Dynamics of Second Order Rational Difference

This self-contained monograph provides systematic, instructive analysis of second-order rational difference equations. Each chapter contains a section of open problems and conjectures, many of them ideal for research projects or Ph.D. theses. The techniques and results in this monograph are extremely useful in analyzing the equations in the mathematical models of various biological systems and other applications. Clear, simple, and direct exposition combined with thoughtful uniformity in the presentation make the book invaluable as a text as a reference for researchers, and as a supplement to every textbook on difference equations.

Preface

ix

Acknowledgements

xi

Introduction and Classification of Equation Types

1

(4)

Preliminary Results

5

(24)

Definitions of Stability and Linearized Stability Analysis

5

(2)

The Stable Manifold Theorem in the Plane

7

(2)

Global Asymptotic Stability of the Zero Equilibrium

9

(1)

Global Attractivity of the Positive Equilibrium

9

(5)

Limiting Solutions

14

(3)

The Riccati Equation

17

(7)

Semicycle Analysis

24

(5)

Local Stability, Semicycles, Periodicity, and Invariant Intervals

29

(18)

Equilibrium Points

29

(1)

Stability of the Zero Equilibrium

30

(2)

Local Stability of the Positive Equilibrium

32

(1)

When is Every Solution Periodic with the Same Period?

33

(2)

Existence of Prime Period-Two Solutions

35

(1)

Local Asymptotic Stability of a Two Cycle

36

(2)

Convergence to Period-Two Solutions When C = 0

38

(3)

Invariant Intervals

41

(2)

Open Problems and Conjectures

43

(4)

(1, 1)-Type Equations

47

(6)

Introduction

47

(1)

The Case α = γ = A = B = 0 :

48

(1)

The Case α = β = A = C = 0 :

49

(1)

Open Problems and Conjectures

49

(4)

(1, 2)-Type Equations

53

(16)

Introduction

53

(1)

The Case β = γ = C = 0 :

54

(1)

The Case β = γ = A = 0 :

55

(2)

The Case α = γ = B = 0 :

57

(1)

The Case α = γ = A = 0 :

58

(1)

The Case α = β = C = 0 :

59

(1)

The Case α = β = A = 0 :

60

(4)

The Case p < 1

61

(1)

The Case p = 1

62

(1)

The Case p > 1

62

(2)

Open Problems and Conjectures

64

(5)

(2, 1)-Type Equations

69

(8)

Introduction

69

(1)

The Case γ = A = B = 0 :

70

(2)

The Case β = A = C = 0 :

72

(1)

Open Problems and Conjectures

72

(5)

(2, 2)-Type Equations

77

(54)

Introduction

77

(1)

The Case γ = C = 0 :

78

(1)

The Case γ = B = 0 :

79

(3)

The Case γ = A = 0 :

82

(7)

Invariant Intervals

83

(1)

Semicycle Analysis

84

(3)

Global Stability and Boundedness

87

(2)

The Case β = C = 0 :

89

(3)

Semicycle Analysis

90

(1)

The Case q < 1

90

(1)

The Case q > 1

91

(1)

The Case β = A = 0 :

92

(9)

Existence and Local Stability of Period-Two Cycles

93

(3)

Invariant Intervals

96

(1)

Semicycle Analysis

97

(3)

Global Behavior of Solutions

100

(1)

The Case α = C = 0 :

101

(8)

Invariant Intervals

102

(1)

Semicycle Analysis of Solutions When q ≤ 1

103

(2)

Semicycle Analysis and Global Attractivity When q = 1

105

(2)

Global Attractivity When q ≤ 1

107

(1)

Long-term Behavior of Solutions When q > 1

107

(2)

The Case α = B = 0 :

109

(4)

Invariant Intervals and Semicycle Analysis

110

(2)

Global Stability of the Positive Equilibrium

112

(1)

The Case α = A = 0 :

113

(10)

Existence of a Two Cycle

114

(3)

Semicycle Analysis

117

(1)

Global Stability Analysis When p ≤ q

117

(6)

Global Stability Analysis When p > q

123

(1)

Open Problems and Conjectures

123

(8)

(1, 3)-Type Equations

131

(6)

Introduction

131

(1)

The Case β = γ = 0 :

131

(1)

The Case α = γ = 0 :

132

(1)

The Case α = γ = 0 :

133

(1)

Open Problems and Conjectures

134

(3)

(3, 1)-Type Equations

137

(4)

Introduction

137

(1)

Open Problems and Conjectures

138

(3)

(2, 3)-Type Equations

141

(26)

Introduction

141

(1)

The Case γ = 0 :

141

(8)

Boundedness of Solutions

142

(1)

Invariant Intervals

143

(1)

Semicycle Analysis

144

(3)

Global Asymptotic Stability

147

(2)

The Case β = 0 :

149

(9)

Existence and Local Stability of Period-Two Cycles

150

(4)

Invariant Intervals

154

(2)

Convergence of Solutions

156

(1)

Global Stability

157

(1)

Semicycle Analysis When pr = q + q2/r

157

(1)

The Case α = 0 :

158

(5)

Invariant Intervals

160

(2)

Convergence of Solutions

162

(1)

Open Problems and Conjectures

163

(4)

(3, 2)-Type Equations

167

(16)

Introduction

167

(1)

The Case C = 0 :

167

(5)

Proof of Part (b) of Theorem 10.2.1

169

(3)

Proof of Part (c) of Theorem 10.2.1

172

(1)

The Case B = 0 :

172

(3)

Invariant Intervals

173

(1)

Global Stability When p ≥ r

173

(1)

Convergence of Solutions

174

(1)

The Case A = 0 :

175

(5)

Invariant Intervals

177

(2)

Convergence of Solutions

179

(1)

Open Problems and Conjectures

180

(3)

The (3, 3)-Type Equation

183

(18)

Linearized Stability Analysis

183

(1)

Invariant Intervals

184

(5)

Convergence Results

189

(3)

Global Stability

191

(1)

Open Problems and Conjectures

192

(9)

Global Attractivity for Higher Order Equations

201

(10)

Bibliography

211

(6)

Index

217

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Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures

Summary

Table of Contents

Supplemental Materials