Biology in Time and Space: A Partial Differential Equation Modeling Approach (Pure and Applied Undergraduate Texts, 50)

How Do Biological Objects Communicate, Make Structures, Make Measurements And Decisions, Search For Food, I.e., Do All The Things Necessary For Survival? Designed For An Advanced Undergraduate Audience, This Book Uses Mathematics To Begin To Tell That Story. It Builds On A Background In Multivariable Calculus, Ordinary Differential Equations, And Basic Stochastic Processes And Uses Partial Differential Equations As The Framework Within Which To Explore These Questions. This Book Tells The Story Of Living Processes That Change In Time And Space. Driven By Scientific Inquiry, Methods From Partial Differential Equations, Stochastic Processes, Dynamical Systems, And Numerical Methods Are Brought To Bear On The Subject, And Their Exposition Seems Effortless In The Pursuit Of Deeper Biological Understanding. With Subjects Ranging From Spruce Budworm Populations To Calcium Dynamics And From Tiger Bush Patterns To Collective Behavior, This Is A Must-read For Anyone Who Is Serious About Modern Mathematical Biology. --mark Lewis, University Of Alberta Prof. Keener Is One Of The Great Minds In Math Biology Who Has Trained Generations Of Fine Scientists And Mathematicians Over The Years. --leah Edelstein-keshet, University Of British Columbia This Is A Fantastic Book For Those Of Us Who Teach Mathematical Modelling Of Spatiotemporal Phenomena In Biology, And For Anyone Who Wishes To Move Into The Field. It Guides The Reader On How One Should Tackle The Art Of Modelling And, In A Very Systematic And Natural Way, Introduces Many Of The Necessary Mathematical And Computational Approaches, Seamlessly Integrating Them With The Biology. It Is A Pleasure To Read. --philip Maini, University Of Oxford Mathematical Biology Has Few Foundational Texts. But This Is One. --michael C. Reed, Duke University Cover Title page Copyright Contents Preface Chapter 1. Background Material 1.1. Multivariable Calculus 1.2. Ordinary Differential Equations 1.3. Stochastic Processes Exercises Chapter 2. Conservation— Learning How to Count 2.1. The Conservation Law 2.2. Examples of Flux—How Things Move Exercises Chapter 3. The Diffusion Equation— Derivations 3.1. Discrete Boxes 3.2. A Random Walk 3.3. The Cable Equation Exercises Chapter 4. Realizations of a Diffusion Process 4.1. Following Individual Particles 4.2. Other Features of Brownian Particle Motion 4.3. Following Several Particles 4.4. Effective Diffusion 4.5. An Agent-Based Approach Exercises Chapter 5. Solutions of the Diffusion Equation 5.1. On an Infinite Domain 5.2. On the Semi-infinite Line 5.3. With Boundary Conditions 5.4. Separation of Variables 5.5. Numerical Methods 5.6. Comparison Theorems 5.7. FRAP Exercises Chapter 6. Diffusion and Reaction 6.1. Birth-Death with Diffusion 6.2. Growth with a Carrying Capacity—Fisher’s Equation 6.3. Resource Consumption 6.4. Spread of Rabies—SIR with Diffusion 6.5. Extras: Facilitated Diffusion Exercises Chapter 7. The Bistable Equation—Part I: Derivations 7.1. Spruce Budworm 7.2. Wolbachia 7.3. Nerve Axons 7.4. Calcium Handling Exercises Chapter 8. The Bistable Equation—Part II: Analysis 8.1. Traveling Waves 8.2. Threshold Behavior 8.3. Propagation Failure Exercises Chapter 9. Advection and Reaction 9.1. Simple Advection 9.2. Advection with Decay 9.3. Structured Populations 9.4. Simulation 9.5. Nonlinear Advection; Burgers’ Equation 9.6. Extras: More Advection-Reaction Models Exercises Chapter 10. Advection with Diffusion 10.1. A Biased Random Walk 10.2. Transport with Switching 10.3. Ornstein–Uhlenbeck Process 10.4. Spread of an Ornstein–Uhlenbeck Epidemic Exercises Chapter 11. Chemotaxis 11.1. Amoeba Aggregation Exercises Chapter 12. Spatial Patterns 12.1. The Turing Mechanism 12.2. Tiger Bush Stripes 12.3. Cell Polarity Exercises Chapter 13. Dispersal-Renewal Theory 13.1. Invasive Species Exercises Chapter 14. Collective Behavior 14.1. Quorum Sensing 14.2. Flocking Behavior Exercises Appendix A. Introduction to Matlab A.1. A Matlab Primer A.2. List of Available Matlab Codes Appendix B. Constants, Units, and Functions B.1. Physical Constants B.2. Functions Used in this Book Appendix C. Selected Answers to Exercises C.1. Selected Answers for Chapter 1 C.2. Selected Answers for Chapter 2 C.3. Selected Answers for Chapter 3 C.4. Selected Answers for Chapter 4 C.5. Selected Answers for Chapter 5 C.6. Selected Answers for Chapter 6 C.7. Selected Answers for Chapter 7 C.8. Selected Answers for Chapter 8 C.9. Selected Answers for Chapter 9 C.10. Selected Answers for Chapter 10 C.11. Selected Answers for Chapter 11 C.12. Selected Answers for Chapter 12 C.13. Selected Answers for Chapter 13 C.14. Selected Answers for Chapter 14 Bibliography Index Back Cover This book tells the story of biological objects by using a multi-variable differential equation approach and it covers both population level dynamics as well as cell biology and physiology. There are fourteen chapters and several appendices in this book. Each chapter contains around ten to thirty exercises with selected solutions provided in an appendix at the end. The Matlab code used in each chapter is also summarized at the end of the book. The first Chapter is devoted to background material including dynamical systems and stochastic processes. Chapter Two explains conservation law. Chapter Three and Chapter Four revolve around diffusion equations and their realization. Chapter Five discusses solutions of diffusion equations both analytically and numerically. Chapter Six is devoted to diffusion and reaction processes. Chapter Seven and Chapter Eight look at the derivation and analysis of the bistable equation. Chapter Nine and Chapter Ten study advection-reaction models and advection with diffusion. Chapter Eleven to Fourteen cover a number of interesting and important biological processes. They include for example amoeba aggregation, tiger bush stripes, cell polarity, invasive species, quorum sensing and flocking. The book is suitable for the level of advanced undergraduate students and above "How do biological objects communicate, make structures, make measurements and decisions, search for food, i.e., do all the things necessary for survival? Designed for an advanced undergraduate audience, this book uses mathematics to begin to tell that story. It builds on a background in multivariable calculus, ordinary differential equations, and basic stochastic processes and uses partial differential equations as the framework within which to explore these questions. This book tells the story of living processes that change in time and space. Driven by scientific inquiry, methods from partial differential equations, stochastic processes, dynamical systems, and numerical methods are brought to bear on the subject, and their exposition seems effortless in the pursuit of deeper biological understanding. With subjects ranging from spruce budworm populations to calcium dynamics and from tiger bush patterns to collective behavior, this is a must-read for anyone who is serious about modern mathematical biology."--Page 4 of cover