Contents Preface Chapter 1 Introduction and Preliminaries 1 1.1 Derivation of the basic chemostat model 1 1.2 Brief review on the modified chemostat models 4 1.3 Preliminaries 7 1.4 Notes 14 Chapter 2 Competition in the Unstirred Chemostat 15 2.1 The model 15 2.2 Simplification 18 2.3 Single population growth and extinction 20 2.4 Coexistence 30 2.5 Notes 33 Chapter 3 Global Bifurcation of the Unstirred Chemostat Model 35 3.1 Introduction 35 3.2 Semitrivial steady-state solutions 37 3.3 Global bifurcation 43 3.4 Stability 50 3.5 Notes 53 Chapter 4 Uniqueness and Stability for the Unstirred Chemostat Model 54 4.1 Preliminaries 55 4.2 Bifurcation from a double eigenvalue 61 4.3 Uniqueness and stability 66 4.4 Notes 81 Chapter 5 Competition between Two Similar Species 82 5.1 Introduction 82 5.2 Stability of semitrivial equilibria 87 5.3 Lyapunov-Schmidt reduction 89 5.4 Stability of coexistence solutions 95 5.5 Dynamical behavior 99 5.6 Discussion 102 Chapter 6 The Unstirred Chemostat Model with Di.erent Di.usion Rates 103 6.1 The model and main results 104 6.2 The global attractor 109？ 6.3 Equilibria 118 6.4 Uniqueness of single population equilibria 125 6.5 Notes 128 Chapter 7 Crowding E.ects on Coexistence Solutions 129 7.1 The model and main results 129 7.2 Single population model 133 7.3 Existence of positive solutions 144 7.4 Bifurcation from a simple eigenvalue 152 7.5 Bifurcation from a double eigenvalue 157 7.6 Crowding e.ects of species 163 7.7 Discussion 165 Chapter 8 The Unstirred Chemostat Model with Two Substitutable Resources 166 8.1 The model 167 8.2 The special case of m = n 169 8.3 Positive solution for general (m; n) 172 8.4 Longtime behavior of the limiting system 180 8.5 Notes 186 Chapter 9 Competition for Two Complementary Resources in the Unstirred Chemostat 187 9.1 Single population model 188 9.2 Two-species model with same di.usion rate 192 9.3 Two-species model with di.erent di.usion rates 203 9.4 Notes 218 Chapter 10 E.ects of Inhibitor in the Unstirred Chemostat 220 10.1 The model 221 10.2 Existence of coexistence solutions 224 10.3 The e.ect of inhibitor 230 10.4 Numerical simulations 264 10.5 Notes 269 Bibliography 270
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