了得網自然科學_非均勻恆化器模型的動力學（英文版）

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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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Chapter 1
Introduction and Preliminaries
The chemostat is a common model in microbial ecology. It is used as an ecological model of a simple lake， as a model of waste water treatment， and as a model for com- mercial production of fermentation processes. Mathematical models of the chemostat are surprisingly amenable to analysis. Predictions based on parameters in the model that can be measured have been tested experimentally and outcomes have been shown to agree rather well with the theory.
The models of exploitative competition in a well-stirred chemostat operated under constant input and dilution， with competition for a non-reproducing substrate， predict competitive exclusion. That is， they predict that at most one competitor population avoids extinction.However， the coexistence of competing populations is obvious in nature. This paradox indicates that the mathematical and experimental approaches had been dangerously over simplified. In order to describe the phenomena of the chemostat precisely， various mathematical models have been developed and analyzed extensively by many di.erent investigators (see， for example， the recent monograph by Smith and Waltman and the references therein).
The main purpose of this chapter is devoted to review some known results on the well-stirred chemostat model and to provide some basic theorems as preliminaries. In Section 1.1， we review the derivation of the basic chemostat model and its competitive exclusion principle， which makes us modify the basic well-stirred chemostat model to produce coexistence. Section 1.2 presents a brief review on the study of a variety of modified models in the well-stirred chemostat. In Section 1.3， some properties of the eigenvalue， the bifurcation theory and the topological fixed point theory are provided as preliminaries.
1.1 Derivation of the basic chemostat model
The basic chemostat consists of three vessels connected by pumps as shown in the schematic diagram Fig.1-1. The first is called the feed vessel and contains all of the nutrients needed for the growth of microorganisms in abundance except one which is limiting， and is referred to as the limiting nutrient. The contents of the feed vessel is pumped at a constant rate into the second vessel， called the culture vessel or bioreactor， which is charged with one or more species of microorganisms. The volume of the culture vessel is kept constant by pumping the contents of culture vessel out at the same rate to the third vessel， called the outflow vessel. It is here that the products of the culture vessel are collected. It is assumed that the culture vessel is well mixed and that all other relevant variables (temperature， pH， etc.) are kept constant. Note that measurements can be made on the contents of the collection vessel without disturbing the action in the culture vessel. Since some nutrient is always in the shortest supply， we focus on that limiting nutrient， hereafter simply called the nutrient， and ignore the others that are present in surplus quantities. We emphasize that Fig.1-1 is a schematic diagram; the actual realization of the device can take many forms.
Fig. 1-1 A schematic diagram of the basic chemostat
A derivation of the basic chemostat model can be found in almost any bioengi- neering literature， for example [2， 5， 7-9]. We give here a brief description and the reader is referred to one of the above references for a more detailed description. Let S(t) denote the concentration of the nutrient in the reaction vessel at time t， F be the flow rate (determined by the pump speed)， V be the volume of the reaction vessel and define the parameter D， called the dilution rate， by D = F=V . The concentration of the nutrient in the feed bottle denoted by S0， is kept constant. If there were no microorganisms， the rate of change of the concentration of the nutrient in the reaction vessel would be given by
the simple statement that change in concentration is proportional to the di.erence between the incoming concentration and the resident concentration. If organisms are consuming the nutrient， then this needs to be corrected for the consumption and the consumed nutrient converted to the growth of the organism. A basic assumption is that growth is proportional to consumption. Based on experimental evidence， nutrient uptake (consumption) is usually taken to be of the Monod (or Michaelis-Menten) form where m is called the maximal growth rate and a is called the Michaelis-Menten constant. Thus， if x(t) denotes the concentration of a microorganism at time t， the differential equations take the form.
The constant r is a yield constant and represents the conversion of nutrient to organism. S0 and D are controlled by the experimenter and can be thought of as environmental variables while m; a， and r are properties of the organism， to be measured in the laboratory. There is also an underlying assumption that all other e.ects are control

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