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Explorations of mathematical models in biology with Maple. (English) Zbl 1309.92010

Hoboken, NJ: John Wiley & Sons (ISBN 978-1-118-03211-4/hbk; 978-1-118-55217-9/ebook). xi, 292 p. (2015).
This nice text invites students in life sciences and related areas to explore possibilities of mathematical modeling using difference equations, matrices, and Maple\(^{\text{\textsc{tm}}}\). There are quite a few good books on mathematical modeling with differential equations, but these may be rather advanced for students who had only one mathematics course covering basics of calculus, linear algebra, and some differential equations. On the other hand, difference equations and matrix algebra constitute quite simple yet powerful tools for the analysis of the dynamics of real biological systems. Aiming at a self-contained exposition, the author includes in the book background material required for the understanding of the main topics.
The material is organized in five chapters. The introductory Chapter 1 provides some basic facts on modeling, difference equations and Maple. Modeling with first-order difference equations is the subject of Chapter 2. The topics cover modeling with first-order linear homogeneous difference equations, nonhomogeneous first-order linear difference equations, and nonlinear logistic difference equations. The chapter concludes with an intuitive introduction to chaos in dynamical systems. Basic material from linear algebra is collected in Chapter 3, where the reader will find useful facts on the solution of linear systems of equations, matrices, determinants, eigenvalues and eigenvectors, and their role in stability analysis of linear models. Realistic modeling applications are discussed in the last two chapters of the book. In Chapter 4, modeling with Markov chains, Leslie’s age-structured matrix population models, and second-order linear difference equations are discussed. The final Chapter 5 introduces techniques for modeling population dynamics, infectious diseases and concludes with a brief insight into modeling with delay logistic difference equations.
Emphasizing the use of graphics and numerical computation for mathematical modeling in biology rather than theoretical methods, the author promotes a “discovery pedagogical approach” which he explains as follows. “To introduce a concept, first we investigate a model numerically and/or graphically and recognize a pattern or certain properties that characterize that concept. Then we give a definition of the concept with examples and applications.” This approach suits well the expectations of students in life sciences who may prefer to skip theoretical foundations, start “intuition-lead” modeling of real systems, and revise theory later, if desired. The book is written in a transparent manner with clear explanations of all necessary steps; relevant Maple codes are provided. Useful supplementary material such as solutions to selected problems, worksheets, Maple codes for many programs in the text, can be found on the publisher’s web page.

MSC:

92-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to biology
92-04 Software, source code, etc. for problems pertaining to biology
92C42 Systems biology, networks
92D30 Epidemiology
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra
34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
00A71 General theory of mathematical modeling

Software:

Maple
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