Biology in Time and Space: A Partial Differential Equation Approach. James P. Keener University of Utah

Transcription

1 Biology in Time and Space: A Partial Differential Equation Approach James P. Keener University of Utah 26 December, 2015

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3 Contents I The Text 9 1 Introduction 11 2 Conservation The Conservation Law Examples of Flux The Diffusion Equation - Derivations Discrete Boxes A Random Walk The Cable Equation Exercises Realizations of a diffusion process Following a single particle Other features of Brownian particle motion - Escape times and splitting probabilities Higher dimensions Following Several Particles Exercises Solutions of the Diffusion Equation 33 3

4 4 CONTENTS 5.1 On the Infinite Line On the semi-infinite line With Boundary Conditions Separation of Variables Numerical Methods Method of Lines Forward Euler Method Backward Euler Method Crank-Nickolson Method Comparison Theorems Exercises Diffusion and Reaction Birth-Death with Diffusion Dimensional Analysis Facilitated Diffusion Facilitated Diffusion in Muscle Respiration Growth with a Carrying capacity - Fisher s Equation On a Bounded Domain Resource Consumption Spread of Rabies - SIR with Diffusion Exercises The Bistable Equation - Part I: Derivations Spruce Budworm Wolbachia

5 CONTENTS Nerve Axons Calcium Handling Exercises The Bistable Equation - Part II: Analysis Traveling Waves Threshold Behavior Propagation Failure Regions of Block Discrete Cells Discrete Release Sites Exercises Advection and Reaction Simple Advection Transport across a Membrane Oxygen exchange between capillaries Structured Populations Red Blood Cells Protein Flexibility Muscle Contraction Sliding Friction Simulation Delay Differential Equations The Method of Characteristics Method of Lines; Upwinding

6 6 CONTENTS 9.8 Exercises Advection with diffusion Axonal transport Transport with Switching Ornstein-Uhlenbeck Process Exercises Chemotaxis Waves and Vibrations Blood Pressure Pulse The Wave Equation Exercises Spatial Patterns The Turing Mechanism Diffusion-Driven Instability Tiger Bush Stripes Cell Polarity Exercises Dispersal-Renewal Theory Invasive Species Exercises Collective Behavior Quorum Sensing

7 CONTENTS camp Oscillations Flocking Behavior Appendix: Background Material Calculus Polar Coordinates Spherical Coordinates Taylor s Theorem Ordinary Differential Equations First Order Equations Systems of first order equations Numerical Simulation Modeling Chemical Reactions Stochastic Processes Decay Processes Several Reactions Exercises Appendix: Matlab Codes A Matlab Primer Particle Decay Method of Lines Discrete Random Walk Motion of a Brownian Particle Diffusion Equation solution via Crank Nicolson Method Fisher Equation and Bistable Equation via Crank-Nicolson

8 8 CONTENTS 17.8 Diffusion Equation with growth/decay solution via Crank Nicolson Solution of Delay differential equation using the Method of Lines (MOL) Solution of the Ornstein-Uhlenbeck equation The Bisection Method Appendix: Physical Constants Diffusion Coefficients Physical constants II Solutions Selected Solutions to Chapter Selected Solutions to Chapter Selected Solutions to Chapter Selected Solutions to Chapter Selected Solutions for Chapter Selected Solutions to Chapter Selected Solutions to Chapter 14 17

9 Part I The Text 9

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11 Chapter 1 Introduction Mathematical biology is, broadly speaking, the study of how biological objects move and interact. The interaction part of biology is often studied using dynamical systems including discrete time maps and ordinary differential equations. Indeed, there is a lot of insight that can be gained from the study of these dynamical descriptions of biological processes. And, there are many excellent books that present this point of view. However, this perspective is limited when it comes to understanding real biological situations, because biological objects are essentially never homogeneously distributed in space, or well-mixed, even in a test tube or on a Petrie dish, let alone in the human body or on a mountain slope. In fact, it is a primary feature of biological objects that there are spatial differences and correspondingly movement of objects from one place to another. So, whether one is studying the spread of an infectious disease in Africa, the spread of an invasive species in the agricultural regions of California, or the movement of an action potential along a nerve axon, it is crucially important to include the effects of spatial differences. This book is about the dynamics of biological objects in time and space. Consequently, it is about partial differential equations. It is intended for an undergraduate audience, and no previous background in partial differential equations is required. In fact, in the Appendices, you will find summaries of the background that is needed from calculus, from ordinary differential equations, and from probability theory and stochastic processes, because these are used a lot. Also, this material is presented from a heavily computational perspective, using Matlab to compute solutions and make plots. So, in the appendix you will find a primer for Matlab as well as Matlab codes for many of the calculations done in this text. The hope is that these codes can be readily modified to be used for Exercises, and that this will facilitate the process of learning how to understand partial differential equations. The content of the book is organized pedagogically around mathematical material. However, all of the models described in this book are derived with a specific biological question in mind. Give a summary of the topics covered. 11