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Introduction to Mathematical Modeling of Crop Growth: How the Equations Are Derived and Assembled Into a Computer Program - Hardcover
Synopsis
Learning 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.
"synopsis" may belong to another edition of this title.
From the Author
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.
Excerpt. © Reprinted by permission. All rights reserved.
Chapter 1. Mathematical modeling
1.1 What is a mathematical model?
A model is a simplified representation of a real system. There are two important concepts in this definition. Firstly, system is a group of objects (components or factors) that interact with one another in an organised manner, and the net result of their interactions produces the system’s behavior, function and purpose. Examples of systems in agriculture are photosynthesis, soil water flow, crop growth and yield, and the interception of sunlight by the plant canopies. Secondly, when a system is represented or described in a simpler form known as a model, the model becomes a tool for us to understand the system by helping us to sift through its complexity and to focus on the important, relevant aspects. Models can be of many types:
1. pictorial (e.g., illustrations, diagrams and flowcharts);
2. conceptual or verbal (descriptions in a natural language);
3. physical (replica or mock-up of the system such as a scale airplane model for wind tunnel experiments, and the helical, double-stranded structure of the DNA molecule); and
4. mathematical (Haefner, 1996).
Consequently, a mathematical model is one type of simplified representation of a real system. It describes the system using mathematical principles in the form of an equation or a set of equations. For example, in a controlled field experiment by Pandey et al. (2000), the maize yield (Y; kg ha-1) responds to the nitrogen fertilizer rate (N; kg ha-1) by the following mathematical/statistical relationship:
Y = 1143 + 31.7*N - 0.084N*N
which reveals that with increasing nitrogen rates, maize yield increases but with diminishing returns, and that past a certain threshold of nitrogen level (189 kg ha-1), the maize yield will instead start to decrease (probably due to nitrogen toxicity).
1.2 Uses of mathematical models
Mathematical models are developed primarily because we want to solve one or more problems in a real system. For example: how do we increase the growth and yield of the rice crops? Why are the current conservation practices not effective in controlling the soil and water losses from the soil? And what is the effect on oil palm yield due to higher planting density on peat soils?
As compared to other model types (e.g., pictorial, conceptual and physical), mathematical models are often the most useful type. This is because they not only tells us what are the essential components of a real system, but more importantly, they also tells us how and to what degree do each of these components interact with each other. Besides helping us to comprehend the system (i.e., the "what, why and how" of a system), mathematical models exclusively allow us, in an organised manner, to predict (i.e., forecast or estimate the outcome) and control the system (i.e., constrain or manipulate the system’s behavior to the desirable outcome). Consequently, mathematical models provide us a tool to solve problems. Specifically, the following are some ways, as adapted from France and Thornley (1984), in which mathematical models can contribute to agricultural research and activities:
1. Modeling helps us to understand, predict and control a system in a more organised or methodological manner because models provide: a) a quantitative description of the system, and b) a
way of bringing together knowledge about the parts to give a coherent and holistic view of the system.
2. Models can help to identify areas where knowledge is lacking, and can help to stimulate new ideas or approaches for research.
3. Modeling leads to less experimentation by trial-and-error, and can reduce the time and cost of experiments. This is particularly relevant for long term growing crops such as oil palm which can take many years to mature. Consequently, actual field experiments for such crops are expensive, time-consuming, and must be planned with great care so that useful results are ensued from such long term experiments. Another good example is the work by Peng et al. (2004), who produced a China "super" rice hybrid that gave yields up to 12 ton ha-1, breaking the yield ceiling of 10 ton ha-1. Peng et al. used a mathematical model to identify the characteristics of a rice plant that were theoretically efficient in photosynthesis, growth and grain production. They then started a rice breeding project to produce rice hybrid varieties that had these idealized plant characteristics.
4. Models can be used to identify interesting and stimulating areas of research, and those short listed can later be implemented in actual experiments.
5. The predictive power of models can be used to answer various "What if?" scenarios such as the yield response of maize under different water management practices, or the effect on oil palm growth and yield on compact, steep lands. Without the use of models, we must conduct actual experiments to predict the outcome of such "What if?" scenarios which can be expensive and time-consuming. In this manner, models can help in the research and development as well as in the management and planning of agriculture activities.
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