Mathematical models of crop growth and yield

340 Pages·2002·1.284 MB·English

by  Allen R Overman, Richard V Scholtz

Mathematical
Models of
Crop Growth
and Yield
Allen R. Overman
Richard V. Scholtz III
University of Florida
Gainesville, Florida, U.S.A.
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BOOKSINSOILS,PLANTS,ANDTHEENVIRONMENT
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AgriculturalEngineering RobertM.Peart,UniversityofFlorida,Gainesville
AnimalScience HaroldHafs,RutgersUniversity,NewBrunswick,
NewJersey
Crops MohammadPessarakli,UniversityofArizona,
Tucson
IrrigationandHydrology DonaldR.Nielsen,UniversityofCalifornia,Davis
Microbiology JanDirkvanElsas,ResearchInstituteforPlant
Protection,Wageningen,TheNetherlands
Plants L.DavidKuykendall,U.S.Departmentof
Agriculture,Beltsville,Maryland
KennethB.Marcum,TexasA&MUniversity, El
Paso,Texas
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Fungal Pathogenesis in Plants and Crops: Molecular Biology and Host
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Handbook of Functional Plant Ecology, edited by Francisco I. Pugnaire and
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Plant Roots: The Hidden Half, Third Edition, Revised and Expanded, edited
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Handbook of Plant Growth: pH as the Master Variable, edited by Zdenko
Rengel
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Mathematical Models of Crop Growth and Yield, Allen R. Overman and
RichardV.ScholtzIII
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Humic Matter: Issues and Controversies in Soil and Environmental Science,
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MolecularHostResistancetoPests,S.SadasivamandB.Thayumanavan
Preface
This book is intended to outline an approach to crop modeling that I have
found to be both mathematically solid and feasible to use in practice. My
strategy is to develop the technical details in a way that offers some insight
into a logical progression from a simple idea toward more complex details.
Thereadershouldsensesomethingoftheuncertaintyinvolvedinthestrug-
gle to interpret data and compose models for describing empirical results.
My search hasbeendrivenby a combinationof practical need and intellec-
tual curiosity.
Sciencecanbeviewedasthesearchfor:(1)patterns,(2)relationships,(3)
connections, (4) consistency, and (5) beauty. Look for patterns (sometimes
trends) in plots of data. Then search for mathematical relationships that
agree with the patterns. Seek to identify connections between or among
various factors in the analysis. Check for consistency from one data set to
others (different investigators and different conditions). Finally, look for
mathematical beauty (such as symmetry) in the models. The last is usually
the most difficult to achieve and is often ignored.
Thelanguageofmodernscienceismathematics.Thishasbecomeincreas-
ingly true in physics, chemistry, and biology. We can only marvel at how
abstract ideas in mathematics have found application in science and engi-
neering. Perhaps that is because many ideas were generalized from particu-
lar problems and observations. Attempts at quantitative descriptions of
plant and animal systems inevitably lead to this same mathematical path-
iii
iv Preface
way. Unfortunately, many people are intimidated by the language and
methods of applied mathematics. By the nature of the subject, this book
necessarily involves many equations of different types. On a relative scale,
the functions employed herein can be mastered with nominal effort and
determination. The functions are all of the analytical type, in contrast to
numerical techniques. Throughout I have followed the practice of starting
withsimplermodelsandthenprogressingtowardsgreaterlevelsofcomplex-
ity and comphrehensiveness.
The reader is encouraged to scan the book to view data and model
simulations before trying to master mathematical details, since the ultimate
criterion for practical usefulness is agreement between the two. There is
some repetition in the contents. Data are presented in both tabular and
graphical form for several cases to help the reader obtain a more compre-
hensive grasp of the material. Graphs are sometimes shown in curvilinear
form(Yvs.X)andtheninlinearizedform(semilogorprobability)toclarify
trends and scatter in data. In addition, the same equation may be listed
more than once in a chapter to enhance the flow of the exposition.
Finally,inmanycasesthesamesymbol(e.g.,A)isusedwithdifferentmean-
ingstoavoidaplethoraofsymbols.Ihavetriedtocarefullydefinevariables
and parameters for each particular usage.
ChildhoodonafarmineasternNorthCarolinagavemetheopportunity
to observe seasons and growth of plants and animals. This led to many
questions about the process of farming. How did farmers know when to
plant their crops? What was the relationship of crop growth to climate,
water, and nutrients? It was obvious that all these factors influenced pro-
duction. Was farming a craft, an art, or a science? In high school I found
science courses more interesting than agricultural courses and was led to
believethatsomeoftheanswerstomyquestionsmightliewithintheframe-
work of science.
AsIreadaboutengineering,thereappearedmanyparallelsbetweenfarm-
ing and engineering. As a result, in deciding on a major for college I chose
agricultural engineering at North Carolina State College. Actually, my first
exposure to this profession occurred at the age of 12 through a 4-H Club
project in tractor maintenance. Eventually, I was to obtain B.S, M.S., and
Ph.D.degreesinthisfieldatNorthCarolinaStateUniversity.Theacademic
program was heavy in math, physics, and chemistry. Course work also
included soil physics and soil chemistry but very little inplant science.
My interest in plant growth and production lay dormant until my post-
doctorate work in agronomy at the University of Illinois. While there I
audited a course on soil fertility taught by Professor Sigurd Melsted.
Through his course I was introduced to the models of Mitsherlich and
multivariate regression. It was very clear that these were empirical models
Preface v
withoutanyphysicalbasis.Afterjoiningtheagriculturalengineeringfaculty
at the University of Florida my interest in crop models was stimulated by
studies on crop response to waste application from agricultural and muni-
cipal sources. It was natural to invoke the Mitscherlich model, partly
because of its mathematical simplicity and ease of calibration. Since it
seemed to work reasonably well in relating crop yield to applied N, P, or
K,Ithenperformedanextensivereviewofnumerousfieldstudiestodevelop
a broader perspective on the subject. It was during this study that I
expanded the Mitscherlich equation to describe plant nutrient uptake as
wellasdrymatterproduction.Thisseemedlogicalenough,anditappeared
to describe the data reasonably well. However, the approach presented a
paradox as soon as I tried to calculate plant nutrient concentration as the
ratio of plant nutrient uptake to plant dry matter. In some cases the ratio
either went negative or blew up at low applied nutrient levels! It was clear
that there was a problem, but the question was how to either avoid the
problem or fix it somehow. The subject was laid aside to incubate while
attention was focused on other more pressing matters.
Progress came from a slightly different direction. I was challenged to
eitherselectordevelop adynamicmodel torelate plant growth(dry matter
andplantnutrientaccumulation)totime.Concernswithgroundwaterqual-
ity related to agriculture and waste management created pressure to incor-
porate practical models into the analysis and engineering design. By now I
hadanideaforarelativelysimplemathematicalapproach.Analysisofsome
field data confirmed the usefulness of the approach. Further analysis with
otherdatagaveadditionalsupportandencouragement.Whiletheapproach
wasstrictlyempirical,thefunctiondidexhibitthecorrectformofresponse.
After reading that science progresses by intuition, not just by cold logic, I
was no longer embarrassed by the ‘‘guessing’’ approach. As often happens,
progresswiththiseffortopenedupquestionsrelatedtootherfactorssuchas
nutrientlevelsandwateravailability.Oneoftheparametersinthedynamic
model depended on level of applied nutrients, the form of which suggested
another mathematical function. In turn, one of its parameters depended on
wateravailability.NowIwasbacktotheleveloftheMitscherlichequation
from earlier times, but with a better function in hand. As discussed in
Chapter 1, the hint for this improved function was identified as early as
1937 by Sir John Russell.
The next challenge was to expand the mathematical models to incorpo-
ratedrymatterandplantnutrientaccumulationasrelatedtomultiplelevels
of applied nutrients (N, P, K) in a mathematically self-consistent manner.
The empirical model was expanded to a phenomenological model with a
physical basis. This model was shown to apply to warm-season perennial
grassforharvestintervalsuptoabout6weeks,butfailedforlongergrowth
vi Preface
intervals and did not apply to annual grasses such as corn. Eventually, the
growth model was modified again to cover these cases.
Many individuals deserve thanks for inspiration and help in my struggle
to understand and describe science and engineering. Francis J. Hassler, J.
van Schilfgaarde, Raymond J. Miller, and Ernest T. Smerdon nurtured the
spiritofintellectualcuriosity.SigurdMelstedintroducedmetotheworldof
soilfertilityandsimplemodels.ThomasP.Smith,WilliamG.Leseman,and
John Dean encouraged research on water reclamation and reuse (which
partially motivated the work on crop models). This led to many conversa-
tions with farmers Walter Vidak and J. L. Morgan about crop and animal
management at the Tallahassee Southeast Farm. I have had the good for-
tune to work with several colleagues in agricultural research, including
Stanley Wilkinson, Frank Martin, Eugene Kamprath, Gerald Evers,
Donald Robinson, William Blue, Charles Ruelke, Fred Rhoads, Robert
Stanley, John Moore, and George Hochmuth. The scope of field plot
research by Glenn Burton at Tifton, Georgia, and William Adams at
Watkinsville, Georgia, has been a particular source of inspiration. Special
thanks are due Elizabeth Angley, who worked with me on the early math-
ematical models and developed computer programs for nonlinear regres-
sion. Marcus Allhands, Bill Reck, Denise Wilson, John Willcutts, Sherry
Allick, Scott Thourot, and Kelly Brock worked on aspects of this subject
during their student days. I especially thank my coauthor, Richard V.
Scholtz III, for extensive discussions on mathematical modeling of crop
systems as well as help with graphics in the text. Without his assistance, I
could not have finished this project.
Allen R. Overman