Springer Undergraduate Mathematics Series

Advisory Board

P.J.Cameron Queen Maryand Westfield CollegeM.A.J. Chaplain University ofDundeeK. Erdmann Oxford UniversityL.c.G. Rogers University ofCambridgeE.Siili Oxford UniversityJ.F.Toland University ofBath

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Nicholas F. Britton

Essential Mathematical Biology

With 92 Figures

Springer

Nicholas Ferris Britton, MA, DPhil Centre for Mathematical Biology, Department of Mathematical Sciences, UniversityofBath, Claverton Down, Bath BA2 7AY, UK

Cover iUwtration elements rtprodueed by kind permWion of: Apte<h Systems, Inc., Pub\ishers of the GAUSS Mathematica! and Statistica! System, 23804 s.s. Kent-Kangley Road. Maple Valley, WA 98038,

USA. Tel: (206) 432 - 7855 Fax (206) 432 - 7832 email: info(o!>aptech.com URI.: www.aptech.com American StatisticaI AS5OCÎation: Olan,e Voi 8 No 1, 1995 artide by KS an<! KW Heiner "free Rings ofthe Northem Shawangunks' page 32 fig 2 Springer-V~ Mathematica in Education and Research Voi 4 Issue 3 1995 artide by Roman E Maeder, Beatrice Amrhein an<! Oliver GIoor

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Mathematica in Education and Research Voi 4 Issue 3 1995 artide by Richard I Gaylord and Kazume Ni.shidate "fratii, Engineering with Cellular Automata' page 35 fig 2. Mathematica in Education an<! Research Voi 5 Issue 2 1996 artide by Michael Trolt 'The Implicitization of a Trefoil Knot' page 14-

Mathematica in Education and Research Voi 5 Issue 2 1996 article by!.ee de Cola 'Coins, Trees, Bars an<! BelIs: Simulation ofth. 8inomial!'ro­ces/i' page 19 fig 3. Mathematica in Education and Research Voi 5 I&sue 2 1996 artide by Richard Gaylord and Kazume Ni.shidate 'Contagious Spreading' page 33 fig 1. Mathematica in Education and Research Voi 5 Issue 2 1996 article by)o. Bubler and Stan Wagon 'Secrets of the Madelung Constant' page SO fig 1.

British Library Cataloguing in Publication Data Britton, N.F.

Essential mathematical biology (Springer undergraduate mathematics series) 1. Biomathematics. 1. Title 570.1'51

Library of Congress Cataloging-in-Publication Data Britton, N.F.

Essential mathematical biology / N.F. Britton p. cm. -- (Springer undergraduate mathematics series)

Indudes index.

1. Biomathematics. 1. Title. II. Series. QH323.5 .B745 2002 570'.1'51-dc21 2002036455

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Springer Undergraduate Mathematics Series ISSN 1615-2085 ISBN 978-1-85233-536-6 ISBN 978-1-4471-0049-2 (eBook)

~ Springer-Verlag London 2003 Origina1ly published by Springer-Verlag London Berlin Heidelberg in 2003

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DOI 10.1007/978-1-4471-0049-2

Preface

This book aims to cover the topics normally included in a first course on Math­ematical Biology for mathematics or joint honours undergraduates in the UK,the USA and other countries. Such a course may be given in th e second yearof an undergraduate degree programme, but more often appears in the thirdyear.

Mathematical Biology is not as hierarchical as many areas of Mathematics,and therefore there is some flexibility over what is included. As a result thebook contains more than enough for two one-semester courses , e.g. one basedon Chapters 1 to 4, mainly using difference equations and ordinary differentialequations, and one based on Chapters 5 to 8, mainly using partial differentialequations. However there are some classic areas that are covered in almost everycourse , th e most obvious being population biology, often including epidemiol­ogy, and mathematical ecology of one or two species. Population genetics is alsoa classic area of appli cation, although it does not appear in every first course . Inspatially non-uniform systems there is even more choice, but reaction-diffusionequations are almost always included , with applications at the molecular andpopulation level. Some large areas have been excluded through lack of space,such as bio-fluid dynamics and most of mathematical physiology, each of whichcould fill an undergraduate textbook on its own.

To cover the whole book the student will need a background in linear alge­bra, vector calculus, difference equations, and ordinary and partial differentialequations, although a one-semester course without vector calculus and partialdifferential equations could easily be constructed. Methods only are required,and the necessary results are collected together in appendices. Some additionalmaterial appears on an associated website, at

http ://www.springeronline.com/1-85233-536-X

v

VI Essential Mathematical Biology

This site includ es more exercises, more detailed answers to the exercises in thebook, and links to ot her useful sites. Some par ts of it requir e more mathemat icalbackground than the book itself, including stochastic processes and continuummechanics.

I would like to th ank Jim Murr ay for his support and advice since myundergraduate days. I am gra teful to all the (academic and non-academic)staff of the Department of Mat hematical Sciences at the University of Bath formaking this a genial place to work , the past and present members of the Centrefor Mathematical Biology at the University of Bath for st imulating discussions,my st udents for their feedback on th e lecture notes from which this book wasdeveloped, and Springer Verlag. Finally I would like to thank Suzy and Rachelfor their love and support .

Contents

List of Figures xiii

1. Single Species Population Dynamics . . . . . . . . . . . . . . . . . . . . . . .. 11.1 Introd uction . . . . . . .. . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . . . . 11.2 Linear and Nonlinear Fir st Order Discrete Time Models . . . . . . . 2

1.2.1 The Biology of Insect Populat ion Dynamics . . . . . . . . . . . . 31.2.2 A Model for Insect Population Dynamics with Competition 4

1.3 Differential Equ ation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Evolutionary Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Harvestin g and Fisheries 171.6 Metapopulat ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.7 Delay Effects 241.8 Fibonacci's Rabbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 271.9 Leslie Matrices: Age-structured Populations in Discrete Tim e .. 301.10 Euler-Lotka Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34

1.10.1 Discrete Time 341.10.2 Continuous Time 38

1.11 The McKendrick Appro ach to Age Structure 411.12 Conclusions 44

2. Population Dynamics of Interacting Species 472.1 Introduction . . . . ... . . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . .. . . .. 482.2 Host-p arasitoid Int eractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 482.3 The Lotka-Volterr a Prey-pr edator Equations 542.4 Modelling the Predator Functional Respons e . . . . . . . . . . . . . . . .. 602.5 Competi tion . . . . . . . . . . . . . . .. . .. . . . . . .. . . ... . .. . . . . . . . . . . . 66

VII

VIII Contents

2.6 Ecosystems Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.7 Interacting Metapopulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74

2.7.1 Competition 752.7.2 Pr edation. . . .. . . . . . . . . . . . .. . .. . . . . . . . . . . . . ... . . . . . 772.7.3 Predator-mediated Coexist ence of Competitors. . . . . . . . . 782.7.4 Effects of Habitat Destruction 79

2.8 Conclusions . . . . . . . . . . . ... . .. . . .. . . ..... . ... . . . . . . .... . .. . 81

3 . Infectious Diseases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.1 Introduction 833.2 The Simple Epidemic and SIS Diseases . . . . . . . . . . . . . . . . . . . . . . 863.3 SIR Epidemics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 903.4 SIR Endemics 96

3.4.1 No Disease-related Death . . . . . . . . . . . . . . . . . . . . . . . . . . .. 973.4.2 Including Disease-related Death . . . . . . . . . . . . . . . . . . . . .. 99

3.5 Eradication and Control 1003.6 Age-structured Popul ations 103

3.6.1 The Equations 1033.6.2 Steady State 105

3.7 Vector -borne Diseases 1073.8 Basic Model for Macroparasitic Diseases 1093.9 Evolutionary Aspects 1133.10 Conclusions 115

4. Population Genetics and Evolution 1174.1 Introduction 1174.2 Mendelian Genetics in Popul at ions with Non-overlapping Gen-

erations 1194.3 Selection Pressure 1234.4 Selection in Some Special Cases 127

4.4.1 Selection for a Dominant Allele 1274.4.2 Selection for a Recessive Allele 1274.4.3 Selection against Dominant and Recessive Alleles 1294.4.4 The Additive Case 129

4.5 Analyt ical Approach for Weak Selection 1304.6 The Balance Between Selection and Mutation 1314.7 Wright 's Adaptive Topography 1334.8 Evolution of the Genetic System 1344.9 Game Theory 1364.10 Replicator Dynamics 1424.11 Conclusions 145

Contents IX

5. Biological Motion 1475.1 Introduction 1475.2 Macroscopic Theory of Motion; A Continuum Approach 148

5.2.1 General Deriva tion 1485.2.2 Some Particular Cases 151

5.3 Directed Motion, or Taxis 1545.4 Steady State Equations and Transi t Times 156

5.4.1 Steady State Equ ations in One Spatial Variab le 1565.4.2 Transit Times 1575.4.3 Macrophages vs Bacteria 160

5.5 Biological Invasions: A Model for Muskr at Dispersal 1605.6 Travelling Wave Solutions of General Reaction-diffusion Equ a-

tions 1645.6.1 Node-saddle Orbits (the Monostable Equ ation) 1665.6.2 Saddle-saddle Orbits (the Bist ab le Equation) 167

5.7 Travelling Wave Solutions of Systems of Reaction-diffusionEquations: Spatial Spread of Epidemics 168

5.8 Conclusions 172

6 . Molecular a nd Cellular Biology 1756.1 Introduct ion 1756.2 Biochemical Kinetics 1766.3 Metabo lic Pathways 183

6.3.1 Activation and Inhibit ion 1846.3.2 Coop erative Phenomena 186

6.4 Neural Modelling 1916.5 Immunology and AIDS 1976.6 Conclusions 202

7. P atter n Form at ion 2057.1 Introduction 2057.2 Turing Instabi lity 2067.3 Turing Bifurcations 2117.4 Activator-inhibitor Systems 214

7.4.1 Conditions for Turing Instability 2147.4.2 Short-range Activation, Long-range Inhibition 2197.4.3 Do Activator-inhibitor Systems Exp lain Biological Pat-

tern Formation? 2237.5 Bifurcations with Domain Size 2247.6 Incorporating Biological Movement 2297.7 Mechanochemical Models 2337.8 Conclusions 233

x Contents

8. Tumour Modelling 2358.1 Introduction 2358.2 Phenomenological Models 2378.3 Nutrients: the Diffusion-limited Stage 2408.4 Moving Boundary Problems 2428.5 Growth Promoters and Inhibitors 2458.6 Vascularis ation 2478.7 Met astasis 2488.8 Immune System Response 2498.9 Conclusions 251

Further Reading 253

A. Some Techniques for Difference Equations 257A.1 First-order Equations 257

A.l.l Graphi cal Analysis 257A.1.2 Linear isation 258

A.2 Bifurcations and Chaos for First-order Equations 260A.2.1 Saddl e-node Bifurcations 260A.2.2 Transcritical Bifurcations 261A.2.3 Pitchfork Bifurcations 262A.2.4 Period-doubling or Flip Bifurcations 263

A.3 Systems of Linear Equations : Jury Conditions 266A.4 Systems of Nonlinear Difference Equ ations 267

A.4.1 Linearisation of Systems 268A.4.2 Bifurcation for Systems 268

B . Some Techniques for Ordinary Differential Equations 271B.1 First-order Ordinary Differenti al Equations 271

B.1.1 Geometric Analysis 271B.1.2 Int egration 272B.1.3 Linear isation 272

B.2 Second-order Ordinary Differential Equations 273B.2.1 Geometri c Analysis (Phase Plane) 273B.2.2 Linearisation 274B.2.3 Poincar e-Bendixson Theory 276

B.3 Some Results and Techniqu es for mth Order Systems 277B.3.1 Linearis ation 278B.3.2 Lyapunov Functions 278B.3.3 Some Miscellaneous Facts 279

B.4 Bifurcation Theory for Ordinary Differenti al Equ ations 279B.4.1 Bifurcations with Eigenvalue Zero 279

Contents XI

B.4.2 Hopf Bifurcations 280

C. Some Techniques for Partial Differential Equations 283C.1 First-order Partial Differential Equations and Characteristics .. 283C.2 Some Results and Techniques for the Diffusion Equation 284

C.2.l Th e Fundamental Solution 284C.2.2 Connection with Probabilities 287C.2.3 Other Coordinate Systems 288

C.3 Some Spectral Theory for Laplace's Equation 289C.4 Separation of Variables in Partial Differential Equations 291C.5 Systems of Diffusion Equations with Linear Kinetics 295C.6 Separating the Spatial Variables from Each Other 297

D. Non-negative Matrices 299D.1 Perron-Frobenius Theory 299

E. Hints for Exercises 301

Index 329

List of Figures

1.1 World human populat ion growth over the last 2000 years . . . . . . . . . . 21.2 Cobweb maps for idealised contest and scramble competit ion. . . . . .. 61.3 Cobweb maps for the Hassell equation 71.4 Parameter space for the Hassell equation . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Cobwebbing variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Logistic population growth 141.7 Yield-effort curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.8 The archetypal met apopulation model 221.9 The archetypal metapopulation growth rat e " 231.10 Inst abil ity in a delay equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 261.11 Fibonacci's rabbi ts 321.12 A Lexis diagram 411.13 Archetypal survivorship curves 43

2.1 Th e functions log Ro and 1 - 1/Ro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.2 Some numerical solut ions of the Nicholson-Bailey host-parasitoid

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 512.3 Negative binomial and Poisson dist ributions 532.4 Results of host -parasitoid model with negative binomial search func-

tion " 532.5 Winter moth host -parasito id simulations and dat a " 542.6 Results of Lotka-Volterr a prey-predator model. . . . . . . . . . . . . . . . . . . 572.7 Hare-lynx population data 582.8 Predator function al response curves " 612.9 Rosenzweig-MacArthur phas e planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.10 Lotka -Volterr a competition phase planes " 68

XIII

XIV List of Figures

2.11 Parameter space for Lotka-Volterr a competition equations 682.12 Numerical solut ion of a plank toni c ecosystem model 732.13 Huffaker 's prey-predator experiments 752.14 State transitions for the metapopulation model with two competitors 752.15 Numerical results for a metapopulation model with competit ion . . .. 762.16 State t ra nsitions for pat ches in a metapopulation model for predation 772.17 Numerical result s for a metapopulation model of a predator-prey

interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.18 Stat e transitions for a metap opulation model with two competing

prey and one predator 792.19 Numerical results for a metapopulation model of predator-mediat ed

coexistence , 80

3.1 Progress of a disease within an individual. , 853.2 Phase planes for th e SIR epidemic. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 913.3 The final size of an SIR epidemic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 933.4 Bombay plague of 1906 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 943.5 Numerical solutions for an SIR endemic. . . . . . . . . . . . . . . . . . . . . . . .. 983.6 A typical t ime series for measles 1003.7 The basic reproductive rati o Ro as a function of the virulence c 114

4.1 Hardy-Weinberg genotype frequencies 1214.2 Diagram of a typical life cycle 1234.3 Numerical solut ions for weak and st rong select ion aga inst a domi-

nant and a recessive allele 1284.4 Central dogma of modern evolut ionary biology 1324.5 Wright 's adapt ive topogra phy 1354.6 Invasibility plots 1404.7 The evolut ionary traj ectori es for t he rock-scissors-paper game 144

5.1 Flux in a genera l direction 1495.2 Conservation in one dimension 1525.3 Data for the invasion of muskrat s in Europe 1615.4 Reaction-diffu sion invasion wave 1625.5 Phase plane for the node-saddl e orbit 1665.6 The Black Death in Europe, 1347-1350 1695.7 Diagramm atic representation of a prop agating epidemic 1695.8 Proof of a tr avelling wave in th e rabies model 171

6.1 Michaelis- Menten reaction dynamics 1786.2 Michaelis-Menten and coopera t ive reaction velocity curves 1866.3 FitzHugh- Nagumo phase plane 194

List of Figures xv

6.4 A numerical solut ion of the excitatory FitzHugh- Nagumo equations 1956.5 A numerical solution of the oscillato ry FitzHugh- Nagumo equations 196

7.1 The early stages of development of a vertebrate egg 2067.2 Turing dispersion relations 2127.3 Spatial eigenvalues for Turing instability 2167.4 Marginal Turing inst ability in diffusion-coefficient space 2197.5 Pure act ivat ion-inhibition and cross-activat ion-inhibit ion 2207.6 Tur ing bifurcations with domain size 2257.7 Numerical solutio ns and real tail pat terns 2277.8 Examples of biological patterns 2297.9 Tur ing bifurcat ion with chemotaxis 231

8.1 Tumour growth models 2388.2 The profile of the t umour growth inhibitor 246

A.1 Some examples of cobweb maps 259A.2 Behaviour of solutio ns of a first-order linear difference equation 260A.3 Saddle-node bifurcations 261A.4 A transcritical bifurcation 262A.5 A pit chfork bifurcat ion 263A.6 Period-doubling bifurcations and the cascade to chaos 265A.7 Naimark - Sacker bifurcat ion 268

B.1 Geometrical method for first-order ordinary differenti al equations 272B.2 An example of a phase plane 274B.3 Properties of the roots of a quadr ati c 276B.4 A second example of a phase plane 277B.5 Sub- and supercr itical Hopf bifurcations 281

E.1 Cobweb maps for the sterile insect cont rol equation 302E.2 Leslie phase plane and time series 306E.3 Phase planes for an ecosystem with photosynthesis 307E.4 Leslie phase plane and time series 308E.5 Population genet ics of sickle cell anaemia 314E.6 Evolu tionary dynamics for the rock-scissors-paper game with reward 317E.7 Turing inst ability in Gierer- Meinhard t par ameter space 323E.8 Mimura- Murray phase plane for ecological pat chiness 324E.9 Bifurcation diagram for tumour growth inhib itor 327E.lO Bifurcation diagram for tumour immune response 328