The chemostat is an experimental device invented in the 1950s, almost simultaneously, by Jacques Monod [MON 50] on the one hand, and by Aaron Novick and Leo Szilard on the other hand [NOV 50]. In his seminal article, Monod presented both chemostat equations and an example of an experimental device that operates continuously with the aim of controlling microbial growth by interacting with the inflow rate. Novick and Szilard, for their part, proposed a simpler experimental device, one of the technical difficulties at the time being to design a system capable of delivering a constant supply to a small volume reactor. Originally named “bactogène” by Monod, Novick and Szilard are the ones who propose the name chemostat for chemical [environment] is static. It is used to study microorganisms and especially their growth characteristics on a so-called “limiting” substrate. The other resources essential to their development and reproduction are assumed to be present in excess inside the reactor. It comprises an enclosure containing the reaction volume, an inlet that enables resources to be fed into the system and an outlet through which all components are withdrawn. This device presents two main characteristics: its content is assumed to be perfectly homogeneous and its volume is kept constant by the use of appropriate technical devices making it possible to maintain continuous and identical in and outflow rates. Its reputation is mainly due to the fact that it is capable of fixing the growth rate of the microorganisms that it contains at equilibrium by means of manipulating the inflow supply. First used by microbiologists to study the growth of a given species of microorganisms (referred to as “pure culture”), its usage greatly diversified over time. In the 1960s, it became a standard tool for microbiologists to study relationships between growth and environment parameters. In the 1970–1980s, it would become the focus of a strong interest in mathematical ecology even though it was somewhat neglected by microbiologists. This was mainly because at the time, the attention of the latter was attracted by the development of molecular biology approaches for the monitoring and understanding of microbial ecosystems. Studies on the competition of microorganisms rekindled interest among researchers for the chemostat in the 1980s, especially in the field of microbial ecology. It is not until the 2000s and the advent of the postgenomic era, which requires knowledge and fine control of reaction media, that a renewed interest was really observed for this device among microbiologists. It is used nowadays in scientific areas related to the acquisition of knowledge that is both fundamental, such as ecology or evolutionary biology, and applied such as water treatment, biomass energy recovery and biotechnologies in a broader sense.

The chemostat has not only been the subject of numerous publications, but also of several books essential in the field of mathematics. A question can be legitimately raised about what additional work can be done concerning a device that ultimately is very simple in principle. To this question, we can provide the several following answers.

The main source of uncertainty when a biological process is being modeled lies in modeling the growth rate of microorganisms. Bearing in mind that practitioners’ concerns must be addressed, it is based on this fundamental question and from an applied point of view that we have built this book. We do realize that the analytic expression of a growth rate is only an approximation thereof and that the properties of the model should not depend on this expression; that is the reason why we will introduce general models involving different growth models which we will subsequently specify. In particular, we study the influence of the type of growth function being considered on the outcome of a competition between several species. Adopting an increasingly complex approach, and after a general introduction which is covered in the first chapter, the second chapter naturally focuses on the growth of a single species of microorganisms on a resource. The properties of this model are analyzed for the three most important classes of growth rate encountered in biotechnology, namely, those limited and/or inhibited by substrate and those known as density-dependent, in which the growth rate does not depend on the resource only, but also on the density of the existing microorganisms. This first situation becomes more complex in Chapter 3 where we address the case of several species competing on a single resource, when growth rates are resource-dependent only. In particular, the competitive exclusion principle at equilibrium is therein exposed, of which demonstrations are given relating to the existence and the local as well as global stability of the equilibria of the system. The fourth chapter specifically addresses the case of several species competing and coexisting on a resource when growth rates are density-dependent. With the study of this type of model being significantly more complex from the mathematical perspective, it emerges that resorting to numerical simulation tools is a means to bring forward the diversity of the situations encountered. Finally, the final chapter addresses models enabling the consideration of the spatial structuring of microorganisms into several classes, here both in planktonic form and flocs or in biofilms.

We restrict our observations to situations in which a single limiting resource is considered. In a real situation, it is obvious that this will not be the case. Nonetheless, the reasoning needed for the study of these more complex situations will always be carried out based on the tools that we are introducing in this book. This publication is written to allow a linear reading in the order of the chapters that constitute it. However, according to the degree of detail, we propose several times that the reader overlooks certain passages, bearing in mind the possibility of returning to them later on, without confusing the progression during reading.

This book is above all dedicated to engineering students and PhD students wishing to study the techniques for the analysis of dynamic systems related to biological systems used in biotechnology and, in particular, to the chemostat (homogeneous system continuously operating). The primary concern is to address the challenge of studying the qualitative properties of a model already available and dedicated to formalizing a situation of interest. The question of confronting this model to data falls outside the scope of this book. We hope that the educational efforts achieved can make its reading accessible to the greatest number of people, including biotechnology students and not only mathematicians. In particular, important techniques are specifically detailed, whereas the elements requiring more significant developments or secondary significance are proposed as exercises. Furthermore, their solutions are included at the end of the book.

A rather significant appendix (Appendix 1) is dedicated to the theory of differential equations. Strictly speaking, this is not a course but a refresher of the principal notions and results to which we will refer. The reader equipped with the knowledge of a preparatory scientific class or a Bachelor of Science should be able to follow. The book contains a very large number of figures that most often will benefit the student when viewed enlarged (when reading the electronic version).