Mathematical Methods in Biology

Mathematik trifft Biologie: Dieser Band lotet ein spannendes Grenzgebiet aus. Er informiert nicht nur über Grundlagen, sondern auch über moderne Spezialthemen wie Populationsdynamik, Fütterungstheorie und Theorie der Lebensgeschichte und enthält Material, dass Sie ansonsten kaum in der Literatur finden. Zunächst vermitteln die Autoren das notwendige mathematische Rüstzeug (biologische Modellierung, Infinitesimalrechnung, Differenzialgleichungen, dimensionslose Variable, deskriptive Statistik). Anschließend untersuchen sie diskrete und kontinuierliche Standardmodelle (Matrizenalgebra ebenso wie Differenzen- und Differenzialgleichungen). Im letzten Teil geht es um Wahrscheinlichkeitsrechnung, Statistik, stochastische Methoden, Bootstrapping und stochastische Differenzialgleichungen.

The last several years have 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