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Dynamical Systems for Biological Modeling

An Introduction

Fred Brauer, Christopher Kribs
Publication Date:
December 22, 2015
Content Details:
478 pages

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  • About the Book

    Book Summary

    Dynamical Systems for Biological Modeling: An Introduction prepares both biology and mathematics students with the understanding and techniques necessary to undertake basic modeling of biological systems. It achieves this through the development and analysis of dynamical systems.

    The approach emphasizes qualitative ideas rather than explicit computations. Some technical details are necessary, but a qualitative approach emphasizing ideas is essential for understanding. The modeling approach helps students focus on essentials rather than extensive mathematical details, which is helpful for students whose primary interests are in sciences other than mathematics need or want.

    The book discusses a variety of biological modeling topics, including population biology, epidemiology, immunology, intraspecies competition, harvesting, predator-prey systems, structured populations, and more.

    The authors also include examples of problems with solutions and some exercises which follow the examples quite closely. In addition, problems are included which go beyond the examples, both in mathematical analysis and in the development of mathematical models for biological problems, in order to encourage deeper understanding and an eagerness to use mathematics in learning about biology.


      • Addresses the needs of biology students without a strong mathematics background*, but who need to understand difference equations and differential equations (*Assumes one semester of calculus but [re]develops calculus ideas in context as needed)
      • Shows the student how to translate mathematical terms into biological concepts and vice versa so that modeling can be undertaken and results interpreted
      • Prepares the student of biology and mathematics to model basic biological systems through the development and analysis of dynamical systems
      • Includes examples along with exercises that follow them closely for practice
      • Discusses many applications in the biological sciences from population biology to epidemiology and more
  • Contents


    Introduction to Biological Modeling
    The Nature and Purposes of Biological Modeling
    The Modeling Process
    Types of Mathematical Models
    Assumptions, Simplifications, and Compromises
    Scale and Choosing Units

    Difference Equations (Discrete Dynamical Systems)
    Introduction to Discrete Dynamical Systems
    Graphical Analysis
    Qualitative Analysis and Population Genetics
    Intraspecies Competition
    Period Doubling and Chaos
    Structured Populations
    Predator-Prey Systems

    First-Order Differential Equations (Continuous Dynamical Systems)
    Continuous-Time Models and Exponential Growth
    Logistic Population Models
    Graphical Analysis
    Equations and Models with Variables Separable
    Mixing Processes and Linear Models
    First-Order Models with Time Dependence

    Nonlinear Differential Equations
    Qualitative Analysis Tools
    Mass-Action Models
    Parameter Changes, Thresholds, and Bifurcations
    Numerical Analysis of Differential Equations


    Systems of Differential Equations
    Graphical Analysis: The Phase Plane
    Linearization of a System at an Equilibrium
    Linear Systems with Constant Coefficients
    Qualitative Analysis of Systems

    Topics in Modeling Systems of Populations
    Epidemiology: Compartmental Models
    Population Biology: Interacting Species
    Numerical Approximation to Solutions of Systems

    Systems with Sustained Oscillations and Singularities
    Oscillations in Neural Activity
    Singular Perturbations and Enzyme Kinetics
    HIV - An Example from Immunology
    Slow Selection in Population Genetics
    Second-Order Differential Equations: Acceleration


    An Introduction to the Use of MapleTM

    Taylor’s Theorem and Linearization

    Location of Roots of Polynomial Equations

    Stability of Equilibrium of Difference Equations

    Answers to Selected Exercises