Delay Differential Equations With Applications in Population Dynamics
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Contents
Preface
Part One: DELAY DIFFERENTIAL EQUATIONS
Chapter 1. Introduction
1.1. Delay Differential Equations from Applications
1.2. Small Delays Can Have Large Effects
1.3. Concluding Remarks
Chapter 2. Basic Theory of Delay Differential Equations
2.1. Preliminaries—Definitions and Notations
2.2. Existence, Uniqueness, Continuous Dependence. and Smoothing Property
2.3. Dynamical Systems and Invariance
2.4. Local Stability Theory
2.5. The Method of Liapunov Functionals
2.6. Razumikhin-Type Theorems
2.7. Infinite Delay and Fading Memory Space
2.8. General Linear Systems
2.9. Hopf Bifurcation and a Periodicity Theorem
Chapter 3. Characteristic Equations
3.1. Discrete Delays—Preliminaries
3.2. Discrete Delays—First Order equations
3.3. Discrete Delays—Second Order Equations
3.4. Discrete Delays—General Theory
3.5. Distributed Delays—Special Cases
3.6. Reducible Systems—The Linear Chain Trick
3.7. Distributed Delays—First Order Equations
3.8. Distributed Delays—Higher Order Equations and Systems
3.9. Remarks and Open Problems
Part Two: APPLICATIONS IN POPULATION DYNAMICS
Chapter 4. Global Stability for Single Species Models
4.1. Introduction
4.2. Wright’s Global Stability Result for ý(t) = -ry(t - 1)[1+ y(t)]
4.3. Global Stability for General Delayed Nonautonomous Logistic Equations
4.4. Asymptotic Theory for Nonautonomous Delay Equations with Negative Feedbacks
4.5. 3/2 Stability Results
4.6. A Model Exhibiting the Allee Effect
4.7. Equations of Type x (t) = f(xt) - g(x(t))—Preliminaries
4.8. Equations of Type x (t) = f(xt) - g(x(t))—When f(x) Is Decreasing
4.9. Equations of Type x (t) = f(xt) - g(x(t))—When f(x) Is Increasing or Has a Hump
4.10. Remarks and Open Problems
Chapter 5. Periodic Solutions, Chaos, Structured Single Species Models
5.1. Global Existence of Periodic Solutions in x (t) = f (x(t - 1)) -g(x(t))
5.2. Periodic Solutions in Delayed Periodic Lotka–Volterra-Type Equations
5.3. A Model of Single Species Growth with Stage Structure
5.4. Reduction of Structured Population Models to Threshold Delay Equations and FDEs
5.5. Chaos
5.6. Remarks
Chapter 6. Global Stability for Multi-Species Models
6.1. Introduction
6.2. Stability via Liapunov Functionals, I
6.3. Stability via Liapunov Functionals, II
6.4. Stability via Razumikhin-Type Theorems—Theory
6.5. Stability via Razumikhin-Type Theorems—Applications
6.6. When Nondelayed Diagonal Terms Do Not Exist
6.7. Remarks and Open Problems
Chapter 7. Periodic Solutions in Multi-Species Models
7.1. Introduction
7.2. Periodic Solutions in Delayed Gause-Type Predator–Prey Systems
7.3. Periodic Solutions in Periodic Systems
7.4. Remarks and Open Problems
Chapter 8. Permanence
8.1. Introduction
8.2. Persistence in Infinite Dimensional Systems
8.3. Permanence in Autonomous Lotka–Volterra-Type Systems
8.4. Permanence in Nonautonomous Systems
8.5. Permanence in Nonautonomous Lotka–Volterra-Type Competition Systems
8.6. Permanence in Nonautonomous Lotka–Volterra-Type Predator–Prey Systems
8.7. Remarks and Open Problems
Chapter 9. Neutral Delay Models
9.1. Models and Preliminaries
9.2. Boundedness of x (t) in the System (1.6)
9.3. Boundedness Results for the System (1.6)
9.4. Convergence in Single Population Models, I
9.5. Convergence in the System (1.6)
9.6. Convergence in Lotka–Volterra Systems
9.7. Convergence in Single Population Models, II
9.8. Boundedness in a Nonautonomous Neutral Logistic Equation
9.9. A Periodic Neutral Logistic Equation
9.10. Remarks and Open Problems
References
Appendix
Index
Mathematics in Science and Engineering