Mathematical Models of Crop Growth and Yield

Allen R. Overman, Richard V. Scholtz III

• Publisher: CRC Press
• Year: 2002
• Print ISBN:
  9781138034747
• eBook ISBN:
  9781135564070
• Topic: Technologie et ingénierie
• Subtopic: Mathématiques appliquées

Highlighting effective, analytical functions that have been found useful for the comparison of alternative management techniques to maximize water and nutrient resources, this reference describes the application of viable mathematical models in data analysis to increase crop growth and yields. Featuring solutions to various differential equations,

1

Introduction

1.1 HISTORICAL BACKGROUND

A dramatic change occurred in human history after the last ice age just over ten thousand years ago. Mankind shifted from hunter/gatherers to the cultivation of plants and domestication of animals (Bronowski, 1973). The last four centuries have witnessed fundamental progress in the management of agriculture (McClelland, 1997; Schlebecker, 1975). Work on a particular machine, the reaper, has been described by Canine (1995). During this latter period rapid advances have been made in the field of science. The twentieth century has been characterized by unprecedented discoveries in physics by a blending of deduction and induction (Born, 1956; Lightman, 2000). Induction leads from specific observations to general laws, while deduction leads from general principles to specific predictions. Our approach in this book is founded on this blend. Ideas at the frontiers of science have been discussed by Bak (1996), Holton (1973), Lindley (2001), Pagels (1988), Penrose (1989), and Polkinghorne (1996).

Response of agricultural crops to management factors (such as available water, applied nutrients, plant density, harvest interval for perennials) has been of interest to farmers and agricultural scientists for a long time. Perhaps the most famous experiments in the world on crop production are those conducted at Rothamsted (England). John Bennet Lawes (1814– 1900) devoted his family estate to the study and improvement of agricultural production. An early account of this work can be found in Hall (1905). In 1834 Lawes initiated chemical experiments in a laboratory in his house. This was followed in 1837 by experiments with plants in pots. Then in 1840 he applied superphosphate to some of the fields on the farm. The results were so satisfactory that in 1842 he patented a process for manufacture of superphosphate. In 1843 Joseph Henry Gilbert (1917–1901), who had taken his PhD with Justus Liebig, was employed as a chemist by Lawes. Their joint work with field plots and various crops continued over the next 57 years, leading to many publications in scientific journals. In 1919 Ronald Aylmer Fisher was hired as statistician at the Rothamsted Experimental Station, and over the next few years developed his analysis of variance and principles of experimental design (Box, 1978).

The approach to modeling in this book is very much motivated and guided by experimental results. Models have been developed and refined step by step rather than from some grand design, which then fills in the details. And while considerable progress has been made, this is very much a picture of a process which is still evolving. In this work we have not attempted to review the broad field of crop modeling, but rather have focused on models which we have found useful in practice. The reader will undoubtedly detect our preference for analytical functions, in contrast to finite difference procedures (Ford, 1999). We have never found finite difference to be an easy alternative to analytical functions. Students should still learn calculus, differential equations, and physics.

1.2 YIELD RESPONSE MODELS

One of the earliest efforts at modeling crop response to management factors was that of E. A. Mitscherlich, which stimulated a lot of interest and generated considerable controversy (Russell, 1912; van der Paauw, 1952). The Mitscherlich equation can be written as

where N = applied nutrient; Y = dry matter yield; Y₀ = dry matter yield at N = 0; Y_m = maximum dry matter yield at high N; c = nutrient response coefficient. Yield values are all assumed to be positive. Equation [1.1] can be rearranged to the alternate form

Equation [1.2] suggests plotting (Y_m – Y) vs. N on semilog paper, which requires an estimate of Y_m. If this plot follows a straight line, then linear regression can be performed on ln(Y_m – Y) vs. N to obtain estimates for (Y_m – Y₀) and c. Figure 1.1 shows data from Mitscherlich (1909), as reported by Russell (1912, p. 25), for response of oats (Avena sativa L.) in pots to applications of P₂O₅. The semilog plot is shown in Fig. 1.2, where the line is given by

with a correlation coefficient of –0.9939. It follows that (Y_m – Y₀ = 51:4 g and that Y₀ = 9:6 g. The line in Fig. 1.2 is drawn from Eq. [1.3] and the curve in Fig. 1.1 from

The fit of the Mitscherlich equation to these data appears quite reasonable.

It turns out that even earlier data do not conform as well to the Mitscherlich equation. Data from Hellriegel and Wilfarth (1888), as reported by Russell (1912, p. 32), for barley (Hordeum vulgare L.) in sand cultures exhibit the trend shown in Fig. 1.3. As pointed out by Russell (1937, p. 135) these data follow an S-shaped curve. We now refer to this form as sigmoid. To illustrate the problem with the Mitscherlich model, the hori-

Figure 1.1 Yield response of oats to applied phosphorus. Data from Mitscherlich (1909) as reported by Russell (1912). Curve drawn from Eq. [1.4].

Figure 1.2 Semilog plot of yield response of oats to applied phosphorus for experiment of Mitscherlich (1909). Line drawn from Eq. [1.3].

Figure 1.3 Yield response of barley to applied nitrogen. Data from Hellriegel and Wilfarth (1888) as reported by Russell (1912). Curve drawn from Eq. [1.6].

zontal and vertical axes have been expanded to include negative values. Figure 1.4 shows the semilog plot, with the line given by

where N = applied N, mg. This leads to (Y_m – Y₀) = 36:6g and to Y₀ = –1.6g. The curve in Fig. 1.3 is drawn from

While negative values of N might signify reduction of soil N below background level, nega...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Preface
• 1: Introduction
• 2: Seasonal Response Models
• 3: Growth Response Models
• 4: Mathematical Characteristics of Models
• 5: Pasture Systems
• 6: Nonlinear Regression for Mathematical Models