The theory of the chemostat. Dynamics of microbial competition.

*(English)*Zbl 0860.92031
Cambridge Studies in Mathematical Biology. 13. Cambridge: Cambridge Univ. Press. xvi, 313 p. (1995).

Probably, the chemostat is one of the best idealizations of nature for the purpose of population dynamics studies. In the monograph under review the basic literature on competition in the chemostat is collected and explained from a common theoretical viewpoint. The chemostat models an open system. Although the exact assumptions of that model may be limited to laboratory environments, it can serve as a paradigm for more complicated naturally occuring systems. The chemostat is mathematically modelled by nonlinear differential equations, and the common feature of all mathematical results of this book represents the description of the global behaviour of their solutions.

In chapter 1 the authors explain the idea of dynamical systems and introduce the basic apparatus. Then it is shown that the mathematical results for the simple chemostat hold in much greater generality (chapter 2). The next seven chapters deal with specific theoretical problems that can be treated in a chemostat setting (chemostat version of the predator-prey problem, competition in the chemostat when an inhibitor is present, the gradostat, time-varying environment, internal-stores or variable-yield model, structured models taking into account the size of the population). The last two chapters lead the reader to new directions and open problems of the theory.

One of the authors’ objectives is to show that much of the theory of chemostat-like models is based on a rigorous mathematical foundation. However, the book is written in a style that a reader who is uninterested in the mathematical fine points may only read the discussion and the statements of the theorems while skipping the proofs. The monograph is in principle mathematically self-contained. In the reviewer’s oppinion the book is very well-written. The mathematics is handled precisely and nevertheless well explained also from the idea side. The whole book is an enrichment for everybody working in dynamical systems or population dynamics.

In chapter 1 the authors explain the idea of dynamical systems and introduce the basic apparatus. Then it is shown that the mathematical results for the simple chemostat hold in much greater generality (chapter 2). The next seven chapters deal with specific theoretical problems that can be treated in a chemostat setting (chemostat version of the predator-prey problem, competition in the chemostat when an inhibitor is present, the gradostat, time-varying environment, internal-stores or variable-yield model, structured models taking into account the size of the population). The last two chapters lead the reader to new directions and open problems of the theory.

One of the authors’ objectives is to show that much of the theory of chemostat-like models is based on a rigorous mathematical foundation. However, the book is written in a style that a reader who is uninterested in the mathematical fine points may only read the discussion and the statements of the theorems while skipping the proofs. The monograph is in principle mathematically self-contained. In the reviewer’s oppinion the book is very well-written. The mathematics is handled precisely and nevertheless well explained also from the idea side. The whole book is an enrichment for everybody working in dynamical systems or population dynamics.

Reviewer: R.Manthey (Jena)

##### MSC:

92D40 | Ecology |

92-02 | Research exposition (monographs, survey articles) pertaining to biology |

92D25 | Population dynamics (general) |

37-XX | Dynamical systems and ergodic theory |