Dynamical System Models in the Life Sciences and Their Underlying Scientific Issues

The book under review is a fairly classical development of differential equations. The applications are all oriented to biology: population models, red blood cell production, virus life cycles, cell death, overfishing and mutation. Most of the applications arise from second and higher order linear systems of differential equations. The book’s preface describes the development of a gateway graduate program in “Mathematical and Computational Biology” (MCB) at the University of California, Irvine in 2007 and the subsequent core curriculum that arose from it. The three-quarter sequence Math 227 A,B,C was created to address the need for a course introducing students to the modeling of biological phenomena through mathematics. This book has evolved from the lecture notes for Math 227A whereas Math 227B addresses modeling using partial differential equations and Math 227 C focuses on stochastic ODEs.

The material could be used profitably by biology majors who have had at least one full year of calculus and a class in multivariable calculus. The book moves through the material at a quick clip. First order techniques are not really developed (minimal treatment of integrating factors, Bernoulli equations appear but only in appendix) and the text spends most of the first chapter analyzing simple population models. This is followed by a more expansive discussion of logistic growth and the analysis of stability of equilibrium solutions. The prototypical bifurcation types for ODEs with a parameter are given some careful attention as well.

Part 2 of the book is subtitled “Interacting Populations”. Here we are faced with linear systems of the form \(\displaystyle\frac{dx}{dt} = Ax + h(t)\), where \(x\) is an \(n\)-vector, \(A\) an \(n\times n\) real matrix, and \(h\) a time-dependent vector function. The inhomogeneous term \(h\) seems to be used in cases where there are periodic (seasonal) influxes into population growth. And boom! We are quickly talking about matrix exponentials, linear independence, Jordan normal form and the full force of linear algebra. It’s all introduced from scratch, but at a fast pace and without supplying proofs for all the results mentioned. Still, what is there is quite correct and there don’t seem to be any mistakes. The worked examples are rich in detail — HIV dynamics, the replicator dynamics, cell lineages and much more.

The next chapters address optimization problems and introduce the calculus of variations to solve a variety of problems. Again, the discussion is rich and touches on many important topics: Hamiltonians, the Pontryagin principle, boundary value problems and heat conduction, etc. The book culminates in an extended discussion of the bacteria Chlamydia Trachomatis which brings all the tools developed to an analysis of the model’s set of critical points and determines optimal conditions for growth and spreading. This is followed by a chapter on genetic instability and carcinogenesis with a discussion of various types of death rates that emerge from the mathematics. The book is rounded out with a collection of problem sets designed for each chapter and a sample midterm and final examination. The problems mostly test facility with the mathematical techniques and the examinations push the discussion further into actual modeling scenarios.

I find myself somewhat “out of breath” at the finish line — this is a compressed and driven journey through a large amount of mathematics. It is certainly different in key respects from a classical approach to differential equations (say Boyce and DiPrima) — there is no development of Laplace transforms or power series solutions. On the other hand, it is very much in line with modern dynamical systems methodology (and certainly this is driven by the plethora of nonlinear models in mathematical biology) and highlights some fascinating applications to modern biology. The examples are all “real life” situations rather than “toy models” and the analysis is correspondingly well-developed. The authors have done a wonderful job at extracting maximum use out of a nicely chosen set of examples. Bravo!

Jeff Ibbotson holds the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.