Contents Preface Main Symbols Chapter 1 Preliminaries-Basic Theory 1 1.1 Maximum Principle and Upper-Lower Solution Method of Partial Differential Equations of Second Order 1 1.2 Eigenvalue Problems and Variational Principle 4 1.3 Theory of Fixed Point Index and Topological Degree in Banach Spaces 5 1.4 Theory of Bifurcation and Stability in Banach Spaces 6 Chapter 2 Autocatalytic Reaction-Diffusion Chemical System in Thermodynamics 9 2.1 Introduction 9 2.2 Characteristic on Coexistent States 12 2.2.1 A Priori Estimate 12 2.2.2 Fundamental Properties 14 2.3 Stability of Constant Coexistent State 19 2.4 Non-Constant Coexistent States 22 2.4.1 Nonexistence 22 2.4.2 Existence 25 2.5 Existence and Uniqueness of Bifurcation Emanating from Constant Coexistence 29 2.5.1 Take the Concentration of Reaction Precursor as the Parameter 29 2.5.2 Take the Diffusion Rate of Reactant or Autocatalyst as the Parameter 36 2.6 Numerical Simulations 40 2.7 Notes 40 Chapter 3 Predator-Stage Structured Reaction-Diffusion Biosystem 43 3.1 The System 43 3.2 A Priori Estimate 46 3.3 Stability 48 3.4 Existence of Non-Constant Coexistent States 53 3.5 Analysis on Hopf Bifurcation 61 3.6 Notes 64 Chapter 4 Periodic Reaction-Diffusion Biosystems of Multi-Species 65 4.1 Competitive Type 65 4.1.1 Introduction 65 4.1.2 A Priori Estimate 67 4.1.3 Asymptotic Behavior 70 4.2 Cooperative Type 79 4.2.1 Introduction 79 4.2.2 A Priori Estimate and Existence of Coexistent States 81 4.2.3 Steady-State System 89 4.3 An Example-Two Species System with Nonlinear Functional Response 96 4.3.1 The System and Preliminaries 96 4.3.2 Existence, Uniqueness and Nonexistence of Coexistent States 97 4.3.3 Numerical Simulations 103 4.4 Notes 105 Chapter 5 Prey-Refuge Presented Reaction-Diffusion Biosystem 107 5.1 The System 107 5.2 Effects of Refuge on Stability 109 5.3 Existence of Predator and Prey with Refuge Consideration 110 5.4 Notes 126 Chapter 6 Reaction-Diffusion Competing Ecosystem with Harvesting and Toxicity 129 6.1 The System 129 6.2 Stability Analysis 130 6.3 Bifurcation Emanating from Double Eigenvalue 135 6.4 Multiplicity of Non-Constant Coexistent States 136 6.5 Coexistent States with Biologic and Economic Considerations 138 6.6 Numerical Simulations 140 6.7 Notes 141 Chapter 7 Reaction-Diffusion Marine Ecosystem of Phytoplankton-Nutrient 142 7.1 Introduction 142 7.2 Long-Time Behavior 144 7.3 Stability 147 7.4 A Priori Estimate 149 7.5 Patterns of Non-Constant Coexistent States 153 7.6 Notes 158 Chapter 8 Reaction-Diffusion Predator-Prey Population System with Fractional Response 159 8.1 The System 159 8.2 Global Attractor and Persistence 160 8.3 Stability of Constant Coexistent State 163 8.4 Bifurcation Emanating from Constant Coexistent State 164 8.5 Existence of Coexistent States 172 8.6 Notes 177 Chapter 9 Population Systems with Cross-Diffusion 178 9.1 Competitive Type 178 9.1.1 The System 178 9.1.2 Stability 180 9.1.3 Boundedness 181 9.1.4 Non-Existence and Existence of Coexistent States 183 9.2 Predator-Prey Type 188 9.2.1 The System 189 9.2.2 A Priori Estimate 191 9.2.3 Stability 193 9.2.4 Existence of Non-Constant Coexistent States 195 9.3 Notes 201 Chapter 10 Global Analysis for Reaction-Diffusion Biological and Biochemical Systems 202 10.1 Ecological System with Non-Selective Harvesting 202 10.1.1 The System 202 10.1.2 Local Bifurcation and Stability 205 10.1.3 Existence of Global Bifurcation 209 10.1.4 Notes 215 10.2 Biochemical Reaction System with Oncolytic Virus 216 10.2.1 Introduction 216 10.2.2 A priori Estimate and Stability 218 10.2.3 Global Bifurcation Analysis 220 10.2.4 Notes 228 Bibliography 229
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