MATHEMATICAL MODELS IN AGRICULTURE

Quantitative Methods for the Plant, Animal and Ecological Sciences

All models are wrong, but some are more wrong than others.(C.W. Clark and M. Mangel (2000) Dynamic State Variable Models in Ecology)

The art of approximation is crucial to most successful applications of theory to real problems.

(J. Tranquada (1999) Physics World )

A model like a map cannot show everything. If it did it would not be a model but a duplicate. Thus the classic definition of art as the purgation of

superfluities also applies to models and the model-maker’s problem isto distinguish between the superfluous and the essential.

(Anon. (1960) Working models in medicine. Journal of the American Medical Association)

Simple things interacting in simple ways can yield surprisingly complex outcomes.(http://serendip.brynmawr.edu/complexity/ (October, 2003))

It is much easier to make measurements than to know exactly what you are measuring.J.W.N. Sullivan (1928)

One should never take observations too seriously until they have been tested by theory.(After Sir Arthur Stanley Eddington (1882–1944))

If a thing is worth doing, it is worth doing well enough for the purpose at hand. And it is probably wrong . . . to do it any better than that.

(After Clifford Swartz (2003) Physics World )

MATHEMATICAL MODELS INAGRICULTUREQuantitative Methods for the Plant,Animal and Ecological Sciences

J.H.M. Thornley

Centre for Ecology and Hydrology, Edinburgh, EH26 0QB, UK;6 Makins Road, Henley-on-Thames, Oxfordshire RG9 1PP, UK(mailing address)

and

J. France

Centre for Nutrition Modelling, Department of Animal & PoultryScience, University of Guelph, Guelph, Ontario N1G 2W1,Canada; School of Agriculture, Policy & Development,University of Reading, Reading, Berkshire, RG6 6AR

CABI is a trading name of CAB International

CABI Head Office CABI North American OfficeNosworthy Way 875 Massachusetts AvenueWallingford 7th FloorOxon OX10 8DE Cambridge, MA 02139UK USA

Tel: +44 (0)1491 832111 Tel: +1 617 395 4056Fax: +44 (0)1491 833508 Fax: +1 617 354 6875E-mail: cabi@cabi.org E-mail: cabi-nao@cabi.orgWeb site: www.cabi.org

© J.H.M. Thornley and J. France 2007. All rights reserved. No part of thispublication may be reproduced in any form or by any means,electronically, mechanically, by photocopying, recording or otherwise,without prior permission of the copyright owners.

A catalogue record for this book is available from the British Library,London, UK.

Library of Congress Cataloging-in-Publication DataThornley, J. H. M.

Mathematical models in agriculture : quantitative methods for the plant,animal and ecological sciences / J.H.M. Thornley & J. France.--2nd ed.

p.cm.France’s name appears first on the earlier editionIncludes bibliographical references and index.ISBN 0-85199-010-X (alk. paper)1. Agriculture--Mathematical models. I. France, J. II. France, J.

Mathematical models in agriculture. III. Title

S494.5.M3T49 2004630�.1�5118--dc22

20040225533

ISBN 0 85199 010 X

Typeset by Columns Design Ltd, Reading.Printed and bound in the UK by Cromwell Press, Trowbridge.

The paper used for the text pages in this book is FSC certified. The FSC(Forest Stewardship Council) is an international network to promoteresponsible management of the world’s forests.

Contents

Preface xv

PART I TECHNIQUES

1 Role of Mathematical Models 1Summary 11.1 Agriculture and Science 11.2 What is a Mathematical Model? 31.3 Hierarchy in Biology 61.4 Types of Models 71.5 Evaluation and Validation of Models 101.6 Possible Modelling Objectives 111.7 Models for Research and Models for Application 131.8 Models: Documentation, Presentation and Reviewing 141.9 Units 17Exercises 18

2 Dynamic Deterministic Models 19Summary 192.1 Variables 19

2.1.1 State variables 202.1.2 Rate variables 202.1.3 Driving variables 212.1.4 Other variables 21

2.2 Parameters and Constants 222.3 Differential Equations 23

2.3.1 Explicit time dependence 242.3.2 Memory and delay 25

2.4 Numerical Integration 262.4.1 Euler’s method – a first-order method 282.4.2 Trapezoidal method – a second-order method 302.4.3 Runge–Kutta method – a fourth-order fixed step

method 312.4.4 Oscillations caused by too large an integration step 322.4.5 Stiff equations 33

2.5 Models and Data: to Fit or Not to Fit 362.5.1 Predictions and measurements 372.5.2 Residual lack of fit 382.5.3 Confidence limits for fitted parameters 392.5.4 Sensitivity analysis 41

2.6 Multiple Steady States 412.6.1 Switches 422.6.2 Catastrophe 472.6.3 Oscillations 492.6.4 Chaos 50

Exercises 54

3 Mathematical Programming 57Summary 573.1 Introduction 573.2 Mathematical Formulation 58

3.2.1 Example 593.3 Graphical Solution 623.4 Computer Solution 653.5 Worked Example 71

3.5.1 Formulation 723.5.2 Solution 74

3.6 Special Topics 763.6.1 Parametric programming 763.6.2 Separable programming 773.6.3 Integer programming 833.6.4 Goal programming 843.6.5 Dynamic programming 85

Exercises 88

4 Basic Biological Processes 91Summary 914.1 Chemical Kinetics 91

4.1.1 First-order reactions 934.1.2 Second-order reactions 964.1.3 Stochastic approach to chemical kinetics 98

4.2 Catalysis 1024.2.1 Arrhenius equation 1034.2.2 Phenomenological temperature function 105

vi Contents

4.3 Biochemical Kinetics 1074.3.1 Michaelis–Menten kinetics 1074.3.2 Sigmoidal kinetics 1094.3.3 Transport plus Michaelis–Menten kinetics 1104.3.4 Bisubstrate Michaelis–Menten equation 1134.3.5 Inhibition 1144.3.6 Activation 1164.3.7 Futile cycles 116

4.4 Transport 1184.4.1 Diffusion – Fick’s law 1184.4.2 Convection 1204.4.3 General equation for transport and chemical

reaction 1204.4.4 Examples 1224.4.5 Lumped representation of transport 1264.4.6 Difference equation representation 128

4.5 Local and Non-local Variables 129Exercises 131

5 Growth Functions 136Summary 1365.1 Introduction 1365.2 Exponential Growth 1395.3 Monomolecular Equation 1405.4 Logistic Equation 1435.5 Gompertz Equation 1455.6 Chanter Equation 1485.7 Exponential Quadratic Equation 1505.8 Von Bertalanffy Equation 1525.9 Richards Equation 1555.10 Schumacher Equation 1575.11 Morgan Equation 1605.12 Other Growth Equations 163Exercises 169

6 Simple Dynamic Growth Models 172Summary 1726.1 Introduction 1726.2 Autocatalytic Growth with Sigmoidal Substrate Limitation 1736.3 Delayed Growth 1746.4 Compensatory Growth 178

6.4.1 Model scheme 1786.4.2 Simulations 182

6.5 Square-root-time Growth Equation 1846.6 ‘Open’ Logistic Growth 1866.7 Logistic Equation Modified for Substrate Supply and

Product Inhibition 188

Contents vii

6.8 Gompertz with Delayed Development 1906.9 Expo-linear-asymptotic Growth 192

6.9.1 Hyperbolic formulation 1936.9.2 Negative-exponential formulation 196

6.10 Allometry and Scaling 1966.10.1 Application of simple geometrical factors 1986.10.2 A branching model for allometric scaling 1996.10.3 Scaling in relation to maturity 202

6.11 Biological Oscillators 2046.11.1 Gene-metabolism oscillator 2046.11.2 Alternative-pathways oscillator 2066.11.3 Grazed-pasture oscillator 208

Exercises 210

7 Simple Ecological Models 213Summary 2137.1 Introduction 2137.2 Difference Equations and Differential Equations 2147.3 Age-structured Models of Population Growth 215

7.3.1 Discrete-age discrete-time model 2167.3.2 Discrete-age continuous-time scheme 2197.3.3 Continuous-age continuous-time model 221

7.4 Morph-structured Growth Model 2237.5 Morph- and Age-structured Growth Model 2267.6 Lotka–Volterra Type Model 2277.7 Disease/Epidemic Models 229

7.7.1 Five-compartment disease model 2297.7.2 Spatial aspects 230

Exercises 233

8 Environment and Weather 235Summary 2358.1 Introduction 2358.2 Time 2368.3 Solar Angles and Day Length 237

8.3.1 Day length switching 2418.4 Representing Weather in Models 243

8.4.1 Need for diurnal data in models 2438.4.2 Matching measured diurnal data to the model 2448.4.3 Generating diurnal data from daily data 2458.4.4 Generating daily data from monthly data

deterministically 2518.4.5 Generating daily data from monthly data

stochastically 2538.4.6 Deterministic sinusoidal seasonal variation 262

8.5 Bright Sunshine Hours and Daily Radiation Receipt 263

viii Contents

8.6 Angular Distribution of Radiation 2648.6.1 Direct solar radiation 2658.6.2 Diffuse radiation from a clear sky 2658.6.3 Overcast skies 265

8.7 Wind 2678.7.1 Diurnal variation 2698.7.2 Seasonal variation 270

8.8 Climate Change 270Exercises 272

PART II CROPS

9 Plant and Crop Processes 275Summary 2759.1 Introduction 2759.2 Light Interception 277

9.2.1 Crops with closed canopies 2789.2.2 Single plant 2829.2.3 Discontinuous canopies 2849.2.4 Mixtures 287

9.3 Photosynthesis 2889.3.1 Overview of photosynthesis 2899.3.2 Leaf photosynthesis 2909.3.3 Canopy photosynthesis 2959.3.4 Integrable closed-canopy models 2969.3.5 Single-plant photosynthesis 3019.3.6 Photosynthesis of discontinuous canopies 3039.3.7 Mixtures 305

9.4 Nitrogen Uptake; Nitrogen Fixation 3059.4.1 Nitrogen uptake 3069.4.2 Nitrogen fixation 309

9.5 Growth and Respiration 3109.5.1 Growth 3119.5.2 Maintenance 3129.5.3 Pathway analysis 3139.5.4 Futile cycles 324

9.6 Allocation 3269.6.1 Empiricism 3279.6.2 Teleonomy 3279.6.3 Mechanism 333

9.7 Development 3419.7.1 Temperature sums 3439.7.2 Generalized developmental rates 3459.7.3 Compartmental models of development 3469.7.4 Survey of recent contributions 348

9.8 Water 3499.8.1 Water potential 350

Contents ix

9.8.2 Soil water 3539.8.3 Root and shoot water 3579.8.4 Transpiration 3629.8.5 Soil surface evaporation 3739.8.6 Rainfall interception by canopy and evaporation 374

9.9 Leaf Stomatal Resistance 3759.9.1 Basic definitions 3769.9.2 Responses of stomata to environment and plant

variables 378Exercises 381

10 Crop Models 385Summary 38510.1 Introduction 38510.2 Crop Model Structure 38710.3 Simple Generic Daily Crop Model 388

10.3.1 Environment 39010.3.2 Temperature functions 39210.3.3 Plant submodel 39310.3.4 Litter and soil submodel 40110.3.5 Hydrology 40310.3.6 Simulations 408

10.4 Supply–Demand Models 41310.4.1 Supply and demand in relation to carbon and

nitrogen 41510.4.2 Supply–demand model of lettuce growth with

osmotic regulation 41610.4.3 Supply–demand models: do they have a role? 425

10.5 Plant Competition 42610.5.1 Two plants compete for a single substrate 42710.5.2 Interaction of two plants simulated 42910.5.3 Crops, weeds and mixtures 43110.5.4 Self-thinning: the three-halves rule 43610.5.5 Intercropping and agroforestry 43910.5.6 Genetically modified crops 441

10.6 Allelopathy and Phytotoxicity 44410.6.1 Stimulus and inhibition 44510.6.2 Dynamics of an allelochemical and its effect 445

Exercises 447

11 Crop Husbandry 450Summary 45011.1 Introduction 45011.2 A Grass Drying Enterprise 45111.3 Allocating Land to Arable Crops 45411.4 Harvesting Plans for Brussels Sprouts 45711.5 Planting Wheat 460

x Contents

11.6 Grazing 46211.6.1 Systems of rotational grazing 46211.6.2 Continuous and rotational grazing 464

11.7 Grassland and Fertilizer Usage 46611.8 Silage 469

11.8.1 Anaerobic phase of ensiling 469Exercises 477

12 Plant Diseases and Pests 481Summary 48112.1 Introduction 48112.2 Estimation of Yield Loss 48312.3 Disease Prediction 484

12.3.1 Potato late blight 48412.3.2 Sugar beet yellows virus 487

12.4 Mechanistic Disease Simulation 48812.4.1 Sugar beet fungal root infection 48812.4.2 Potato late blight 494

12.5 Pests 50412.5.1 Plant, pests and parasites 50412.5.2 Plants and aphids 508

Exercises 518

PART III ANIMALS

13 Animal Processes 522Summary 52213.1 Introduction 52213.2 Tissue and Whole-body Protein Synthesis 52313.3 Production of Volatile Fatty Acids in the Rumen 52513.4 Viability of the Fungal Population in the Rumen 52913.5 Leucine Kinetics in the Udder 53413.6 Degradation of Feed in the Rumen 539

13.6.1 In sacco system 53913.6.2 In vitro system 543

13.7 Passage of Digesta through the Gastro-intestinal Tract 547Exercises 553

14 Animal Organs 560Summary 56014.1 Introduction 56014.2 The Mammary Gland 560

14.2.1 Hormone, H 56214.2.2 Division of undifferentiated cells, Cu 56314.2.3 Production and loss of secretory cells, Cs 56314.2.4 Secretion and removal of milk, M 56514.2.5 Averaged amount of milk in animal, M 56614.2.6 Substrate, S 567

Contents xi

14.2.7 Mathematical summary 56714.2.8 Application 56814.2.9 More detailed mammary models 570

14.3 The Rumen 57014.3.1 Model description 57014.3.2 Model application 57314.3.3 Model developments 574

14.4 Other Organs 57514.5 Blood 575

14.5.1 Dual indicator method for estimating blood flow 57714.5.2 Blood flow and uptake of nutrients by the udder 579

Exercises 583

15 Whole-animal Models 593Summary 59315.1 Introduction 59315.2 The Veal Calf 593

15.2.1 Digestion 59415.2.2 Protein metabolism 59515.2.3 Energy metabolism 59615.2.4 Body ash and auxiliary variables 59615.2.5 Model application 597

15.3 The Lactating Dairy Cow 59915.3.1 Rumen module 60015.3.2 Body metabolism module 60215.3.3 Model application 60315.3.4 Other dairy cow models 604

15.4 The Pig 60515.4.1 Auspig 60515.4.2 Lactating sow model 607

Exercises 610

16 Animal Products 620Summary 62016.1 Introduction 62016.2 Milk Yield by the Dairy Cow 620

16.2.1 Gaines equation 62116.2.2 Wood equation 62216.2.3 Dijkstra equation 62716.2.4 Other lactation equations 630

16.3 Efficiency of Energy Utilization for Milk Production 63216.4 Meat Produced by the Growing Animal 635

16.4.1 Blaxter equation 63616.4.2 Allometry and body composition 640

16.5 Efficiency of Energy Utilization for Growth 64116.5.1 Intake and liveweight gain 64216.5.2 Feed consumption and liveweight 644

xii Contents

16.6 Egg Production by the Laying Hen 64516.6.1 McMillan equation 64616.6.2 Other egg production equations 649

16.7 Amino Acid Requirements of Laying Hens 65016.7.1 Hurwitz equation 65016.7.2 Reading model 651

16.8 Calcium and Phosphorus Flows in Laying Hens 65416.8.1 Model formulation 65516.8.2 Operation 66116.8.3 Results 661

16.9 Wool Growth 66316.9.1 Physiology of wool growth 66416.9.2 Research models of wool growth 66416.9.3 Simplified models of wool growth for decision

support 670Exercises 672

17 Animal Husbandry 677Summary 67717.1 Introduction 67717.2 Ration Formulation 67717.3 Allocating Pregnant Ewes to Feeding Groups 68217.4 Effects of Feeding Level on Milk Production and

Live-weight 68517.5 Pattern of Calving 68817.6 Replacement 690Exercises 691

18 Animal Diseases 694Summary 69418.1 Introduction 69418.2 Bovine Spongiform Encephalopathy 695

18.2.1 Deterministic five-pool BSE model 69518.2.2 Age-stratified BSE model 698

18.3 Rabbit Haemorrhagic Disease 70318.4 Foot and Mouth Disease 706

18.4.1 Single-farm disease dynamics 70818.4.2 Spatial disease spread 71118.4.3 Secondary infections 712

Exercises 715

Solutions to Exercises 717

Mathematical Glossary 803Bessel Functions 803Binomial and Poisson Distributions 805Coordinate Axes Systems 808

Contents xiii

Determinants, see Matrices and DeterminantsDifferentiation 810Dirac Delta Function 812Duality 813Eigenvalues 814Error Function 815Fourier Series 816F-test 818Gamma Function and Gamma Distribution 818Geometric Series 822Hôpital’s Rule 822Hyperbolas: Rectangular and Non-rectangular 823Integration by Parts 825Linear Differential Equations 825Matrices and Determinants 827Newton–Raphson Method 830Normal and Log-normal Distribution Functions 831Numerical Differentiation 835Partial Fractions 835Poisson Distribution, see Binomial and Poisson DistributionsQuadratic Forms 836Taylor Series 837t-distribution 838Vectors 839

Appendix: Constants and Conversions 841

Bibliography: Further Reading and References 843

Index 887

xiv Contents

Agriculture and related biological disciplines such as ecology are rapidlychanging. Quantitative experimentation has been the norm for manyyears, but now, increasingly, we are seeking explanation and under-standing rather than description alone. An accurate description ofsystem response, say, animal growth rate to feed intake or crop yield tofertilizer application, is undoubtedly useful, although it is even moreuseful if at the same time we understand the mechanisms or processeswhich determine the response. Links need to be made between system-level response, animal growth rate or crop yield, and the lower level ofmechanisms and processes, such as biting parameters, digestion,metabolism, mineral uptake and photosynthesis. Our ideas about howthings work must be formulated in such a way that we can makepredictions about what will happen and compare these with experimentand observation, what actually does happen. It is now widely acceptedthat mathematics is the most appropriate tool for achieving this end, andindeed, provides the only way of doing this. Mathematics permits us todo two things simultaneously: first, to describe mechanisms and theiroutcome quantitatively and, secondly, to integrate many contributingmechanisms from different subsystems of the system under study. Thesesubsystems may comprise items such as digestion, metabolism, lactation,reproductive growth, plant, soil, litter, water, etc. In the language of theepistemologist, who studies the philosophy and methodology of science,it seems that agriculture and ecology are entering a phase where the‘normal’ mode of scientific research dominates. Normal science ischaracterized by a reasonable consensus about the body of theory(working hypotheses) which may be used to explain and predict theobserved responses. Progress is a consequence of a continuingproductive interaction between experiment and hypothesis, observationand theory, always accompanied by increasing precision, generality and

Preface

explanatory power. These methods began to be applied to agriculturalproblems some 30 to 40 years ago, but there is still much to be done inboth research and applications. This work promises to change greatlyboth perceptions and practice in many areas of agriculture, ecology andenvironmental science.

This book is a textbook rather than a research monograph. Ourobjective is to facilitate the developments described above. We aim toteach students of agriculture and ecology how to express their ideasmathematically, how to solve the resulting mathematical model andcompare its predictions with experimental data. We have made our booksuitable for self-study. Problems are provided, together with outlinesolutions. The mathematical presentation is kept as simple as possible.Where mathematical difficulties might arise, we provide additionalexplanation, either in the text, in the Glossary or via Exercises/Solutions.While the level of mathematics is dictated by the problem, rather thanthe wishes of the student or the pedagogue, we choose a qualifiedsimplicity rather than a less digestible rigour. We appreciate the need ofmany students to grasp every step of a derivation. A basic knowledge ofalgebra, calculus and ordinary differential equations is sufficient for allthe material presented.

The advent of computers has meant that long and difficultmathematical analyses are not usually needed. Problem formulation, oftenrequiring creative approximation, is the first step in the process. Problemsolution may be tedious but is mostly straightforward. Errors must beavoided. It can be difficult to combine objectives such as user-friendlinesswith ease of model modification. Model evaluation, possibly involvingexercising the model and exploring its behaviour on the computer, can befun, and it is often an essential step for deciding the next move.

A few ‘big’ models are presented, in outline form, of plantecosystems and animal growth, as examples of the genre. Such modelsare commonly viewed as being ‘difficult’ or ‘complex’. This is notgenerally true. Usually, they consist of a large number of simpleelements, assembled together. However, to construct a large model takestime (to learn the relevant parts of science), diligence, patience andcontinuity of effort. The research scene currently makes it rather difficultto construct and document adequately big models, giving these models areputation they do not merit. Big models seem to provide the only wayof grappling conceptually with the broad canvas, but it should not beforgotten that the big model is composed from bits and pieces. Nospecial mathematics is needed for big or complex models, although newconcepts may emerge. These modelling bits and pieces are the principalconcern of this book. When constructing a big model, each componentmust be researched and assessed. The decision to leave something outcan be as important as what to put in and how to do it. Big models tendto have a short shelf-life, often determined by changes in our knowledgeof one or two components of the model. Their detailed exposition istherefore unsuited to a textbook, apart from space constraints.

xvi Preface

This book is a revised version of our successful 1984 book of thesame name. In the intervening years, the boundaries of traditionalagriculture have continued to blur, as wider ecological andenvironmental concerns assume greater importance. The value of modelsfor integrating the elements of the enlarging knowledge base isincreasingly recognized. These changes are reflected both in the choiceof topics and how they are treated.

The contents, inevitably, have been influenced by our experience,our competence and the time we could devote to the project. Statisticalmethods are not covered; there are many excellent textbooks coveringthis area. There are eight method-oriented chapters, the last chapterdealing with weather and the environment, followed by ten topic-oriented chapters. The first seven method-oriented chapters should beread sequentially, optionally followed by the weather/environmentchapter. The topic-oriented chapters stand more alone, at least withineach part. The lack of chapters dealing specifically with soils and watermay be regarded as significant lacunae. These two topics are brieflyintroduced in Chapters 9 and 10.

Every chapter begins with a summary. The exercises, outlinesolutions and glossary are additional resources. Our main objectiveswhile revising the 1984 text have been to broaden, up-date and simplify,concentrating on the nuts and bolts of the subject, avoiding longreference lists and accounts of very recent work which is still beingevaluated and whose staying power is uncertain. Above all, we attemptto provide students and teachers of agriculture, ecology, environmentalsciences and some areas of biology with a book which can form the basisof a course of study.

We are greatly indebted to Jeffrey Amthor, Hamlyn Jones andJonathan Newman for commenting on draft material. We also thank EricAudsley, Donald Aylor, Lee Baldwin, Roger Brugge, Les Crompton, DanDhanoa, Jan Dijkstra, William Fry, Walter Gerrits, Niklaus Grünwald,Mark Hanigan, Ian Johnson, Ermias Kebreab, Steve Leeson, SecundinoLopez, Jon Mills, Heather Neal, Paul Waggoner, Marcel van Oijen andChris Yates for help in various ways. We accept total responsibility forour text. One of the personal rewards of working in science is theinvariably generous and warm response given to requests for help.

JHMT particularly thanks the Centre for Ecology and Hydrology inEdinburgh and Professor Melvin Cannell for invaluable support,scientific and otherwise, over many years. He is also indebted to theDepartment of Zoology, University of Oxford and the Institute ofGrassland & Environmental Research, Aberystwyth for important help.

We would appreciate it if errors, comments, notable omissions andsuggestions for improvement could be brought to our notice.

J.H.M. Thornley (Henley-on-Thames, UK)J. France (Guelph, Canada)

Preface xvii

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1 Role of Mathematical Models

Summary

Agriculture and science are put into context and ‘explained’. Two simplemathematical models are presented in order to introduce modellingterminology and ideas. Hierarchy in biology is a concept crucial tounderstanding differences between various model types. Description isdifferent from, but related to, understanding or explanation. Empirical,mechanistic and teleonomic models are three important model types.Model evaluation and ‘validation’ are discussed. Objectives are criticalin a modelling project. Models for research are usually different frommodels for application. Benefits from a model can be diverse.Suggestions are made on the presentation and reviewing of models.

1.1 Agriculture and Science

Agriculture (= ager + cultura, literally the cultivation of fields) comprisesthose activities which take place mostly on farms and result in theproduction of materials for food, clothes and industrial processes. Thefarms operate within a wider ecological context, so that agriculture andecology interact.

The practice of agriculture rests on three components, described hereas tradition, (scientific) knowledge and conjecture. Tradition is inheritedknow-how: things are done in a certain way because it is known that thisworks quite well, although it may not be known why or whether doingthings a bit differently would work better. The knowledge-basedcomponent is grounded on the formal learning contained in books andpapers which gives a rationale for taking decisions and doing things in acertain way. The conjecture component of agriculture is present because

© J.H.M. Thornley and J. France 2007. Mathematical Models in Agriculture, 2nd edn (J.H.M. Thornley and J. France) 1

every situation, day and season is unique to an extent and lies a littleoutside tradition and knowledge, but of course something has to bedone.

The purpose of much agricultural research is to increase theknowledge-based component of agricultural production, at the expenseof tradition and conjecture, and thereby increase the efficiency ofagricultural production. The current efficiency of agricultural productionprovides a baseline from which, other things remaining the same, it ispossible to move only forwards. The farmer can always continue withpresent practices and present efficiency. Increased knowledge does notnecessarily lead to increased efficiency, but it may. Perhaps moreimportant is the fact that increased knowledge permits a more rationalresponse when other things do not remain the same, when theenvironment changes significantly, possibly in a way which is a threat tofuture agricultural production.

Agricultural and ecological research is concerned with adding to thepart of scientific knowledge which is relevant to the practice ofagriculture and with understanding the natural environment. Scientificknowledge is not just about observed or measured data, but includeshaving a scheme of concepts or a predictive theory which correspondswith those data. There is a continual interaction between hypotheses –how we think things work – and observed data – how they actually dowork. This interaction generates progress, as illustrated in Fig. 1.1. Astime goes on, measurements become more precise and more extensive.Our theories become more detailed and more general, althoughsometimes we realize when a useful simplification is possible. Whenconfronting theory with experiment, we attempt to connect the two at asmany points and with as much accuracy as possible. This is pertinent tothis book, because, if the measurements are numerical, then the

2 Chapter 1

Fig. 1.1. Cyclic nature of scientific enquiry: interaction of experiment and theory.

hypotheses need to be expressed numerically in order that a properconnection can be made. This can mean using mathematics forrepresenting our theories and making predictions, which can becompared with observational data. Any branch of science, as itprogresses from the qualitative towards the quantitative, is likely oneday to reach the point where using mathematics to connect experimentand theory is increasingly fruitful and indeed essential. The ideas andhypotheses being modelled are from biology. Mathematics provides thelanguage in which the theories are expressed so that quantitativepredictions which can be compared with the real world are possible.

1.2 What is a Mathematical Model?

A mathematical model is an equation or a set of equations whichrepresents the behaviour of a system. There is a correspondence betweenthe variables of the model and observable quantities. Figure 1.2illustrates a growth curve which might result from an experiment inwhich an animal is supplied with food at varying rates (F, kg dry matterday–1), and responds by growing at different rates of growth (G, kg drymatter day–1). The data points in Fig. 1.2 may be approximatelyrepresented by a mathematical equation:

Role of Mathematical Models 3

Fig. 1.2. Dependence of animal growth rate G on food intake rate F. Experimental data: ●.Equation (1.1) is represented by the continuous line. The dashed lines represent starvation (F = 0), the half-maximal response (F = K) and the asymptote (F → ∞).

(1.1)

Both F and G are variables. F is an independent variable because theexperimenter fixes F at certain values – the feeding rates he wishes tocover. Growth rate G is a dependent variable because it is not under thedirect control of the experimenter, and the value of G is a consequence ofthe level of feeding rate F which has been chosen. The three quantitiesG1, G2 and K are parameters. They take the fixed values given inequation (1.1) for the curve drawn in Fig. 1.2. Each of the threeparameters describes an easily recognizable feature of the curve. If thefeeding rate F is zero, then growth rate G = – G2; the animal is losingweight. If the feeding rate is very large, then F/(K + F) approaches unity,and growth rate G approaches G1 – G2, the asymptote. The thirdparameter, K, defines the steepness of the curve: if the feeding rate F = K,then growth rate G = G1/2 – G2; G is just half way between its minimumvalue (at F = 0, G = – G2) and its maximum value (at F is large, G = G1 –G2). A small K gives a steep curve; a large K gives a shallow curve,without changing the initial value of G (= –G2) or its maximumasymptotic value (G1 – G2). The parameters G1, G2 and K describe theasymptote, intercept and half-maximal response of equation (1.1)(Exercise 1.1).

Equation (1.1) is often referred to as a rectangular hyperbola. This isbecause it has two asymptotes which are at right angles to each other:one of these asymptotes, the straight line G = G1 – G2, is shown in Fig.1.2; this asymptote is approached as F approaches infinity. The secondasymptote (not shown) is the straight line F = –K, which is approachedas F approaches –K.

Figure 1.2 shows a classic ‘diminishing returns’ response, frequentlyencountered in many areas of biology: e.g. the response of crop yields tofertilizers, the response of the photosynthetic rates of leaves, plants andcrops to light, and the response of the rate of an enzyme-catalysedreaction to substrate concentration, known, when the intercept G2 = 0, asthe Michaelis–Menten equation (section 4.3.1).

Note that the line in Fig. 1.2 (which represents equation (1.1)) doesnot go exactly through the experimental data points. The mathematicalmodel of equation (1.1) only gives an approximate representation of thedata. Also, equation (1.1) with its three parameters has summarized,albeit approximately, some eight data points. Equation (1.1) has done nomore than redescribe and summarize the results of an experiment. It hasnot added anything new to our knowledge about the situation. Later inthis chapter, the important distinction between description andunderstanding (or explanation) and how this distinction is reflected inthe construction of a model are considered further.

G G FK F

G .

G G K

=+

−1 2

1 2–1 –1, = 0.8, 0.1 kg dry matter day , = 6 kg dry matter day .

4 Chapter 1

To draw the continuous curve in Fig. 1.2, equation (1.1) has been‘fitted’ to the data: the three parameters G1, G2 and K have been adjustedso that the curve and the data points coincide as closely as possible (seesection 2.5). If the experiment is carried out on, say, animals of differentage, species or size, and the same equation is used to fit the data in eachcase (always assuming that the same equation works acceptably well),then the values of the parameters are likely to be different for thedifferent cases. The way in which the parameters change may be of greatinterest to the researcher. The use of a descriptive equation such asequation (1.1) (which may be termed an ‘empirical’ model) can bevaluable in uncovering such effects.

A simple dynamic model

Figure 1.2 and equation (1.1) depict a static model: it does not containthe time variable t. Static models are an important class of models.However, there is another important class of models, dynamic models,which describe time-dependent behaviour. A dynamic model canrepresent the time course of events: e.g. consider the three-parameterequation containing the exponential function

(1.2)

M denotes organism dry matter; M0 is its initial value at time t = 0;Mf is the final (asymptotic) value when t → ∞; and the rate constant kdetermines the time scale of growth (a high value of k means rapidgrowth). This equation is drawn in Fig. 1.3 and, like equation (1.1) (Fig.1.2), it describes diminishing growth as the time variable t increases.This equation may apply to an organism (animal, plant, microorganism)whose growth, rather than being autocatalytic (section 5.2), is perhapslimited by a decreasing food supply (section 5.3) or a decreasing abilityto utilize an abundant food supply.

In equation (1.2), the time variable t appears explicitly on the rightside of the equation. It generally gives more insights when a dynamicmodel is written as a differential equation for the rate of change of M(dM/dt), rather than as an equation giving mass M directly. If equation(1.2) is differentiated with respect to the variable t, and t is eliminatedusing equation (1.2), then the equation

(1.3)

can be derived (Exercise 1.2). The differential equation form of a growthmodel, as in equation (1.3), is a more valuable way of presenting agrowth model than the integrated form in equation (1.2) for severalreasons. The processes causing growth are relatively explicit in the

dd

( )Mt

k M – Mf=

M M M M .

M M

fk t= +0 0

0–1

( ) (1 e )

, = 2, 10 kg dry matter, = 0.1 day .

– – –

f k

Role of Mathematical Models 5

differential equation form (e.g. see section 5.3 for a biological basis forequation (1.3)). Also, equation (1.3) can only be integrated to giveequation (1.2) if k and Mf are constant; these parameters could vary asnutrition or other environmental factors such as temperature alter.Finally, although it can be helpful to have analytical solutions, in fact,most biological models cannot be solved analytically, as in equation(1.2), but numerical solutions are needed. (Do Exercise 1.3.)

1.3 Hierarchy in Biology

Biology, including the animal and plant sciences, differs from physicsand chemistry in that there are many different levels of description ororganization. The existence of these many levels gives rise to thediversity of the biological world. For example, for the plant and animalsciences the scheme is typically as follows:

Level Description of level… … i + 1 Crop Group of animalsi Plant Animali – 1 Organs etc.i – 2 Tissues… Cells… Organelles… Macromolecules… Molecules, atoms

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Fig. 1.3. A dynamic growth model. Equation (1.2) represents dry matter M of an organism withits dependence on time t.

For agriculture, the levels of principal interest are those labelled i + 1, iand i – 1. Hierarchical systems have certain important properties:

1. Each level has its own descriptors, which belong to that level alone.For example, at the crop level, the terms crop yield and leaf area indexare meaningful, but they have little or no meaning at the lower levels.Thus, going to a higher level gives rise to emergent properties –behaviour or responses which do not exist at lower levels.2. A given level is dependent upon what is happening at the lowerlevels. The response at level i can be related to processes at lower levels.This is scientific reductionism and leads to mechanistic models.3. From level i we look to the next higher level, i + 1, for the constraints(boundary values, driving functions) which impinge on level i.4. On descending to lower levels, generally both the spatial and temporalscales become smaller, corresponding to smaller size and faster processes.

1.4 Types of Models

Models can be deterministic or stochastic, dynamic or static,mechanistic or empirical. An important subcategory of model isteleonomic, which may relate to the whole or parts of a model and toany of the above categories except the empirical category.

Deterministic models make definite predictions for quantities suchas plant dry matter or animal intake without any associated probabilitydistribution. This may be acceptable in many instances. But for rathervariable quantities or processes, such as rainfall or migration (ofdiseases, pests or predators), this may not be satisfactory.

Stochastic models include a random element as part of the model, sothat the predictions have a distribution. A problem with stochasticmodels is that they can be technically difficult to construct and hard totest or falsify. The apparent stochasticity of a system may merely reflectour ignorance about what is going on. A deterministic crop simulator, asin section 10.3, is driven by determined inputs for rainfall, wind,radiation, etc. These inputs can be given values that occur in actualweather, thus allowing unambiguous connections to be made betweenoutputs (predictions) and inputs. When investigating a system, it isusually worth building a deterministic model first, to see if this will givethe desired results. It may not be necessary to attempt what may be adifficult stochastic problem. After all our weather forecasts are mostlypredicted by large deterministic simulators, in spite of the variability ofthe weather in many locations. In this book, we only treat deterministicmodels.

Dynamic models predict how quantities vary with time, so adynamic model is generally presented as a set of ordinary differentialequations with time, t, as the independent variable, such as equation(1.3). Sometimes the model is presented as a set of difference equations,

Role of Mathematical Models 7

relating, say, the live weight of the animal on day i + 1, Wi+1, to the liveweight on day i, Wi. Thus Wi+1 = Wi + … .

Static models do not contain time as a variable and do not maketime-dependent predictions: e.g. a model predicting fruit dry matter atharvest or total animal input over the season may be a static model.Figure 1.2 and equation (1.1) are examples of a static model – the animalis not growing.

Most models in agriculture and ecology are dynamic, describing thetime course of events over periods from a few days or a growing seasonto many years.

Empirical models aim principally to describe the responses of asystem, often using mathematical or statistical equations without anyscientific content and unconstrained by any scientific principles.Depending upon one’s objectives, this may be the best kind of model toconstruct. Generally, an empirical model describes the responsesbelonging to a single level of the descriptional or organizationalhierarchy (see above). Figure 1.2 and equation (1.1) illustrate anempirical model.

Mechanistic models provide a degree of understanding orexplanation of the phenomena being modelled. To achieve this, themodel must be constructed on (at least) two levels of description (e.g. theplant and organ levels). The approach of scientific reductionism isemployed. The term ‘understanding’ implies a causal relationshipbetween the quantities and mechanisms (processes) which arerepresented on the lower level and the phenomena which are predictedat the upper level. For example, plant/crop growth rates (upper-levelphenomena) can be interpreted in terms of the operation of the processesof photosynthesis, allocation of substrates, respiration, nitrogen uptakeand transpiration (lower-level processes). A scientific explanation has tostop somewhere, and a mechanistic model is always incomplete in somerespect or other. However, a well-constructed mechanistic model istransparent and open to modification and extension, more or lesswithout limit. It is advisable not to use more than two levels in thereduction process, otherwise the model tends to become unmanageable.The processes (with their mathematical equations) which describe thelower level can be regarded as ‘empirical’ – that is, they should describereasonably accurately the process being modelled at the lower level –and the equation describing the process might or might not have sometheoretical basis from analysis at a still lower level. A mechanistic modelis based on our ideas of how the system works, what the importantelements are and how they relate to each other. As with other types ofmodels, the careful formulation of objectives is primary and determinesthe scope of the model constructed. Mechanistic models tend to be moreresearch-oriented than application-oriented, although this is changing asour mechanistic models become more reliable. Evaluation of suchmodels is essential, although it is often, and inevitably, rather subjective(see below). Mechanistic models can represent what we know

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scientifically about the system and its components; the knowledge baseis ever expanding; the models can be developed rationally and step bystep, integrating different items/areas of knowledge. A mechanisticmodel, built using a lower level of assumptions, has the potential ofproviding a view of system behaviour which, compared with anempirical model constructed on a single level using mathematical/statistical relationships unconstrained by science, is richer and offersmore possibilities.

This book is mostly concerned with mechanistic modelling, whilerecognizing that a mechanistic model comprises and depends onempirical elements, whose formulation and inter-relationships are farfrom being a trivial matter.

Teleonomic is a term which describes models (submodels) which areapparently goal-seeking: ‘apparently’ because science denies the trueexistence of goal-seeking behaviour. That is, some responses may appeargoal-seeking, but there is always an alternative and deeper view of thephenomenon which is ‘objective’ or following a free time course(Monod, 1974, p. 30). However, a teleonomic model may provide asimple and easily applicable (within a larger model) component which isof value. The teleonomic model at level i (see section 1.3 above) may bean interpretation of an otherwise empirical model. The goals of level imay be viewed as the requirements imposed by the constraints of leveli + 1. The higher-level constraints, via evolutionary pressures, select forcombinations of lower-level mechanisms of biochemistry and molecularbiology, which lead to apparently goal-directed behaviour at level i.Thus, it can be convenient and insightful to regard grazing behaviour asseeking certain goals of food acquisition or the disposition of leaves in acanopy as seeking optimum light interception. However, either of theseprocesses of food acquisition or light interception could be representedin terms of mechanisms, using ‘free’ differential equations to determinethe outcome. A teleonomic model requires four layers of contingentassumptions: first, an optimum function (goal) whose choice may appearplausible but which is often rather subjective; secondly, evolutionarypressures exist to fulfill the goal; thirdly, there are mechanisms whichcan fulfill the teleonomic goal and are accessible by adaptation; andfourthly, there has been time for the required adaptations to occur. Whena teleonomic submodel fails, as all models do eventually, there seems nological way forward other than by rather arbitrary adjustment ofhypotheses. It is important that a teleonomic model should be firmlybased on observational data, rather than on speculation on whereevolution might have led. It must be remembered that a teleonomicmodel is at best a useful approximation of limited validity. The modellermust be aware that there is an alternative to a teleonomic model: becausegoals can only be achieved by means of mechanisms and because amechanistic model permits step-by-step elaboration and refinement astheory and experiment improve, it is always possible, and sometimespreferable, to proceed mechanistically.

Role of Mathematical Models 9

Recently, it has been proposed that ‘frame-based’ models mayprovide a new way of dealing with highly complex systems (Starfield etal., 1993; Hahn et al., 1999). The intention is to facilitate theconstruction of ‘parsimonious’ systems models. The need arises becauseconventional mechanistic models of complex systems are highly time-consuming to build, difficult to parameterize and most unfriendly in use.Frames are defined to represent distinct and typical states of the system.For each frame an independent model is constructed, simulating keyprocesses for that frame. Rules are formulated for switching betweenframes. Frame-based models attempt to combine conventionalmechanistic differential equation models (section 2.3) with the cellularautomata approach (e.g. Toffoli and Margolus, 1987). For example, forrangeland in South Africa, Hahn et al. (1999) employ six frames: A,grassland with scattered mature trees; B, grassland with many bushseedlings and scattered mature trees; C, dense bush cover with less grass;D, thicket; E, degraded; and F, recently burnt with many bush seedlingsand/or resprouts. Transitions between some of these frames occur underdefined rules. Defining the rules is crucial and not straightforward.Schwinning and Parsons (1999) discuss similar problems of stability andspatial heterogeneity in grazed grassland (section 6.11.3). While it is tooearly to say whether frame-based and indeed cellular automata modelsare more than an expedient stopgap and the real work must be done bythe conventional approaches, there is little doubt that discreteassumptions of frames and states can provide valuable conceptualinsights. In terms of our categorization of models above, frame-basedmodels are dynamic, can be deterministic or stochastic and can liealmost anywhere on the empirical–mechanistic continuum.

1.5 Evaluation and Validation of Models

Evaluation is the term we use to include all methods of critiquing amodel. Evaluation is certainly not a wholly objective process, and thisfact gives many problems between authors of manuscripts and reviewersand editors. As mentioned above, a mechanistic model is alwaysincomplete and therefore usually does some things well and other thingsbadly or not at all. While an initial evaluation of a model should alwaysstart from the objectives of the modeller and includes questioning themodelling objectives, it is reasonable to progress to a wider evaluation.Model evaluation can and should proceed both at the level of predictedoutcomes (upper level) and at the level of the assumptions (lower level).Parameters should be determined by investigations at the lower level, thatis, at the level of the model’s assumptions. Unfortunately, this is notalways possible, and some ‘tuning’ or ‘calibration’ of parameters isusually needed. The wider evaluation may consider properties of themodel such as: simplicity, plausibility of assumptions, elegance,generality, applicability and the qualitative and quantitative accuracy of

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predictions. Note that some of these are properties of the model standingalone, whereas others depend on the relationship between the model andother matters: e.g. ‘applicability’ depends on the application beingconsidered.

Validation is a demonstration that a model, within a specifieddomain of application, has acceptable predictive accuracy over thatdomain (Rykiel, 1966). The validity (or otherwise) of models is still, forsome, an issue. (Predictive accuracy is discussed in section 2.5, wherefitting dynamic models to data is addressed.) The quantitative accuracyof predictions is often referred to as the ‘validity’ of the model. It isusually concerned with a model’s ability to predict results for someparticular experimental or observational scenario accurately. A model isacceptable (or ‘valid’) for certain situations (e.g. southern Britain) andnot acceptable in other situations (e.g. north-west Africa). Validity isclearly not a property of a model alone. Neither is it an all-or-nothingconcept, but may be regarded as lying in the range zero to unity.A farmer considering using a model for helping him take managementdecisions on his farm is primarily concerned with an affordable,transparent, modifiable and easy-to-use tool and acceptable predictionsfor his environment (e.g. Douthwaite, 2002). These requirements arerightly far more important to him than the quality of the sciencerepresented in the model or the elegance of the model. Attempts tovalidate formally a mechanistic model are rarely conclusive orproductive. A mechanistic model is always incomplete (see above) andtherefore, from some viewpoint, not acceptable. Using mechanisticmodels as gigantic regression equations is, arguably, misguided (Hopkinsand Leipold, 1996). However, both activities (attempts at conclusivevalidation and using mechanistic models as regression equations)continue to attract some support, although not usually from themechanistic modellers themselves.

1.6 Possible Modelling Objectives

The need for mathematical models is driven by the increasinglyquantitative nature of many biological data, the requirement forintegrating the behaviour of different parts of a complex system andunderstanding the whole, and it is facilitated by the rapid advances incomputer technology. There are many possible reasons for building amodel, just as there are several different types of model which can bebuilt. It is useful to summarize what the various motivations for amodelling project might be:

1. Models can provide a convenient data summary and be useful forinterpolation, cautious extrapolation and prediction. ‘Empirical’ modelsare often suitable for this purpose.2. Models can make good use of quantitative data, which are becomingincreasingly precise but are often rather expensive to obtain.

Role of Mathematical Models 11

3. Modelling may lead to a reduction in the amount of experimentation,because experiments can sometimes be designed to answer focussedquestions.4. We want to be able to make predictions. A model of some sort, notnecessarily mathematical, is always needed in order to makepredictions. However, it is not essential to understand how somethingworks in order to be able to predict its behaviour. Indeed, in many areasit has been common for humans to be able to make predictions longbefore an understanding is achieved.5. For problems such as the impact of climate change on ecology andagriculture, the time scale of change may be many hundreds of years.Direct experimentation is rather difficult, but a model provides a way,albeit imperfect, of exploring possible outcomes.6. We want to integrate existing knowledge about an agricultural orecological system and see where we get to.7. We want to understand how (say) the grassland system works. Thatis, curiosity is the motivating impulse. ‘Understanding’ is taken to meandefining the relationships between the responses of a system and themechanisms that are assumed to operate within the system. For thispurpose a mechanistic model is required.8. Attempting to build a model can help pinpoint areas whereknowledge and/or data are lacking, and sometimes stimulate new ideasand experimental approaches. Models can sharpen up the questions.9. A model may be used to indicate priorities for applied research anddevelopment, and also to help the farm manager to take decisions.However, only carefully tested and evaluated models are suitable forsuch purposes and, even then, much caution must be exercised.

Because, often, biologists and agronomists are wary of modelling, itmay need emphasizing that modelling and mathematics are the servantsof science, not its master. A theory does not change what the theory isabout. The hypotheses expressed in mathematics and computerprograms are derived from biological concepts. The model and computerprogram provide a framework for representing, exploring and applyingour ideas about how we think the system works. The best modelling maybe done by biologists and modellers working together. Dedicatedmodellers are needed because the requirements of modelling include abroad background in basic science, a feeling for and fluency withmathematics and the mathematical expression of scientific ideas andprinciples, and some competence with numerical analysis andcomputation. Biologists are needed because biology is what it is allabout.

When building a model, finding an appropriate blend of limitedreductionism with phenomenology and empiricism appears to be afruitful strategy. To summarize: define objectives and scope; start simplebut try to keep the model open (to modification, which will surely beneeded); when ‘simple’ fails, identify cause of failure and modify; etc.

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1.7 Models for Research and Models for Application

We have stressed the importance of objectives in relation to modellingwork where so many agendas are feasible. Focussed work is generallymore likely to be productive, as well as avoiding misunderstandings andunreal expectations. In these discussions, it can be helpful to divideresearch and development activity into three categories: basic research,applied research and development.

Basic research is simply concerned with adding to the currentknowledge base. Development is devising a solution to a practicalproblem of technology whose solution is reckoned to be possible usingcurrent knowledge, perhaps with some quite minor additions. Appliedresearch is where work is directed towards solving a practicaltechnological problem, but it is thought that an appreciable increment tothe knowledge base will be required before the problem can be solved.

Contrary to a widespread view, basic research is in some ways aneasier activity than applied research. This is because the basic researchercan choose his problem. He chooses a problem he thinks is soluble,given his competence and situation. No credit it obtained for attemptingan insoluble problem. This perhaps explains the tendency of researchersto gravitate towards ‘fundamental’ problems. Because the basicresearcher chooses his own problems, if he consistently fails to achieveresults, then he has some explaining to do, and perhaps is not wellsuited to the demands of basic research. Also a research model can behighly speculative, and it is permissible for it to fail. Indeed, its failuremay well be very instructive and lead to new insights.

In applied research and development, the problem is given from‘outside’: there is some perceived need or deficiency. There may be aneed to spend funds on some high-profile objective. Often scantconsideration may be given as to whether, given current knowledge, theproblem is soluble in a meaningful sense. Research aimed at buildingmodels is not the same as research aimed at building some device orfinding a cure for a disease. A model can always be constructed. Wherethere are areas of ignorance or difficulty, they are bridged withassumptions. Modellers rarely fail to get their model built. If the modelis of little value, it may well be because too much effort is directedtowards stitching together bits of the existing knowledge base (which istoo small in this instance), rather than extending the knowledge base.

The agricultural/ecological modeller is in an interesting position inseveral respects. Most modelling work has a practical objective andcould be classified as applied research or development. Typicalobjectives might be concerned with animal feeding systems, cropmanagement or the impact of climate change on various ecosystems.Very occasionally, enough is known for the modeller to solve theproblem, i.e. build a model, of what might be called ‘engineering’strength – that is, every component is tried and tested. More usually,several items are imperfectly or even almost completely unknown.

Role of Mathematical Models 13

However, the modeller is undeterred, even encouraged, by suchdifficulties. He makes assumptions, which might be conservative,plausible or quite speculative. This is necessary in order that he cancomplete the modelling project which he has undertaken. Although it ispossible in principle to test his assumptions experimentally, suchmeasurements are often impossible or too difficult, which is why thestate of ignorance exists in the first place. The model can thus only betested at the level of its predictions at the highest hierarchical level inthe model (section 1.3). Testing at the highest level is the bluntest ofinstruments and is rarely conclusive. The farmer and the agriculturaladviser, being practical men, who are happily and rightly littleconcerned with science content or elegance, will simply measure whatthe model tells them against how they currently do the job and theirusually good common sense. If they are satisfied that the model givesinformation which enables them, overall, to do a better job, then theywill take the model on board. This approach is sometimes referred to asthe ‘champion-challenger’ concept, the idea being that a worthychampion model will emerge, propelled by Adam Smith’s invisiblehand. In practice, market forces do not operate very well; environmentsare variable and cannot be fully represented in the model; the model isnecessarily of a restricted size and items (important for some) areomitted; the data with which the predictions are compared are rathervariable; there are too many models to be considered, given theresources available; the models are continually changing while thecomparison process is underway. Model comparisons are popular withsome, although generally, to date, they have been expensive andinconclusive and have failed to add to the knowledge base.

Researchers are sometimes asked to estimate costs and benefits tojustify their work and requests for cash. For basic research, the costs areinfinitely elastic; we spend what we choose to spend; the benefits cannotbe quantified because the outcomes of basic research cannot bepredicted. All that can be said is that, historically, basic research hasbeen of enormous value, and it is often the unexpected contributionswhich make the most impact. For development work, both costs andbenefits can be evaluated, although errors, sometimes substantial, are thenorm rather than otherwise. For applied research, the benefits can beestimated, but the costs are more uncertain because there is doubt if theresearch component can be successfully completed within a givenbudget.

1.8 Models: Documentation, Presentation and Reviewing

The progress of science depends on communication. Publication is thetraditional and still the principal means by which scientists put theirwork before the scientific community, which is then free to applaud,ignore, refute, modify and make use of the contribution. Science has

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experienced many instances of attempts by the scientific ‘establishment’to suppress innovative contributions. Fortunately the fragmented andpluralistic nature of science eventually thwarts such efforts. The adventof the web is bringing about changes and, increasingly, material, archiveand otherwise, is made available on the web.

Documentation of small models is straightforward using traditionalpublication and the mathematical definition of the model. Large models,perhaps of ecosystems or biochemically based animal models, are anunsolved problem. Traditional publication is largely unsuitable,although some new journals are more catholic. Some modellers arguethat the computer program is the best documentation, since this worksand is, so far as it goes, complete. Others take the position that computerprograms are rather indigestible, programming languages change quickly,we all have our favourite (and sometimes idiosyncratic) programminglanguage, and, therefore, customary mathematical notations are preferredas they are sufficient to provide a relatively timeless and succinctstatement of the model. Examples of both approaches are found inThornley (1998) and Müller (1999).

Many modellers in agriculture, ecology and some parts of biologyexperience problems in attempting to publish modelling papers. Theproblems are arguably worse for modellers than for biologists pursuingmore traditional approaches. As one moves ‘down’ the organizationalhierarchy towards biochemistry and molecular biology, chemistry andphysics, models are generally better accepted as an important part of theresearch scene. No-one knows where science (or life) is going next, andtherefore it is best to proceed according to rather general guidingprinciples and treat each problem/opportunity on its merits. However,some journals like to have a detailed policy to decide what is and whatis not acceptable to the journal. It is always, of course, difficult for anyjournal to strike the right balance between material which is not worthpublishing (for such undoubtedly exists), mainstream material andmaterial which nurtures the growing points of the subject (Lock, 1986).The latter is always a minority interest, few may be competent toevaluate it and it may seem to be of little value to many.

It is not always appreciated that numerous activities can contributeto science: observation, experiment, speculation, hypothesis, analysis,integration and deduction (possibly using a mathematical model). Thereare now many examples where theoretical work has thrown up ideaswhich, although they may have remained untested for some years, havebeen enormously stimulating to the subject.

The editors and reviewers of some biological journals sometimesdemand of modelling papers the satisfaction of criteria which are seldomapplied to non-modelling papers, as well as requiring the fulfilling ofobjectives of their own choosing, rather than the author’s objectives indoing the work. This may cause frustration when neither the editors northe reviewers have had direct hands-on experience of modelling, withthe perspective which this would bring. For instance, an extensive

Role of Mathematical Models 15

‘validation’ may be demanded (section 1.5). This might ignore the factthat models of a given system may be constructed for different purposes(section 1.6) and that fitting the model closely to data might not be asignificant objective. More importantly, while accepting the importanceof comparing theory and experiment, such a demand may ignore: (i) thedifficulties of finding suitable data; (ii) the operational difficulties ofcomparing possibly complex mechanistic models with data; and (iii) theinconclusiveness of such procedures even when carried out. Theobjectives stated by the author should be examined closely and assessedas to their legitimacy and to the extent to which they are subsequentlymet.

First we suggest that the initial position of a reviewer should be:excellent; another contribution to science; how can it be improved?

Then we propose that authors, editors and reviewers should beconcerned principally with five items. These are:

● clarity;● economy;● methodological correctness;● not a trivial repetition of already published work;● accuracy.

Judgements which go beyond these five criteria may be assuming anauthority which belongs properly to the scientific community as a whole.

Clarity is needed if the work is to be understood, at least by a fewwho may communicate it further. Readability and clarity can generallybe assessed by researchers in the same general area. A specialist is notnecessarily required. Indeed, specialists may be so concerned with whatis being said that they find it difficult to consider how it is being said.

Economy is required simply because journal space is expensive.Clarity and economy usually, but not always, go together.

Methodological correctness is usually easy to assess. In many areasof mathematics and experiment, there is wide acceptance of basicmethodology. Here the work should be free from error. For themathematical modeller, his use of algebra, analysis, calculus andnumerical methods should be free from error.

Item four – not a trivial repetition of already published work – isself-explanatory. All work stands on what has gone before and, forcontinuity, context and comprehensibility, some repetition is needed.However, there must be some important aspects of the new work whichare different from previous work. This could be labelled as ‘originality’.For instance, a synthesis of existing concepts may not seem to beoriginal, although it could lead to novel insights. Modelling is oftenabout the integration of concepts, the simulation of complexity – thewhole being more than the sum of the parts, although explainable interms of the parts and how they fit together.

Accuracy is de rigueur. As reviewers, we are often surprised thatsome authors do not realize that, if they allow minor errors of typing,

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referencing, style and layout to appear frequently in their submittedmanuscript, this casts doubts on the correctness of the work, especiallythose parts which are not easily checked. Such errors may greatly lessenthe credibility of a submitted manuscript.

Above all, it should be remembered that models come in all shapesand sizes and with widely differing objectives. Many different types ofcontribution can play a role in furthering science. The scientificcommunity has its own often inscrutable but usually effective methodsof separating wheat from chaff. There is no way of eliminating chaffalone from the scientific literature. Arguably, open reviewing would helpthe process.

1.9 Units

It might be thought that now, with wide acceptance of the internationalsystem of units (SI) (Royal Society, 1975), units were no longer an issue.Regrettably, for the agricultural modeller, units can still present aproblem.

This is partly due to scaling: models may try to relate biochemistryto ecosystem, laboratory bench to whole farm. Grams and centimetresmay need to be translated into kilograms, tonnes and hectares. Seconds,the preferred SI unit of time, may be far less appropriate than the naturaltime units of day and year. Conversion is, of course, straightforward, butoften tedious and, most importantly, gives much scope for errors.

Those who set up SI did posterity a disservice when they failed topropose a new name for the base mass unit of SI, the kilogram. This wascompounded when they ‘screwed up’ (there is no other word for it) thedefinition of the mole, the SI unit for amount of substance. For instance,the most common isotope of carbon has a dimensionless relativemolecular mass (‘molecular weight’) of 12. The SI mole is defined so that1 mole of carbon-12 has a mass of 0.012 kg, rather than 12 kg, whichwould be preferable because it uses the base mass unit of SI rather than10�3 × the base unit (Thornley and Johnson, 1991). This can result in itbeing necessary to insert conversion factors of 1000 in the equations of amodel [e.g. equation (9.186)]. The precursor of SI, metre-kilogram-second (mks), was set up to avoid conversion factors which at one timelittered some of the equations of physics and caused confusion anderrors. An ideal system of units would enable the equations of science,from the neutrino to the cosmos, to be written without arbitraryconversion factors. Perhaps a revision of SI will put things right.

There are also areas where, quite reasonably, scientists have decidedto differ, while still remaining within SI. Examples are water potential(J kg–1 or Pa), stomatal conductance (m s–1 or mol m–2 s–1) and radiation(J m–2 or mol m–2). These potential points of confusion are dealt with asthey arise in the text. Some conversion factors are given in theAppendix.

Role of Mathematical Models 17

The golden rule for the modeller is to write the core model in asingle consistent set of units. It is probably best to stay as close to SI aspossible. Then only convert to more convenient units for outputpurposes but always outside the core model.

Exercises

Exercise 1.1

The first term on the right side of equation (1.1) is a rectangularhyperbola. It can be generally written as y = ax/(b + x), where x is theindependent variable, y is the dependent variable, and a and b areconstants. A hyperbola is an example of what mathematicians call aconic section, defined by the intersection of a plane and a cone. Otherconics are circles, ellipses, parabolas and straight lines. It is rectangularbecause the asymptotes are at right angles. Derive an equation for x interms of y. Satisfy yourself that the two straight-line asymptotes are: x =–b, y is infinite; y = a, x is infinite. Derive an expression for the slopedy/dx. Show that the initial slope (dy/dx at x = 0) is a/b. Derive anequation for the rate of change of slope d2y/dx2. Show that, for x � 0, theslope, dy/dx, is maximum at x = 0.

Exercise 1.2

Fill in the missing steps between equations (1.2) and (1.3). Hint:differentiate both sides of equation (1.2) with respect to t to give anequation dM/dt = … . Rearrange equation (1.2) in the form e–kt = … .Substitute for e–kt in the dM/dt = … equation.

Exercise 1.3

During a growth experiment, mass M is measured at various values oftime t. The data are fitted well by the equation M = at/(b + t), where aand b are constants. Differentiate with respect to time t to obtain agrowth equation of the form dM/dt = g(t), where g(t) denotes somefunction of time t. From the equation M = at/(b + t), derive an equationfor t in terms of M, and use this equation to substitute for t in the dM/dt= … equation.

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