Introduction

The previous two chapters showed that competitive exclusion holds under a variety of conditions in the chemostat and its modifications. In this chapter it will be shown that if the competition is moved up one level – if the competition occurs among predators of an organism growing on the nutrient – then coexistence may occur. The fact that the competitors are at a higher trophic level allows for oscillations, and the coexistence that occurs is in the form of a stable limit cycle. Along the way it will be necessary to study a three-level food-chain problem which is of interest in its own right; it is the chemostat version of predator–prey equations. The presentation follows that of [BHW1].

Although most of the results can be established with mathematical rigor, there are some elusive problems. These center around the possibility of multiple limit cycles and the difficulty of determining the stability of such limit cycles. At this point one must simply make a hypothesis and resort to numerical evidence in any specific case. Determining the number of limit cycles is a deep mathematical problem, and even in very simple cases the solution is not known. Hilbert's famous sixteenth problem, concerning the number of limit cycles of a second-order system with polynomial right-hand sides, remains basically unresolved. In principle, the stability of a limit cycle can be determined from the Floquet exponents (see Section 4), but this is a notoriously difficult computation – indeed, generally an impossible one.

Introduction

The models considered in previous chapters of this monograph have ignored the fact that populations of microorganisms contain individuals with differing body size and that individuals of different size have different characteristics. Body size is clearly an important factor in determining an organism's energy requirements and its ability to uptake resources. Furthermore, if an organism can grow as well as reproduce, then it becomes important to determine how the organism allocates its energy resources between growth and reproduction.

Data from [Wi] – particularly figures 3, 6, 8, 10, 18, 19, and 21 – leave no doubt that, at least for certain populations of algae, individual cell volume varies significantly during the course of experiments in the chemostat. These data also suggest that steady-state size distributions are reached which have remarkably stable shapes with respect to changes in the control parameters for the chemostat (flow rates, temperature, CO2).

The purpose of this chapter is to present a model of competition in the chemostat for a single resource that takes these factors into account, and to determine the extent to which these factors influence the outcome of competition. The model presented here is a special case of a more general class of models formulated in [DMKH] and treated in [MD, sec. 1.3]. This simplified model is one of several considered in an elegant paper by Cushing [Cu2]. Most of the results of this chapter are taken from [Cu2].

Introduction

In the previous chapter it was shown that the simple chemostat produces competitive exclusion. It could be argued that the result was due to the two-dimensional nature of the limiting problem (and the applicability of the Poincaré–Bendixson theorem) or that this was a result of the particular type of dynamics produced by the Michaelis–Menten hypothesis on the functional response. This last point was the focus of some controversy at one time, inducing the proposal of alternative responses. In this chapter it will be shown that neither additional populations nor the replacement of the Michaelis–Menten hypothesis by a monotone (or even nonmonotone) uptake function is sufficient to produce coexistence of the competitors in a chemostat. This illustrates the robustness of the results of Chapter 1. It will also be shown that the introduction of differing “death rates” (replacing the parameter D by Di in the equations) does not change the competitive exclusion result.

A brief mathematical digression on Liapunov stability theory will set the stage for the results of this chapter. Those familiar with the LaSalle corollary to Liapunov stability theory, also called the “invariance principle” by some authors, can skip immediately to Section 3.

Liapunov Theory

A more sophisticated mathematical approach is required for the results of this chapter because the equations considered cannot be reduced to planar systems. The theorem described in this section and used throughout this chapter is one of great power.

The chemostat is a basic piece of laboratory apparatus, yet it occupies a central place in mathematical ecology. Its importance stems from the many roles it plays. It is a model of a simple lake, the ideal place to study competition in its most primitive form – exploitative competition. It is also used as a model of the wastewater treatment process. In its commercial form the chemostat plays a central role in certain fermentation processes, particularly in the commercial production of products by genetically altered organisms (e.g., in the production of insulin). The theoretical literature is scattered; papers appear in mathematical, biological, and chemical engineering journals. In addition to being known as a chemostat, other common names are “continuous culture” and, in the engineering literature, CSTR (continuously stirred tank reactor). This monograph is devoted to the theoretical description of ecological models based on the chemostat and to problems that can be described in that context, drawing from literature in all fields and presented within a common framework and with consistent notation.

In order to understand the mathematical importance of the chemostat, one must look at the broader picture of the subject of nonlinear differential equations. Linear differential equations have been studied for more than two hundred years; their solutions have a rich structure that has been well worked out and exploited in physics, chemistry, and biology. A vast and challenging new world opens up when one turns to nonlinear differential equations.

Introduction

In the previous chapter the gradostat was introduced as a model of competition along a nutrient gradient. The case of two competitors and two vessels with Michaelis–Menten uptake functions was explored in considerable detail. In this chapter the restriction to two vessels and to Michaelis–Menten uptake will be removed, and a much more general version of the gradostat will be introduced. The results in the previous chapter were obtained by a mixture of dynamical systems techniques and specific computations that established the uniqueness and stability of the coexistence rest point. When the number of vessels is increased and the restriction to Michaelis–Menten uptake functions is relaxed, these computations are inconclusive. It turns out that unstable positive rest points are possible and that non-uniqueness of the coexistence rest point cannot be excluded. The main result of this chapter is that coexistence of two microbial populations in a gradostat is possible in the sense that the concentration of each population in each vessel approaches a positive equilibrium value. The main difference with the previous chapter is that we cannot exclude the possibility of more than one coexistence rest point.

Throughout this chapter we rely extensively on the results contained in Appendices A, B, and C. The presentation here follows closely that in [STW] (see also the review [SW2]).

The most straightforward generalization of the work of Chapter 5 would be simply to extend the number of vessels from two to an arbitrary number, say n.

Introduction

In the preceding chapter we saw how the chemostat could be modified to account for a new phenomenon – the presence of an inhibitor. In this chapter we extend the idea behind the simple chemostat to a new apparatus in order to model a property of ecological systems that it is not possible to model in the simple chemostat. The idea is to capture the essentials of the new phenomenon without destroying the tractability of the chemostat either as a mathematical model or as an experimental one. A very simple situation will be described here; a more complicated one – with a less explicit (in the sense of less computable) analysis – will be discussed in the next chapter. Just as the chemostat is a basic model for competition in the simplest situation, the apparatus here shows promise of being the model for competition along a nutrient gradient.

The “well mixed” hypothesis for the chemostat does not allow a nutrient gradient to be generated. A basic tenet is that the nutrient concentration is the same everywhere; hence any advantage in nutrient consumption is present everywhere. The model that incorporates a true gradient would be one involving partial differential equations; a new variable, space, must be accommodated. Systems of nonlinear partial differential equations are difficult mathematical objects to understand and analyze. Even numerical solutions pose added and significant difficulties. Moreover, even if an experimental gradient is devised, measurements that do not disturb the local environment take on new difficulties.

Introduction

In the first two chapters the general theory of the chemostat was developed, and it was shown that competitive exclusion is the expected outcome. In Chapter 3, coexistence was shown to occur when the competition was at a higher trophic level; the mechanism was simply the oscillation of the object of the competition – the prey in the case being considered. In this chapter, we return to the basic chemostat model but add another factor, the presence of an inhibitor. The inhibitor affects the nutrient uptake rate of one of the competitors but is taken up by the other without ill effect. The use of Nalidixic acid in the experiments of Hansen and Hubbell [HH], discussed in Chapter 1, is an example. Its effect on one strain of E. coli was essentially nil while the growth rate of the other was severely diminished.

The interest in this subject goes far beyond laboratory examples. It is common for one strain of bacteria to be affected by an antibiotic while another is resistant. The antibiotic acts as the inhibitor in our discussion, and we want to know whether the resistant strain will eliminate the nonresistant strain. If it can, the antibiotic will not be effective in treating the disease. In commercial applications, genetically altered organisms are used to generate “products”.