# Introduction to mathematical modeling of crop growth: How the equations are derived and assembled into a computer program

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## About the author

Christopher Teh Boon Sung completed his Ph.D. in agriculture at University of Reading, U.K. in 2001, and he is now a lecturer at Universiti Putra Malaysia, Serdang. His fields of specialty are environmental biophysics, crop modeling and C++ programming. He has numerous publications in local and international journals as well as in conference proceedings. He served as the Chief Editor for the Proceedings of the Malaysian Soil Science Conference 2005. He is currently the Honorary Secretary for the Malaysian Soil Science Society (MSSS).

## Book Synopsis

Mathematical modeling need not be difficult. Unlike other books, this book not only lists the equations one-by-one, but explains in detail how they are each derived, used, and finally assembled into a computer program for model simulations. This book shows how mathematics is applied in agriculture, in particular to modeling the growth and yield of a generic crop. Topics covered are agriculture meteorology, solar radiation interception and absorption, evapotranspiration, energy and soil water balance, soil water flow, photosynthesis, respiration, and crop growth development.

Rather than covering many modeling approaches but in superficial detail, this book selects one or two widely-used modeling approaches and discusses about them in depth. Principles learned from this book equips readers when they encounter other modeling approaches or when they develop their own crop models.

## Table of contents

```
FIGURES
TABLES
CODES
PREFACE
CHAPTER 1. MATHEMATICAL MODELING
1.1 What is a mathematical model?
1.2 Uses of mathematical models
1.3 Characteristics of mathematical models
1.4 Modeling methodology: Example of a leaf count model
1.4.1 Review of the modeling methodology
1.5 Types of mathematical models
1.5.1 Mechanistic and empirical models
1.5.2 Static and dynamic models
1.5.3 Discrete and continuous models
1.5.4 Deterministic and stochastic models
1.6 Looking ahead: Crop production levels
CHAPTER 2. METEOROLOGY
2.1 Weather components
2.2 Solar position and time
2.3 Day length and times of sunrise and sunset
2.4 Solar radiation
2.4.1 Daily irradiance
2.4.2 Partitioning of daily direct and diffuse irradiance
2.4.3 Hourly irradiance
2.4.4 Partitioning of hourly direct and diffuse irradiance
2.4.5 Net radiation
2.5 Air vapor pressure
2.6 Wind speed
2.7 Air temperature
CHAPTER 3. THE PLANT-RADIATION REGIME
3.1 Interception of hourly direct solar radiation
3.1.1 Scattering and reflection of solar radiation
3.1.2 Discontinuous canopies
3.2 Interception of hourly diffuse solar radiation
3.3 Daily interception
3.4 PAR absorption
3.4.1 Sunlit and shaded leaves
CHAPTER 4. PLANT TRANSPIRATION AND SOIL EVAPORATION
4.1 Energy balance
4.2 Latent and sensible heat fluxes
4.3 Electrical resistance analogy: Penman-Monteith model
4.4 Dual source evaporation: Shuttleworth-Wallace model
4.4.1 Computing evapotranspiration
4.5 Aerodynamic resistances
4.5.1 Air turbulence
4.5.2 Vertical wind speed profile
4.5.3 Formulation of the aerodynamic resistances
4.6 Boundary layer and stomatal resistances
4.7 Soil surface resistance
CHAPTER 5. WATER BALANCE
5.1 Soil water balance
5.1.1 Expressions of soil water content
5.2 Solving the water balance in a two-layered soil
5.2.1 Soil water movement
5.2.2 Actual evapotranspiration
CHAPTER 6. PHOTOSYNTHESIS (C3 PLANTS)
6.1 Modeling trends
6.2 Light and dark reactions
6.3 Enzyme kinetics: Michaelis-Menten equation
6.3.1 Competitive inhibition
6.4 Limitations to CO2 assimilation
6.4.1 Rubisco-limited rate
6.4.2 Light-limited rate
6.4.3 Sink-limited rate
6.4.4 Gross assimilation rate
6.5 Photosynthesis parameters
6.5.1 Leaf temperature
6.5.2 Internal CO2 concentration
6.6 Scaling from leaf to canopy CO2 assimilation
6.7 Re-expressing gross photosynthesis
CHAPTER 7. RESPIRATION AND GROWTH DEVELOPMENT
7.1 Conceptual model of respiration
7.1.1 Maintenance respiration
7.1.2 Growth respiration
7.2 Growth development
7.3 Leaf area index
7.4 Leaf death
7.5 Plant height growth
7.6 Root elongation (rooting depth)
7.7 Growth reduction due to water stress
CHAPTER 8. PUTTING IT TOGETHER: THE COMPUTER MODEL
8.1 Source code and header files
8.2 Data files
8.2.1 Weather data
8.2.2 Tabulated data
8.2.3 Model data
8.3 Global variables
8.4 Main program flow
8.5 Model output
8.6 Source code listings
8.7 Examples of data files
APPENDICES
A List of main symbols
B Spherical trigonometry
C Numerical integration: Gauss method
D Derivation of the Shuttleworth-Wallace evapotranspiration model
E Sample weather data
INDEX
```

## Why this book was written

**Why this book is written:**

I wrote this book because I wanted a book that explains clearly and in depth how mathematics can be applied in agriculture. I wanted a book that takes the trouble to explain how each equation is derived and how it is used. And finally, I wanted a book that shows how these equations work together and are finally assembled into a computer program for model simulations. I believe my wish list is not unique; it is shared by other agriculturists who desire a book that speaks to them rather at them. Mathematical modeling need not be difficult, but we require a book that does not only list the equations one-by-one but shows how they are derived and used.

**What this book is about:**

This book is to show how mathematics is applied in agriculture, in particular to modeling the growth and yield of a generic crop. Principles learn from the growth and yield of a generic crop can then be applied to “real” crops such as maize, rice and oil palm. This book explains how each equation is derived and used. Finally, this book shows how all the equations work together for model simulations by assembling them into a C++ computer program.

**What this book is not about:**

This book is not meant to be a comprehensive discussion about all modeling approaches or widely-used crop models. Rather than covering many modeling approaches but in superficial detail, this book selects one or two widely-used modeling approaches and discusses about them in detail. By discussing a particular method in detail, this equips readers with the necessary principles required when they encounter other modeling approaches or crop models. This book is also not about C++ computer programming. This would require a separate book. Nonetheless, the computer programs in this book are written deliberately in a simple manner so that readers only require a basic knowledge in C++ to understand them. Finally, this book is not a leisure reading material. Although, I have tried my best to explain how each equation is derived, readers are still expected to roll up their sleeves and do some hard work. Readers must think deeply and thoroughly, ask questions, and refer to other reading materials for further clarification or for a deeper understanding.

**Who should read this book:**

This book is intended for senior undergraduates, postgraduates, academicians and scientists in the field of agriculture. Basic knowledge in mathematics, in particular algebra, trigonometry and calculus, is required. Additional knowledge in basic C++ programming will be helpful to build the computer model.

## Book Details

ISBN-10: 1581129998

ISBN-13: 9781581129991

Publisher: BrownWalker Press, Boca Raton, Florida, USA

Year: 2006

No. of pages: 276

Type: Paperback and e-book

Dimensions: 5.5 by 8.5 inches (paperback)

## Download

1. Source code (C++) - Gg.zip (44 kb) | |

2. Sample weather file (Netherlands, 1990) - 1990neth.txt (14 kb) | |

3. More sample weather files (Serdang, 1974-2004) - serdang.zip (121 kb) |

## Contact the book author

Homepage | ResearchGate | Google Scholar | Tel: +60-3-8947 4858