An Introduction to Delay Differential Equations With Applications to the Life Sciences

by Smith, Hal; Sirovich, L.

• ISBN13: 9781441976451
• ISBN10: 1441976450
• Format: Hardcover
• Copyright: 2010-10-06
• Publisher: Springer Nature

Summary

This book is intended to be an introduction to Delay Differential Equations for upper level undergraduates or beginning graduate mathematics students who have a reasonable background in ordinary differential equations and who would like to get to the applications quickly.The author has used preliminary notes in teaching such a course at Arizona State University over the past two years.This book focuses on the key tools necessary to understand the applications literature involving delay equations and to construct and analyze mathematical models involving delay differential equations. The book begins with a survey of mathematical models involving delay equations.

Author Biography

Hal Smith is a Professor at the School of Mathematical and Statistical Sciences at Arizona State University.

Table of Contents

Introduction	p. 1
Examples of Delay Differential Equations	p. 1
Some Terminology	p. 9
Solving Delay Equations Using a Computer	p. 11
Delayed Negative Feedback: A Warm-Up	p. 13
Preliminaries	p. 13
The Simplest Delay Equation	p. 16
Oscillation of Solutions	p. 20
Solutions Backward in Time	p. 22
Existence of Solutions	p. 25
The Method of Steps for Discrete Delay Equations	p. 25
Positivity of Solutions	p. 27
A More General Existence Result	p. 29
Continuation of Solutions	p. 36
Remarks on Backward Continuation	p. 37
Stability Definitions	p. 38
Linear Systems and Linearization	p. 41
Autonomous Linear Systems	p. 41
Laplace Transform and Variation of Constants Formula	p. 43
The Characteristic Equation	p. 45
Small Delays Are Harmless	p. 48
The Scalar Equation x′(t) = Ax(t) + Bx(t - r)	p. 49
Principle of Linearized Stability	p. 54
Absolute Stability	p. 56
Semidynamical Systems and Delay Equations	p. 61
The Dynamical Systems Viewpoint	p. 61
Semiflows and Omega Limit Sets	p. 64
Semi Dynamical Systems Induced by Delay Equations	p. 65
Monotone Dynamics	p. 70
Delayed Logistic Equation	p. 73
Delayed Microbial Growth Model	p. 76
Liapunov Functions	p. 78
Logistic Equation with Instantaneous and Delayed Density Dependence	p. 80
Hopf Bifurcation	p. 87
A Canonical Example	p. 87
Hopf Bifurcation Theorem	p. 89
Delayed Negative Feedback	p. 91
Computation of the Hopf Bifurcation	p. 92
Series Expansion of Hopf Solution	p. 94
The Logistic Equation	p. 97
A Second-Order Delayed Feedback System	p. 99
Delayed Feedback Dominates Instantaneous Feedback	p. 101
Instantaneous Feedback Dominates Delayed Feedback	p. 104
Stabilizing the Straight-Up Steady State of the Pendulum	p. 106
Gene Regulation by End-Product Repression	p. 111
A Poincaré-Bendixson Theorem for Delay Equations	p. 115
Distributed Delay Equations and the Linear Chain Trick	p. 119
Infinite Delays of Gamma Type	p. 119
Characteristic Equation and Stability	p. 120
The Linear Chain Trick	p. 123
A Model of HIV Transmission	p. 126
An ODE Approximation to a Delay Equation	p. 129
Phage and Bacteria in a Chemostat	p. 131
Model	p. 131
Positivity and Boundedness of Solutions	p. 133
Basic Reproductive Number for Phage	p. 134
Persistence of Host and Phage Extinction	p. 135
The Coexistence Equilibrium	p. 137
Another Formulation of the Model	p. 141
Results from Real and Complex Analysis	p. 149
Analytic Functions	p. 149
Implicit Function Theorem for Complex Variables	p. 151
Rouché's Theorem	p. 152
Ascoli-Arzela Theorem	p. 153
Fluctuation Lemma	p. 154
General Implicit Function Theorem	p. 155
Gronwall's Inequality	p. 155
Hopf Bifurcation for Delayed Negative Feedback	p. 157
Basic Setup and Preliminaries	p. 157
The Solution	p. 160
Solve for q	p. 161
Solve for ¿ and ¿	p. 163
References	p. 167
Index	p. 171
Table of Contents provided by Ingram. All Rights Reserved.