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The last several years has witnessed a revolution in the connections between mathematics and biology, and this book differs from most others on the topic in that it covers both deterministic and probabilistic models. The first chapter is a long introduction and review of ideas about biological modeling, calculus, differential equations, dimensionless variables, and descriptive statistics. The next three chapters examine standard discrete and continuous models using difference and differential equations, and matrix algebra (there is a long appendix in Chapter 3 on matrices). The final three chapters cover probability, statistics, and stochastic processes, including bootstrap methods and stochastic differential equations. The book focuses mostly in one area of the life sciences, namely, theoretical ecology.
Ecology has become extremely quantitative, and the mathematical techniques used in ecology are applicable to most other areas in the life sciences. Ecology provides an especially accessible context for study by mathematics majors. Moreover, the authors chose ecology for the book's motivations and examples in light of their own interests and research in the area. Additional topical coverage includes an introduction to ecological modeling, population dynamics for single species, structure and interacting populations, interactions in continuous time, concepts of probability, statistical inference, and stochastic processes.
Preface. 1. Introduction To Ecological Modeling. 1.1 Mathematical Models. 1.2 Rates of Change. 1.3 Balance Laws. 1.4 Temperature in the Environment. 1.5 Dimensionless Variables. 1.6 Descriptive Statistics. 1.7 Regression and Curve Fitting. 1.8 Reference Notes. 2. Population Dynamics for Single Species. 2.1 Laws of Population Dynamics. 2.2 Continuous Time Models. 2.3 Qualitative Analysis of Population Models. 2.4 Dynamics of Predation. 2.5 Discrete Time Models. 2.6 Equilibria, Stability, and Chaos. 2.7 Reference Notes. 3. Structure and Interacting Populations. 3.1 Structure--Juveniles and Adults. 3.2 Structured Linear Models. 3.3 Nonlinear Interactions. 3.4 Appendix--Matrices. 3.5 Reference Notes. 4. Interactions in Continuous Time. 4.1 Interacting Populations. 4.2 Phase Plane Analysis. 4.3 Linear Systems. 4.4 Nonlinear Systems. 4.5 Bifurcation. 4.6 Reference Notes. 5. Concepts of Probability. 5.1 Introductory Examples and Definitions. 5.2 The Hardy-Weinberg Law. 5.3 Continuous Random Variables. 5.4 Discrete Random Variables. 5.5 Joint Probability Distributions. 5.6 Covariance and Correlation. 5.7 Reference Notes. 6. Statistical Inference. 6.1 Introduction. 6.2 Interval Analysis. 6.3 Estimating Proportions. 6.4 The Chi-Squared Test. 6.5 Hypothesis Testing. 6.6 Bootstrap Methods. 6.7 Reference Notes. 7. Stochastic Processes. 7.1 Introduction. 7.2 Randomizing Discrete Dynamics. 7.3 Random Walk. 7.4 Birth Processes. 7.5 Stochastic Differential Equations. 7.6 SDEs from Markov Models. 7.7 Solving SDEs. 7.8 The Fokker-Planck Equation. 7.9 Reference Notes. A. Hints and Solutions to Exercises
J. David Logan, PhD, is Willa Cather Professor of Mathematics at the University of Nebraska Lincoln. He has written more than eighty research articles in his areas of research interest, which include mathematical physics, combustion and detonation, hydrogeology, and mathematical biology. Dr. Logan is the author of Applied Mathematics, Third Edition and An Introduction to Nonlinear Partial Differential Equations, Second Edition, both published by Wiley. William R. Wolesensky, PhD, is Associate Professor in the Department of Mathematics at Doane College. Dr. Wolesensky has written numerous journal articles on the use of mathematical modeling techniques in scientific research.
Admirably, the volume is written with bits of MATLAB code inserted at appropriate places and has exercises interspersed throughout the text (as well as hints for solutions to the exercises at the end of the book). The Quarterly Review of Biology, June 2010) "The mathematical and reasoning sophistication increases as the chapters proceed." (Book News, December 2009)