Differential Equations and Mathematical Biology

Differential Equations and Mathematical Biology

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The conjoining of mathematics and biology has brought about significant advances in both areas, with mathematics providing a tool for modelling and understanding biological phenomena and biology stimulating developments in the theory of nonlinear differential equations. The continued application of mathematics to biology holds great promise and in fact may be the applied mathematics of the 21st century. "Differential Equations and Mathematical Biology" provides a detailed treatment of both ordinary and partial differential equations, techniques for their solution, and their use in a variety of biological applications.The presentation includes the fundamental techniques of nonlinear differential equations, bifurcation theory, and the impact of chaos on discrete time biological modelling. The authors provide generous coverage of numerical techniques and address a range of important applications, including heart physiology, nerve pulse transmission, chemical reactions, tumour growth, and epidemics. This book is the ideal vehicle for introducing the challenges of biology to mathematicians and likewise delivering key mathematical tools to biologists. Carefully designed for such multiple purposes, it serves equally well as a professional reference and as a text for coursework in differential equations, in biological modelling, or in differential equation models of biology for life science students.show more

Product details

  • Hardback | 408 pages
  • 152.4 x 236.22 x 27.94mm | 748.42g
  • Taylor & Francis Inc
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • New edition
  • New edition
  • 84 black & white illustrations, 3 black & white tables
  • 1584882964
  • 9781584882961
  • 2,427,468

Table of contents

INTRODUCTIONPopulation GrowthAdministration of DrugsCell DivisionDifferential Equations with Separable VariablesGeneral PropertiesEquations of Homogeneous TypeLinear Differential Equations of the First OrderLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTSIntroductionFirst-Order Linear Differential EquationsLinear Equations of the Second OrderFinding the Complementary FunctionDetermining a Particular IntegralForced OscillationsDifferential Equations of the Order nUniquenessAppendix: Existence TheorySIMULTANEOUS EQUATIONS WITH CONSTANT COEFFICIENTSSimultaneous Equations of the First OrderReplacement of One Differential Equation by a SystemThe General SystemThe Fundamental SystemMatrix NotationInitial and Boundary Value ProblemsSolving the Inhomogeneous Differential EquationAppendix: Symbolic ComputationMODELLING BIOLOGICAL PHENOMENAIntroductionHeart BeatBlood FlowNerve Impulse TransmissionChemical ReactionsPredator-Prey ModelsFIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONSExistence and UniquenessEpidemicsThe Phase PlaneLocal StabilityStabilityLimit CyclesForced OscillationsAppendix: Existence TheoryAppendix: Computing TrajectoriesMATHEMATICS OF HEART PHYSIOLOGYThe Local ModelThe Threshold EffectPhase Plane Analysis and the Heart Beat ModelPhysiological Considerations of the Heart Beat CycleA Model of the Cardiac PacemakerMATHEMATICS OF NERVE IMPULSE TRANSMISSIONExcitability and Repetitive FiringTravelling WavesQualitative Behaviour of Travelling WavesCHEMICAL REACTIONSWavefronts for the Belousov-Zhabotinskii ReactionPhase Plane Analysis of Fisher's EquationQualitative Behaviour in the General CasePREDATOR AND PREYCatching FishThe Effect of FishingThe Volterra-Lotka ModelPARTIAL DIFFERENTIAL EQUATIONSCharacteristics for Equations of the First OrderAnother View of CharacteristicsLinear Partial Differential Equations of the Second OrderElliptic Partial Differential EquationsParabolic Partial Differential EquationsHyperbolic Partial Differential EquationsThe Wave EquationTypical Problems for the Hyperbolic EquationThe Euler-Darboux EquationEVOLUTIONARY EQUATIONSThe Heat EquationSeparation of VariablesSimple Evolutionary EquationsComparison TheoremsPROBLEMS OF DIFFUSIONDiffusion through MembranesEnergy and Energy EstimatesGlobal Behaviour of Nerve Impulse TransmissionsGlobal Behaviour in Chemical ReactionsTuring Diffusion Driven Instability and Pattern FormationFinite Pattern Forming DomainsBIFURCATION AND CHAOSBifurcationBifurcation of a Limit CycleDiscrete BifurcationChaosStabilityThe Poincar? PlaneAveragingAppendix: ProgramsGROWTH OF TUMOURSIntroductionA Mathematical Model of Tumour GrowthA Spherical TumourStabilityEPIDEMICSThe Kermack-McKendrick ModelVaccinationAn Incubation ModelSpreading in SpaceANSWERS TO EXERCISESINDshow more

Review quote

"A strength of [this book] is its concise coverage of a broad range of topics. It is truly remarkable how much material is squeezed into the slim book's 400 pages." - SIAM Review, Vol. 46, No. 1 "It is remarkable that without the classical scheme (definition, theorem and proof) it is possible to explain rather deep results like properties of the Fitz-Hugh-Nagumo model or the Turing model . This feature makes the reading of this text pleasant business for mathematicians. [This book] can be recommended for students of mathematics who like to see applications, because it introduces them to problems on how to model processes in biology, and also for theoretical oriented students of biology, because it presents constructions of mathematical models and the steps needed for their investigations in a clear way and without references to other books." - EMS Newsletter "The title precisely reflects the contents of the book, a valuable addition to the growing literature in mathematical biology from a deterministic modeling approach. This book is a suitable textbook for multiple purposes Overall, topics are carefully chosen and well balanced. The book is written by experts in the research fields of dynamical systems and population biology. As such, it presents a clear picture of how applied dynamical systems and theoretical biology interact and stimulate each other--a fascinating positive feedback whose strength is anticipated to be enhanced by outstanding text like the work under review." -Mathematical Reviews, Issue 2004gshow more

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