-use efficacy (PNUE).  The suite allowed plants to maintain rather high growth rates (RGR, relative growth rates) to extremely low concentrations of nutrients in solution.  The FB model explained the relative value of each acclimation.  It also showed that adjustments in nutrient content were passive and that RGR varied as the square root of tissue N content, not linearly with it.  This model also explains a lot of the variation of RGR and tissue nutrient content at high CO2: a doubling of CO2 gives a doubled PNUE, an increase of RGR closely as the square root of CO2 content, and a reduction in tissue N content as 1/(square root of CO2 content).  Finally, it raised some major questions about regulation of root allocation, many only partially answered to date (water may be the factor, not nutrients).

      An algebraic model of similar (low) degree of complexity was the model that Thierry Simonneau and I (2002) developed to merge the understanding of how leaf stomatal conductance was controlled by the aerial environment (photosynthetic rate, humidity, CO2 level) and the root environment (soil water potential).  The aerial control is described well by the Ball-Berry model, which we combined with the model of Tardieu and his group for responses to soil water stress, mediated by the concentration of the ABA hormone sent to the leaves in xylem sap. If we start from measured photosynthetic rate, air humidity, and such, we have a data-fitting exercise.

       If we start from the basic aerial environment - PAR irradiance, air temperature, air humidity and CO2 content, windspeed, and other radiant fluxes (near infrared, thermal infrared) - in order to predict stomatal conductance, photosynthesis, and transpiration, then we have to solve four coupled nonlinear algebraic equations, one for the enzyme kinetics of photosynthesis, one for CO2 transport across stomata and the leaf boundary layer, one for leaf energy balance, and the Ball-Berry equation for stomatal conductance.  I developed very efficient, foolproof solution methods to solve these simultaneously.  They are described in several publications (see, e.g., p 258 in Wang et al., 2007) and reports.  The Fortran code is available.

      Another algebraic model, much more complex, is my model of the diversity of plant responses to elevated CO2.  Plants change a number of physiological and developmental traits at elevated (or, in the past, depressed) CO2 levels.  Stomatal conductance is described rather well by the Ball-Berry equation.  It predicts a rise in photosynthetic rate, a fall in stomatal conductance and transpiration, and a rise in water-use efficiency.  These changes are tempered by physical relations (leaf energy balance - closed-down leaves get hotter) and by additional physiological and developmental responses.  These include functional balance (described above), which predicts a lowering of leaf photosynthetic capacity,  and changes in nutrient uptake rates by roots (genetically controlled, widely variant among species, and rarely adaptive - adaptive variation has been lost by genetic drift over the last 20 million years of low CO2).   The simultaneous solution of all these equations was more complex, and the method of estimating plant physiological parameters was pretty complicated, though fully defined (see Fig. 1 in the paper).  A conclusion of the model is that the genetically programmed responses to elevated CO2 are so diverse that the changes in photosynthesis (and growth) rates, efficacies of using water and N, and sparing of water and N are also very diverse.  Predictions of biogeographic ranges that rely on temperature and precipitation "preferences" of species are likely to be very far off the mark.

      Yet another model comes down to algebraic equations, implicitly, but it has a solution arrived at using some interesting spatial transformations.  This is my calculation of the probability distribution of diffuse radiation (skylight)levels (irradiances) on leaves of random leaf angle distribution. 

  • discrete time or space.  I tend to avoid these, as most of these models are better formulated as differential equations; nature rarely comes in discrete steps.  Still, I have done some discrete layer models of plant canopies, as in looking at leaf clumping effects on light interception.  Another model I tried was inverting the measured light penetration from different sky angles to estimate total leaf area.  This is the problem that was solved, at the same time, as Welles and Norman (1991) developed an even better theory that led to the LAI-2000 canopy analyzer.  I discovered that the problem is extremely ill-posed, as a set of linear equations in which the rows of the matrices are nearly linearly dependent on each other.  Some clever techniques for stabilizing the solutions (constrained linear inversion; Twomey, 1977) for the distribution of air temperature with height weren't quite good enough.
  • differential equations, ordinary (one variable) or partial (two or more variables).  One early example was my model of the capture of excitations of chlorophyll molecules in the photosynthetic unit - alas, not published, under the direction of my postdoc advisor, hamstrung in this and other work by perfectionism (remember Voltaire's dictum, translated as, "The best is the enemy of the good [enough]."  A published example is the model that Frits Wiegel and I published on the optimal distribution of leaf mass (representing photosynthetic capacity) with depth in a canopy.  We solved this with Lagrange multipliers, which generate a differential equation in depth, L.  The results were surprisingly close to observed values for a number of plant species. 

           Another differential-equation model was embedded in a larger model.  This was a model of the distribution of light fluxes inside a leaf, accounting for absorption and scattering of light, including light entering the leaf from top and bottom.  It was used in a model of whole canopies, in which I estimated the value of sharing light with the lower canopy by having lighter-colored, less absorptive leaves.  This is detailed in the section on the diversity of purposes for models.  I'll keep the intriguing results for that section.

  • linear or nonlinear.  Alas, nature is often nonlinear.  I've described above some truly nonlinear models, such as the Ball-Berry model.


Some references

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. 1991. Joining leaf photosynthesis models and canopy photon-transport models. In: Photon-Vegetation Interaction: Applications in Optical Remote Sensing and Plant Ecology ,eds. R. B. Myneni and J. Ross. Springer Verlag, Berlin. Pp. 501-535.

V. P. Gutschick. 2007. Plant acclimation to elevated CO2 - from simple regularities to biogeographic chaos. Ecological Modelling 200: 433-451. (PDF here)

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 T. Simmoneau. 2002. Modelling stomatal conductance of field-grown sunflower under varying soil water status and leaf environment: comparison of three models of response to leaf environment and coupling with an ABA-based model of response to soil drying. Plant Cell Environ. 25: 1423-1434. (PDF here)

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.

S. Twomey. 1977. Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements. Elsevier, Amsterdam.

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)

J. M. Welles and J. M. Norman. 1991. Instrument for indirect measurement of canopy architecture. Agron. J. 83: 818-825.