Table of Contents
1 Introduction 1
1.1 Examples of Delay Differential Equations 1
1.2 Some Terminology 9
1.3 Solving Delay Equations Using a Computer 11
2 Delayed Negative Feedback: A Warm-Up 13
2.1 Preliminaries 13
2.2 The Simplest Delay Equation 16
2.3 Oscillation of Solutions 20
2.4 Solutions Backward in Time 22
3 Existence of Solutions 25
3.1 The Method of Steps for Discrete Delay Equations 25
3.2 Positivity of Solutions 27
3.3 A More General Existence Result 29
3.4 Continuation of Solutions 36
3.5 Remarks on Backward Continuation 37
3.6 Stability Definitions 38
4 Linear Systems and Linearization 41
4.1 Autonomous Linear Systems 41
4.2 Laplace Transform and Variation of Constants Formula 43
4.3 The Characteristic Equation 45
4.4 Small Delays Are Harmless 48
4.5 The Scalar Equation x′(t) = Ax(t) + Bx(t - r) 49
4.6 Principle of Linearized Stability 54
4.7 Absolute Stability 56
5 Semidynamical Systems and Delay Equations 61
5.1 The Dynamical Systems Viewpoint 61
5.2 Semiflows and Omega Limit Sets 64
5.3 Semi Dynamical Systems Induced by Delay Equations 65
5.4 Monotone Dynamics 70
5.5 Delayed Logistic Equation 73
5.6 Delayed Microbial Growth Model 76
5.7 Liapunov Functions 78
5.7.1 Logistic Equation with Instantaneous and Delayed Density Dependence 80
6 Hopf Bifurcation 87
6.1 A Canonical Example 87
6.2 Hopf Bifurcation Theorem 89
6.3 Delayed Negative Feedback 91
6.3.1 Computation of the Hopf Bifurcation 92
6.3.2 Series Expansion of Hopf Solution 94
6.3.3 The Logistic Equation 97
6.4 A Second-Order Delayed Feedback System 99
6.4.1 Delayed Feedback Dominates Instantaneous Feedback 101
6.4.2 Instantaneous Feedback Dominates Delayed Feedback 104
6.4.3 Stabilizing the Straight-Up Steady State of the Pendulum 106
6.5 Gene Regulation by End-Product Repression 111
6.6 A Poincaré-Bendixson Theorem for Delay Equations 115
7 Distributed Delay Equations and the Linear Chain Trick 119
7.1 Infinite Delays of Gamma Type 119
7.1.1 Characteristic Equation and Stability 120
7.1.2 The Linear Chain Trick 123
7.2 A Model of HIV Transmission 126
7.3 An ODE Approximation to a Delay Equation 129
8 Phage and Bacteria in a Chemostat 131
8.1 Model 131
8.2 Positivity and Boundedness of Solutions 133
8.3 Basic Reproductive Number for Phage 134
8.4 Persistence of Host and Phage Extinction 135
8.5 The Coexistence Equilibrium 137
8.6 Another Formulation of the Model 141
A Results from Real and Complex Analysis 149
A.1 Analytic Functions 149
A.2 Implicit Function Theorem for Complex Variables 151
A.3 Rouché's Theorem 152
A.4 Ascoli-Arzela Theorem 153
A.5 Fluctuation Lemma 154
A.6 General Implicit Function Theorem 155
A.7 Gronwall's Inequality 155
B Hopf Bifurcation for Delayed Negative Feedback 157
B.1 Basic Setup and Preliminaries 157
B.2 The Solution 160
B.2.1 Solve for q 161
B.2.2 Solve for μ and δ 163
References 167
Index 171
