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Differential Equations and Mathematical Biology: Chapman & Hall/CRC Mathematical and Computational Biology

Autor Michael Plank, B. D. Sleeman, D. S. Jones
en Limba Engleză Hardback – 12 noi 2009
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
  • A section on spiral waves
  • Recent developments in tumor biology
  • More on the numerical solution of differential equations and numerical bifurcation analysis
  • MATLAB® files available for download online
  • Many additional examples and exercises
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 Poincaré phase plane. They also analyze the heartbeat, nerve impulse transmission, chemical reactions, and predatorߝprey 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 growth and the spread of infectious diseases.
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Specificații

ISBN-13: 9781420083576
ISBN-10: 1420083570
Pagini: 444
Ilustrații: 100 black & white illustrations, 3 black & white tables
Dimensiuni: 157 x 236 x 28 mm
Greutate: 0.77 kg
Ediția:Revizuită
Editura: Chapman & Hall/CRC
Seria Chapman & Hall/CRC Mathematical and Computational Biology

Locul publicării:United States

Public țintă

Undergraduate students in mathematics, biology, and the physical sciences; graduate students and researchers in mathematical biology.

Cuprins

Introduction
Population growth
Administration of drugs
Cell division
Differential equations with separable variables
Equations of homogeneous type
Linear differential equations of the first order
Numerical solution of first-order equations
Symbolic computation in MATLAB
Linear Ordinary Differential Equations with Constant Coefficients
Introduction
First-order linear differential equations
Linear equations of the second order
Finding the complementary function
Determining a particular integral
Forced oscillations
Differential equations of order n
Uniqueness
Systems of Linear Ordinary Differential Equations
First-order systems of equations with constant coefficients
Replacement of one differential equation by a system
The general system
The fundamental system
Matrix notation
Initial and boundary value problems
Solving the inhomogeneous differential equation
Numerical solution of linear boundary value problems
Modelling Biological Phenomena
Introduction
Heartbeat
Nerve impulse transmission
Chemical reactions
Predatorߝprey models
First-Order Systems of Ordinary Differential Equations
Existence and uniqueness
Epidemics
The phase plane and the Jacobian matrix
Local stability
Stability
Limit cycles
Forced oscillations
Numerical solution of systems of equations
Symbolic computation on first-order systems of equations and higher-order equations
Numerical solution of nonlinear boundary value problems
Appendix: existence theory
Mathematics of Heart Physiology
The local model
The threshold effect
The phase plane analysis and the heartbeat model
Physiological considerations of the heartbeat cycle
A model of the cardiac pacemaker
Mathematics of Nerve Impulse Transmission
Excitability and repetitive firing
Travelling waves
Qualitative behavior of travelling waves
Piecewise linear model
Chemical Reactions
Wavefronts for the BelousovߝZhabotinskii reaction
Phase plane analysis of Fisher’s equation
Qualitative behavior in the general case
Spiral waves and λω systems
Predator and Prey
Catching fish
The effect of fishing
The VolterraߝLotka model
Partial Differential Equations
Characteristics for equations of the first order
Another view of characteristics
Linear partial differential equations of the second order
Elliptic partial differential equations
Parabolic partial differential equations
Hyperbolic partial differential equations
The wave equation
Typical problems for the hyperbolic equation
The EulerߝDarboux equation
Visualization of solutions
Evolutionary Equations
The heat equation
Separation of variables
Simple evolutionary equations
Comparison theorems
Problems of Diffusion
Diffusion through membranes
Energy and energy estimates
Global behavior of nerve impulse transmissions
Global behavior in chemical reactions
Turing diffusion driven instability and pattern formation
Finite pattern forming domains
Bifurcation and Chaos
Bifurcation
Bifurcation of a limit cycle
Discrete bifurcation and period-doubling
Chaos
Stability of limit cycles
The Poincaré plane
Averaging
Numerical Bifurcation Analysis
Fixed points and stability
Path-following and bifurcation analysis
Following stable limit cycles
Bifurcation in discrete systems
Strange attractors and chaos
Stability analysis of partial differential equations
Growth of Tumors
Introduction
Mathematical model I of tumor growth
Spherical tumor growth based on model I
Stability of tumor growth based on model I
Mathematical model II of tumor growth
Spherical tumor growth based on model II
Stability of tumor growth based on model II
Epidemics
The KermackߝMcKendrick model
Vaccination
An incubation model
Spreading in space
Answers to Selected Exercises
Index

Recenzii

…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 Edition
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 FitzߝHughߝNagumo 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 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 texts like the work under review.
Mathematical Reviews, Issue 2004g

Notă biografică

D.S. Jones, FRS, FRSE is Professor Emeritus in the Department of Mathematics at the University of Dundee in Scotland.
M.J. Plank is a senior lecturer in the Department of Mathematics and Statistics at the University of Canterbury in Christchurch, New Zealand.
B.D. Sleeman, FRSE is Professor Emeritus in the Department of Applied Mathematics at the University of Leeds in the UK.

Descriere

This 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.