Deepen students¡¯ understanding of biological phenomena

Suitable for courses on differential equations with applications to mathematical biology or as an introduction to mathematical biology, Differential Equations and Mathematical Biology, Second Edition introduces students in the physical, mathematical, and biological sciences to fundamental modeling and analytical techniques used to understand biological phenomena. In this edition, many of the chapters have been expanded to include new and topical material.

New to the Second Edition

This textbook shows how first-order ordinary differential equations (ODEs) are used to model the growth of a population, the administration of drugs, and the mechanism by which living cells divide. The authors present linear ODEs with constant coefficients, extend the theory to systems of equations, model biological phenomena, and offer solutions to first-order autonomous systems of nonlinear differential equations using the Poincar� phase plane. They also analyze the heartbeat, nerve impulse transmission, chemical reactions, and predator–prey problems. After covering partial differential equations and evolutionary equations, the book discusses diffusion processes, the theory of bifurcation, and chaotic behavior. It concludes with problems of tumor gr2owth and the spread of infectious diseases.

IntroductionPopulation growthAdministration of drugsCell divisionDifferential equations with separable variablesEquations of homogeneous typeLinear differential equations of the first orderNumerical solution of first-order equationsSymbolic computation in MATLABLinear Ordinary Differential Equations with Constant CoefficientsIntroductionFirst-order linear differential equationsLinear equations of the second orderFinding the complementary functionDetermining a particular integralForced oscillationsDifferential equations of order nUniquenessSystems of Linear Ordinary Differential EquationsFirst-order systems of equations with constant coefficientsReplacement of one differential equation by a systemThe general systemThe fundamental systemMatrix notationInitial and boundary value problemsSolving the inhomogeneous differential equationNumerical solution of linear boundary value problemsModelling Biological PhenomenaIntroductionHeartbeatNerve impulse transmissionChemical reactionsPredator–prey modelsFirst-Order Systems of Ordinary Differential EquationsExistence and uniquenessEpidemicsThe phase plane and the Jacobian matrixLocal stabilityStabilityLimit cyclesForced oscillationsNumerical solution of systems of equationsSymbolic computation on first-order systems of equations and higher-order equationsNumerical solution of nonlinear boundary value problemsAppendix: existence theoryMathematics of Heart PhysiologyThe local modelThe threshold effectThe phase plane analysis and the heartbeat modelPhysiological considerations of the heartbeat cycleA model of the cardiac pacemakerMathematics of Nerve Impulse TransmissionExcitability and repetitive firingTravelling wavesQualitative behavior of travelling wavesPiecewise linear modelChemical ReactionsWavefronts for the Belousov–Zhabotinskii reactionPhase plane analysis of Fisher¡¯s equationQualitative behavior in the general caseSpiral waves and ¥ë − ¥ø systemsPredator 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 equationVisualization of solutionsEvolutionary EquationsThe heat equationSeparation of variablesSimple evolutionary equationsComparison theoremsProblems of DiffusionDiffusion through membranesEnergy and energy estimatesGlobal behavior of nerve impulse transmissionsGlobal behavior in chemical reactionsTuring diffusion driven instability and pattern formationFinite pattern forming domainsBifurcation and ChaosBifurcationBifurcation of a limit cycleDiscrete bifurcation and period-doublingChaosStability of limit cyclesThe Poincaré planeAveragingNumerical Bifurcation AnalysisFixed points and stabilityPath-following and bifurcation analysisFollowing stable limit cyclesBifurcation in discrete systemsStrange attractors and chaosStability analysis of partial differential equationsGrowth of TumorsIntroductionMathematical model I of tumor growthSpherical tumor growth based on model IStability of tumor growth based on model IMathematical model II of tumor growthSpherical tumor growth based on model IIStability of tumor growth based on model IIEpidemicsThe Kermack–McKendrick modelVaccinationAn incubation modeSpreading in spaceAnswers to Selected ExercisesIndex

much progress by these authors and others over the past quarter century in modeling biological and other scientific phenomena make this differential equations textbook more valuable and better motivated than ever. …The writing is clear, though the modeling is not oversimplified. Overall, this book should convince math majors how demanding math modeling needs to be and biologists that taking another course in differential equations will be worthwhile. The coauthors deserve congratulations as well as course adoptions.�SIAM Review, Sept. 2010, Vol. 52, No. 3

� Where this text stands out is in its thoughtful organization and the clarity of its writing. This is a very solid book � The authors succeed because they do a splendid job of integrating their treatment of differential equations with the applications, and they don’t try to do too much. � Each chapter comes with a collection of well-selected exercises, and plenty of references for further reading.�MAA Reviews, April 2010

Praise for the First EditionA 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. 1It 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 theoretically 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 NewsletterThe 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 texts like the work under review.—Mathematical Reviews, Issue 2004g