R. V. Scholtz

Mathematical Models of Crop Growth and Yield

INTRODUCTION Historic Background Yield Response Models Growth Models Environmental Input Summary Exercises References SEASONAL RESPONSE MODELS Background Extended Logistic Model Extended Multiple Logistic Model Water Availability Legume/Grass Interaction Summary Exercises References GROWTH RESPONSE MODELS Background Empirical Growth Model Extended Empirical Growth Model Phenomenological Growth Model Expanded Growth Model Exercises References MATHEMATICAL CHARACTERISTICS OF MODELS Background Phenomenological Growth Model Expanded Growth Model Rational Basis for Logistic Model Coupling Among Applied Soil and Plant Components Accumulation of Dry Matter and Plant Nutrients Exercises References PASTURE SYSTEMS Background Quadratic Model Linear Exponential Model Summary Exercises References NONLINEAR REGRESSION FOR MATHEMATICAL MODELS Background Logistic Model Probability Model Confidence Contours Sensitivity Analysis Dimensionless Plots Correlation Coefficient Exercises References Index

CORN RESPONSE TO IRRIGATION AND APPLIED NITROGEN

Yield of corn (Zea mays L.) is very dependent on water availability and applied nitrogen (N). In this article data from the literature are used to establish dependence of yield upon each of these factors and the two in combination. The three separate studies involved surface irrigation of corn. From the first study an exponential relationship is established between dry matter yields (grain and total) and evapotranspiration (ET). Yield at the highest ET rate was shown to be about 80% of the maximum potential yield for three soils studied. Grain was shown to constitute about 50% of total dry matter at all ET rates. Analysis of data from the second study confirmed the utility of the logistic equation for yield response to applied N. The third study included three irrigation rates and six rates of applied N. Analysis showed that the model could be written as the product of a logistic term for applied N and an exponential term for ET. This represents an improvement over the previous linear model for dependence of yield on water availability. *Florida Agricultural Experiment Station Journal Series No. R-08371.

Model for accumulation of dry matter and plant nutrients by corn

Abstract Accumulation of dry matter by warm‐season annuals depends upon time of season, including planting time. A mathematical model has been developed to simulate the growth process. The model contains a Gaussian environmental function and a linear‐exponential intrinsic growth function. Previous work has shown the applicability of the model to data for the perennials bahiagrass (Paspalum notatum) and bermudagrass (Cynodon dactylon). This article applies the model to field data for the annual corn (Zea mays) from four locations. Only two of the five parameters are varied for the different studies to match dry matter simulation with data. A hyperbolic relationship between plant nutrient accumulation [nitrogen (N), phosphorus (P), or potassium (K)] and dry matter accumulation has been included. Parameters for the hyperbolic equation for plant N agree closely for the three locations where plant N was measured. Results for P and K varied. Since the total plant dry matter accumulates at a faster rate than pla...

Langmuir‐hinshelwood model of soil phosphorus kinetics

Abstract A mathematical model is needed to relate the dynamics of soil phosphorus (P) chemistry in a batch reactor to the soil/solution ratio and to initial P in the reactor. The Langmuir‐Hinshelwood model appears to describe this system quite well. According to this model reversible adsorption of P from solution to the colloidal surface (Langmuir component) is followed by an irreversible reaction of the surface species (Hinshelwood component). The system is an example of heterogeneous catalysis. Adsorption follows 2nd order kinetics related to solution P concentration and concentration of available surface sites for adsorption. Desorption and reaction are assumed to follow 1st order kinetics. The system is described by three simultaneous equations, two of which are 1st order nonlinear ordinary differential equations. Numerical integration is by the Euler finite difference method. Data from the literature are used to calibrate the model and to demonstrate its characteristics for a sandy soil. Rapid drop i...

Model of Yield Response of Corn to Plant Population and Absorption of Solar Energy

Biomass yield of agronomic crops is influenced by a number of factors, including crop species, soil type, applied nutrients, water availability, and plant population. This article is focused on dependence of biomass yield (Mg ha−1 and g plant−1) on plant population (plants m−2). Analysis includes data from the literature for three independent studies with the warm-season annual corn (Zea mays L.) grown in the United States. Data are analyzed with a simple exponential mathematical model which contains two parameters, viz. Ym (Mg ha−1) for maximum yield at high plant population and c (m2 plant−1) for the population response coefficient. This analysis leads to a new parameter called characteristic plant population, xc = 1/c (plants m−2). The model is shown to describe the data rather well for the three field studies. In one study measurements were made of solar radiation at different positions in the plant canopy. The coefficient of absorption of solar energy was assumed to be the same as c and provided a physical basis for the exponential model. The three studies showed no definitive peak in yield with plant population, but generally exhibited asymptotic approach to maximum yield with increased plant population. Values of xc were very similar for the three field studies with the same crop species.

In Defense Of The Extended Logistic Model Of Crop Production

The extended logistic model has been used extensively to relate seasonal crop production to applied nutrients (such as nitrogen). Model estimates include dry matter yield, plant nutrient uptake, and plant nutrient concentration. It has been applied to perennials such as bermudagrass (Cynodon dactylon L.), bahiagrass (Paspalum notatum Flügge), dallisgrass (Paspalum dilatatum Poir), ryegrass (Lolium perenne L.), tall fescue (F. arundinacea Schreb.), and annuals such as corn (Zea mays L.). Dependence of response to such factors as water availability and harvest frequency (for perennials) can be incorporated into the model. On occasion readers still question the utility of the logistic model over other models (such as polynomials). This article attempts to clarify this point and offers a defense of the logistic model for analysis, design, and management of crop production systems. †Florida Agricultural Experiment Station Journal Series No. R-08664.

Accumulation of Biomass and Mineral Elements with Calendar Time by Cotton: Application of the Expanded Growth Model

Accumulation of plant biomass (Mg ha−1) with calendar time (wk) occurs as a result of photosynthesis for green land-based plants. A corresponding accumulation of mineral elements (kg ha−1) such as nitrogen, phosphorus, and potassium occurs from the soil through plant roots. Field data from literature for the warm-season annual cotton (Gossypium hirsutum L.) are used in this analysis. The expanded growth model is used to describe accumulation of biomass and mineral elements with calendar time. The growth model predicts a simple linear relationship between biomass yield and the growth quantifier, which is confirmed with the data. The growth quantifier incorporates the unit processes of distribution of solar energy which drives biomass accumulation by photosynthesis, partitioning of biomass between light-gathering and structural components of the plants, and an aging function. A hyperbolic relationship between plant nutrient uptake and biomass yield is assumed, and is confirmed for the mineral elements nitrogen, phosphorus, and potassium. It is concluded that the rate limiting process in the system is biomass accumulation by photosynthesis and that nutrient accumulation occurs in virtual equilibrium with biomass accumulation. The expanded growth model describes field data from California and Alabama rather well. Furthermore, all model parameters were common for the two sites with the exception of the yield factor A which accounts for differences in soil types, environmental conditions, fertilizer levels, and plant population.

Model Analysis of Corn Response to Applied Nitrogen and Plant Population Density

Abstract The extended logistic model relates seasonal dry matter and plant nutrient uptake to applied nutrient level. It has been shown to apply to data for annuals such as corn (Zea mays L.) and perennials such as bermudagrass (Cynodon dactylon L.) and bahiagrass (Paspalum notatum Flügge). The linear parameters in the model have been shown to depend on water availability and harvest interval (for perennials). Further work is needed to relate model parameters to plant characteristics. In this article, data from a field experiment with corn at six nitrogen levels (0, 0.5, 1.0, 2.0, 3.0, and 5.0 g N plant−1) and three plant population densities (3, 6, and 9 plants m−2; 3, 6, and 9 104 plants ha−1) are used to provide insight into this question. It turns out that all five model parameters are dependent on plant density, approaching maximum values at 8.3 plants m−2. Three of the parameters approach zero as density approaches zero, which seems intuitively correct. It is concluded that lower and upper limits of plant nitrogen concentration are independent of population density and are functions of the particular plant species. Detailed procedures are described for estimation of model parameters.

Model Analysis of Response of Bermudagrass to Applied Nitrogen

The extended logistic model of crop response to applied nutrients provides quantitative coupling of seasonal dry matter (Y), plant N uptake (N u ), and plant N concentration (N c ) with applied nutrient (N). It predicts a hyperbolic relationship between Y and N c with N u . Analysis of data from numerous studies has confirmed the model. In this article the model was applied to data for Midland and Tifton 44 bermudagrass (Cynodon dactylon L.) grown on the same soil in Oklahoma. Results showed that the intercept parameters b and b n , as well as the nitrogen response coefficient c, were common to the two cultivars. The difference was accounted for in yield parameter A and plant N uptake parameter A n . Applied N required to achieve 50% of maximum yield was 92 kg ha−1 for both cultivars. #Florida Agricultural Experiment Station Journal Series No. R-08756.

Model Analysis for Growth Response of Corn

Abstract The expanded growth model was developed to describe accumulation of dry matter and plant nutrients with time for annual and perennial crops. It incorporates an environmental driving function and an intrinsic growth function. Previous analysis has shown that the model applies to the annual corn (Zea mays L.) and the warm-season perennial bermudagrass (Cynodon dactylon L. Pers.). In this article the model is used to describe accumulation of dry matter and plant nutrients [nitrogen (N), phosphorus (P), potassium (K), calcium (Ca), magnesium (Mg)] by corn. The model describes dry matter accumulation by the vegetative component of the plant followed by accumulation of dry matter and plant nutrients with time by ears. The time lag between planting and full vegetative growth is 14 − 9.3 = 4.7 wk, followed by a lag of 20 − 14 = 6 wk between vegetative and ear growth. After 20 wk there is a rapid decline in plant N in the vegetative component of the plant. During ear formation plant N, P, and Mg concentrations remained essentially constant, while plant K and Ca showed a hyperbolic linkage to dry matter accumulation. Several model parameters were the same for vegetative and ear components.