Interactions of Tetrahymena pyriformis, Escherichia coli, Azotobacter vinelandii, and glucose in a minimal medium

The role of mathematical modelling in understanding prokaryotic predation

With increasing levels of antimicrobial resistance impacting both human and animal health, novel means of treating resistant infections are urgently needed. Bacteriophages and predatory bacteria such as Bdellovibrio bacteriovorus have been proposed as suitable candidates for this role. Microbes also play a key environmental role as producers or recyclers of nutrients such as carbon and nitrogen, and predators have the capacity to be keystone species within microbial communities. To date, many studies have looked at the mechanisms of action of prokaryotic predators, their safety in in vivo models and their role and effectiveness under specific conditions. Mathematical models however allow researchers to investigate a wider range of scenarios, including aspects of predation that would be difficult, expensive, or time-consuming to investigate experimentally. We review here a history of modelling in prokaryote predation, from simple Lotka-Volterra models, through increasing levels of complexity, including multiple prey and predator species, and environmental and spatial factors. We consider how models have helped address questions around the mechanisms of action of predators and have allowed researchers to make predictions of the dynamics of predator-prey systems. We examine what models can tell us about qualitative and quantitative commonalities or differences between bacterial predators and bacteriophage or protists. We also highlight how models can address real-world situations such as the likely effectiveness of predators in removing prey species and their potential effects in shaping ecosystems. Finally, we look at research questions that are still to be addressed where models could be of benefit.

Determination of phage load and administration time in simulated occurrences of antibacterial treatments

The use of phages as antibacterials is becoming more and more common in Western countries. However, a successful phage-derived antibacterial treatment needs to account for additional features such as the loss of infective virions and the multiplication of the hosts. The parameters critical inoculation size (V F ) and failure threshold time (T F ) have been introduced to assure that the viral dose (V ϕ) and administration time (T ϕ) would lead to the extinction of the targeted bacteria. The problem with the definition of V F and T F is that they are non-linear equations with two unknowns; thus, obtaining their explicit values is cumbersome and not unique. The current study used machine learning to determine V F and T F for an effective antibacterial treatment. Within these ranges, a Pareto optimal solution of a multi-criterial optimization problem (MCOP) provided a pair of V ϕ and T ϕ to facilitate the user's work. The algorithm was tested on a series of in silico microbial consortia that described the outgrowth of a species at high cell density by another species initially present at low concentration. The results demonstrated that the MCOP-derived pairs of V ϕ and T ϕ could effectively wipe out the bacterial target within the context of the simulation. The present study also introduced the concept of mediated phage therapy, where targeting booster bacteria might decrease the virulence of a pathogen immune to phagial infection and highlighted the importance of microbial competition in attaining a successful antibacterial treatment. In summary, the present work developed a novel method for investigating phage/bacteria interactions that can help increase the effectiveness of the application of phages as antibacterials and ease the work of microbiologists.

A computational framework for finding parameter sets associated with chaotic dynamics

Many biological ecosystems exhibit chaotic behavior, demonstrated either analytically using parameter choices in an associated dynamical systems model or empirically through analysis of experimental data. In this paper, we use existing software tools (COPASI, R) to explore dynamical systems and uncover regions with positive Lyapunov exponents where thus chaos exists. We evaluate the ability of the software's optimization algorithms to find these positive values with several dynamical systems used to model biological populations. The algorithms have been able to identify parameter sets which lead to positive Lyapunov exponents, even when those exponents lie in regions with small support. For one of the examined systems, we observed that positive Lyapunov exponents were not uncovered when executing a search over the parameter space with small spacings between values of the independent variables.

Higher-Order Interaction between Species Inhibits Bacterial Invasion of a Phototroph-Predator Microbial Community

The composition of an ecosystem is thought to be important for determining its resistance to invasion. Studies of natural ecosystems, from plant to microbial communities, have found that more diverse communities are more resistant to invasion. In some cases, more diverse communities resist invasion by more completely consuming the resources necessary for the invader. We show that Escherichia coli can successfully invade cultures of the alga Chlamydomonas reinhardtii (phototroph) or the ciliate Tetrahymena thermophila (predator) but cannot invade a community where both are present. The invasion resistance of the algae-ciliate community arises from a higher-order interaction between species (interaction modification) that is unrelated to resource consumption. We show that the mode of this interaction is the algal inhibition of bacterial aggregation, which leaves bacteria vulnerable to predation. This mode requires both the algae and the ciliate to be present and provides an example of invasion resistance through an interaction modification.

Synthetic microbial ecology and the dynamic interplay between microbial genotypes

Assemblages of microbial genotypes growing together can display surprisingly complex and unexpected dynamics and result in community-level functions and behaviors that are not readily expected from analyzing each genotype in isolation. This complexity has, at least in part, inspired a discipline of synthetic microbial ecology. Synthetic microbial ecology focuses on designing, building and analyzing the dynamic behavior of ‘ecological circuits’ (i.e. a set of interacting microbial genotypes) and understanding how community-level properties emerge as a consequence of those interactions. In this review, we discuss typical objectives of synthetic microbial ecology and the main advantages and rationales of using synthetic microbial assemblages. We then summarize recent findings of current synthetic microbial ecology investigations. In particular, we focus on the causes and consequences of the interplay between different microbial genotypes and illustrate how simple interactions can create complex dynamics and promote unexpected community-level properties. We finally propose that distinguishing between active and passive interactions and accounting for the pervasiveness of competition can improve existing frameworks for designing and predicting the dynamics of microbial assemblages.

Competition between a nonallelopathic phytoplankton and an allelopathic phytoplankton species under predation

We propose a model of two-species competition in the chemostat for a single growth-limiting, nonreproducing resource that extends that of Roy [38]. The response functions are specified to be Michaelis-Menten, and there is no predation in Roy's work. Our model generalizes Roy's model to general uptake functions. The competition is exploitative so that species compete by decreasing the common pool of resources. The model also allows allelopathic effects of one toxin-producing species, both on itself (autotoxicity) and on its nontoxic competitor (phytotoxicity). We show that a stable coexistence equilibrium exists as long as (a) there are allelopathic effects and (b) the input nutrient concentration is above a critical value. The model is reconsidered under instantaneous nutrient recycling. We further extend this work to include a zooplankton species as a fourth interacting component to study the impact of predation on the ecosystem. The zooplankton species is allowed to feed only on the two phytoplankton species which are its perfectly substitutable resources. Each of the models is analyzed for boundedness, equilibria, stability, and uniform persistence (or permanence). Each model structure fits very well with some harmful algal bloom observations where the phytoplankton assemblage can be envisioned in two compartments, toxin producing and non-toxic. The Prymnesium parvum literature, where the suppressing effects of allelochemicals are quite pronounced, is a classic example. This work advances knowledge in an area of research becoming ever more important, which is understanding the functioning of allelopathy in food webs.

Dynamics of two phytoplankton populations under predation

The aim of this paper is to investigate the manner in which predation and single-nutrient competition affect the dynamics of a non-toxic and a toxic phytoplankton species in a homogeneous environment (such as a chemostat). We allow for the possibility that both species serve as prey for an herbivorous zooplankton species. We assume that the toxic phytoplankton species produces toxins that affect only its own growth (autotoxicity). The autotoxicity assumption is ecologically explained by the fact that the toxin-producing phytoplankton is not mature enough to produce toxins that will affect the growth of its nontoxic competitor. We show that, in the absence of phytotoxic interactions and nutrient recycling, our model exhibits uniform persistence. The removal rates are distinct and we use general response functions. Finally, numerical simulations are carried out to show consistency with theoretical analysis. Our model has similarities with other food-chain models. As such, our results may be relevant to a wider spectrum of population models, not just those focused on plankton. Some open problems are discussed at the end of this paper.

Unraveling the contribution of pancreatic beta-cell suicide in autoimmune type 1 diabetes

In type 1 diabetes, an autoimmune disease mediated by autoreactive T-cells that attack insulin-secreting pancreatic beta-cells, it has been suggested that disease progression may additionally require protective mechanisms in the target tissue to impede such auto-destructive mechanisms. We hypothesize that the autoimmune attack against beta-cells causes endoplasmic reticulum stress by forcing the remaining beta-cells to synthesize and secrete defective insulin. To rescue beta-cell from the endoplasmic reticulum stress, beta-cells activate the unfolded protein response to restore protein homeostasis and normal insulin synthesis. Here we investigate the compensatory role of unfolded protein response by developing a multi-state model of type 1 diabetes that takes into account beta-cell destruction caused by pathogenic autoreactive T-cells and apoptosis triggered by endoplasmic reticulum stress. We discuss the mechanism of unfolded protein response activation and how it counters beta-cell extinction caused by an autoimmune attack and/or irreversible damage by endoplasmic reticulum stress. Our results reveal important insights about the balance between beta-cell destruction by autoimmune attack (beta-cell homicide) and beta-cell apoptosis by endoplasmic reticulum stress (beta-cell suicide). It also provides an explanation as to why the unfolded protein response may not be a successful therapeutic target to treat type 1 diabetes.

Shiga toxin as a bacterial defense against a eukaryotic predator, Tetrahymena thermophila

Bacterially derived exotoxins kill eukaryotic cells by inactivating factors and/or pathways that are universally conserved among eukaryotic organisms. The genes that encode these exotoxins are commonly found in bacterial viruses (bacteriophages). In the context of mammals, these toxins cause diseases ranging from cholera to diphtheria to enterohemorrhagic diarrhea. Phage-carried exotoxin genes are widespread in the environment and are found with unexpectedly high frequency in regions lacking the presumed mammalian "targets," suggesting that mammals are not the primary targets of these exotoxins. We suggest that such exotoxins may have evolved for the purpose of bacterial antipredator defense. We show here that Tetrahymena thermophila, a bacterivorous predator, is killed when cocultured with bacteria bearing a Shiga toxin (Stx)-encoding temperate bacteriophage. In cocultures with Tetrahymena, the Stx-encoding bacteria display a growth advantage over those that do not produce Stx. Tetrahymena is also killed by purified Stx. Disruption of the gene encoding the StxB subunit or addition of an excess of the nontoxic StxB subunit substantially reduced Stx holotoxin toxicity, suggesting that this subunit mediates intake and/or trafficking of Stx by Tetrahymena. Bacterially mediated Tetrahymena killing was blocked by mutations that prevented the bacterial SOS response (recA mutations) or by enzymes that breakdown H(2)O(2) (catalase), suggesting that the production of H(2)O(2) by Tetrahymena signals its presence to the bacteria, leading to bacteriophage induction and production of Stx.

Complex dynamics of microbial competition in the gradostat

We examine the conditions necessary for the emergence of complex dynamic behavior in systems of microbial competition. In particular, we study the effect of spatial heterogeneity and substrate-inhibition on the dynamics of such a system. This is accomplished through the study of a mathematical model of two microbial populations competing for a single nutrient in a configuration of two interconnected chemostats. Microbial growth is assumed to follow substrate-inhibited kinetics for both species. Such a system with sterile feed has been shown in a previous work to exhibit stable periodic states. In the present work we study the system for the case of non-sterile feed, i.e., when the two species are present in the feed of the chemostats. The analysis is done by numerical bifurcation theory methods. We demonstrate that, in addition to periodic states, the system possesses stable quasi-periodic states resulting from Neimark-Sacker bifurcations of limit cycles. Also, periodic states may undergo successive period doublings leading to periodic states of increasing period and indicating that chaotic states might be possible. Multistability is also observed, consisting in the coexistence of several stable steady states and possibly stable periodic or quasi-periodic states for given operating conditions. It appears that substrate-inhibition, spatial heterogeneity and presence of microorganisms in the inflow are all necessary conditions for complex dynamics to arise in a microbial system of pure and simple competition.

Is there really insufficient support for Tilman's R* concept? A comment on Miller et al

Miller et al. (2005), in the American Naturalist (165:439-448), critically reviewed the applicability of Tilman's resource-ratio hypothesis. One of their conclusions was that only eight experimental papers support the R* concept, while five do not. We are familiar with some of the latter studies, and we question this conclusion. Our evaluation shows that 12 of the 13 articles investigated by Miller et al. support R* prediction, while one article does not fit the experimental conditions for a proper test. Moreover, the microbial and aquatic literature contains many more competition experiments consistent with the R* prediction. We therefore conclude that there is strong experimental support for the R* concept, at least from studies with bacteria, phytoplankton, and zooplankton.

Colored-noise-induced Hopf bifurcations in predator-prey communities

A broad class of (N+1) -species ratio-dependent predator-prey stochastic models, which consist of one predator population and N prey populations, is considered. The effect of a fluctuating environment on the carrying capacities of prey populations is taken into account as colored noise. In the framework of the mean-field theory, approximate self-consistency equations for prey-populations mean density and for predator-population density are derived (to the first order in the noise variance). In some cases, the mean field exhibits Hopf bifurcations as a function of noise correlation time. The corresponding transitions are found to be reentrant, e.g., the periodic orbit appears above a critical value of the noise correlation time, but disappears again at a higher value of the noise correlation time. The nonmonotonous dependence of the critical control parameter on the noise correlation time is found, and the conditions for the occurrence of Hopf bifurcations are presented. Our results provide a possible scenario for environmental-fluctuations-induced transitions between the oscillatory regime and equilibrium state of population sizes observed in nature.

Experimental demonstration of chaos in a microbial food web

Discovering why natural population densities change over time and vary with location is a central goal of ecological and evolutional disciplines. The recognition that even simple ecological systems can undergo chaotic behaviour has made chaos a topic of considerable interest among theoretical ecologists. However, there is still a lack of experimental evidence that chaotic behaviour occurs in the real world of coexisting populations in multi-species systems. Here we study the dynamics of a defined predator-prey system consisting of a bacterivorous ciliate and two bacterial prey species. The bacterial species preferred by the ciliate was the superior competitor. Experimental conditions were kept constant with continuous cultivation in a one-stage chemostat. We show that the dynamic behaviour of such a two-prey, one-predator system includes chaotic behaviour, as well as stable limit cycles and coexistence at equilibrium. Changes in the population dynamics were triggered by changes in the dilution rates of the chemostat. The observed dynamics were verified by estimating the corresponding Lyapunov exponents. Such a defined microbial food web offers a new possibility for the experimental study of deterministic chaos in real biological systems.

Crossing the hopf bifurcation in a live predator-prey system

Population biologists have long been interested in the oscillations in population size displayed by many organisms in the field and laboratory. A wide range of deterministic mathematical models predict that these fluctuations can be generated internally by nonlinear interactions among species and, if correct, would provide important insights for understanding and predicting the dynamics of interacting populations. We studied the dynamical behavior of a two-species aquatic laboratory community encompassing the interactions between a demographically structured herbivore population, a primary producer, and a mineral resource, yet still amenable to description and parameterization using a mathematical model. The qualitative dynamical behavior of our experimental system, that is, cycles, equilibria, and extinction, is highly predictable by a simple nonlinear model.

Qualitative behavior of a variable-yield simple food chain with an inhibiting nutrient

A simple food chain which consists of nutrient, prey and predator in which nutrient is growth limiting at low concentrations but growth inhibiting at high concentrations is investigated in this study. It is assumed that the nutrient concentration is separated into internal and external nutrient concentration and only the internal nutrient level is capable of catalyzing cell growth. It is shown that the dynamics of the system depend on thresholds R(0) and R(1). With inhibition, there exist initial conditions for which the predator becomes extinct but not the prey when R(0)<1. If R(0),R(1)1, the system is uniformly persistent even in the inhibited environment.

Chaotic dynamics of a food web in a chemostat

We analyze a mathematical model of a simple food web consisting of one predator and two prey populations in a chemostat. Monod's model is employed for the dependence of the specific growth rates of the two prey populations on the concentration of the rate-limiting substrate and a generalization of Monod's model for the dependence of the specific growth rate of the predator on the concentrations of the prey populations. We use numerical bifurcation techniques to determine the effect of the operating conditions of the chemostat on the dynamics of the system and construct its operating diagram. Chaotic behavior resulting from successive period doublings is observed. Multistability phenomena of coexistence of steady and periodic states at the same operating conditions are also found.

Coexistence of three microbial populations competing for three complementary nutrients in a chemostat

We study a model of three microbial populations competing for three complementary nutrients in a single chemostat. By using methods of numerical bifurcation theory we analyze the model equations and determine the effect of the model parameters on the dynamics of the system. The main question to be answered is whether there exist conditions under which the three populations can coexist in a stable state in the chemostat. The analysis shows that coexistence can be obtained as a stable steady state but also as a stable periodic state for a wide range of operating conditions of the chemostat.

Computing operating diagrams of bioreactors

The operating diagram of a bioreactor is an illustrative way to present the effect that the operating conditions have on its long-term behavior. It can be constructed if a mathematical model of the bioreactor is available. The procedure for constructing the operating diagram consists in analyzing the dynamic behavior of the system of the differential equations of the model. Some methods are described that can be used in computing operating diagrams of bioreactors. They are based on numerical bifurcation techniques for systems of differential equations. Both cases of bioreactors with constant and periodically varying operating conditions are considered.

On the coexistence of three microbial populations competing for two complementary substrates in configurations of interconnected chemostats

We examine the question of coexistence of three microbial populations competing for two complementary rate-limiting substrates in configurations of interconnected chemostats. It is known that coexistence of two populations competing for two rate-limiting substrates is possible in a single chemostat, but coexistence is not possible when three populations are involved. We examine whether coexistence of three populations becomes possible by considering configurations of two or three interconnected chemostats, thus allowing for effects of spatial heterogeneity. Computational analysis of the model equations indicates that in the case of two chemostats coexistence is possible only for specific discrete parameter values where the system is structurally unstable and therefore the coexistence state is not practically attainable, whereas in the case of three chemostats coexistence is possible for a whole range of parameter values where the system is structurally stable and therefore the coexistence state can be realized in practice.

Oscillations of two competing microbial populations in configurations of two interconnected chemostats

It is known that, when two microbial populations competing for a single rate-limiting nutrient are grown in a spatially uniform environment, such as a single chemostat, with competition being the only interaction between them, they cannot coexist, but eventually one of the two populations prevails and the other becomes extinct. Spatial heterogeneity has been suggested as a means of obtaining coexistence of the two populations. A configuration of two interconnected chemostats is a simple model of a spatially heterogeneous environment. It has been shown that, when Monod's model is used for the specific growth rates of the two populations, steady-state coexistence can be obtained in such systems for wide ranges of operating conditions. In the present work, we study a model of microbial competition in configurations of interconnected chemostats and we show that, if a substrate inhibition model is used for the specific growth rates of the two populations, coexistence in a periodic state is also possible. The analysis of the model is done by numerical bifurcation theory methods.

A community model of ciliate Tetrahymena and bacteria E coli: Part II. interactions in a batch system

Premised on relatively simple assumptions, mathematical models like those of Monod, Pirt or Droop inadequately explain the complex transient behavior of microbial populations. In particular, these models fail to explain many aspects of the dynamics of a Tetrahymena pyriformis-Escherichia coli community. In this study an alternative approach, an individual-based model, is employed to investigate the growth and interactions of Tetrahymena pyriformis and E. coli in a batch culture. Due to improved representation of physiological processes, the model provides a better agreement with experimental data of bacterial density and ciliate biomass than previous modeling studies. It predicts a much larger coexistence domain than rudimentary models, dependence of biomass dynamics on initial conditions (bacteria to ciliate biomasses ratio) and appropriate timing of minimal bacteria density. Moreover, it is found that accumulation of E. coli sized particles and E. coli toxic metabolites has a stabilizing effect on the system.

Coexistence of three competing microbial populations in a chemostat with periodically varying dilution rate

Coexistence of three microbial populations engaged in pure and simple competition is not possible in a chemostat with time-invariant operating conditions under any circumstances. It is shown that by periodic variation of the chemostat dilution rate it is possible to obtain a stable coexistence state of all three populations in the chemostat. This is accomplished by performing a numerical bifurcation analysis of a mathematical model of the system and by determining its dynamic behavior with respect to its operating parameters. The coexistence state obtained in the periodically operated chemostat is usually periodic, but cases of quasi-periodic and chaotic behavior are also observed.

Microbial predation in coupled chemostats: a global study of two coupled nonlinear oscillators

Predator-prey systems in continuously operated chemostats exhibit sustained oscillations over a wide range of operating conditions. When two such chemostats interact through flow exchange, the interplay of the oscillation frequencies gives rise to a wealth of dynamic behavior patterns. Using numerical bifurcation techniques, we perform a detailed computational study of these patterns and the transitions between them as the coupling strength and relative frequencies of the two chemostats vary. We concentrate on certain strong resonance phenomena between the two frequencies as well as their mutual extinction and provide a representative sampling of possible phase portraits for our model system. Our observations corroborate recent mathematical results and case studies of coupled nonlinear chemical oscillators in which regions of mutual extinction as well as the Arnol'd structure for two-parameter families of maps of the plane have been observed. We highlight certain unexpected features of the operating diagram discovered through our computational study and discuss their implication for the dynamic response of the chemostat system.

Periodic, quasi-periodic, and chaotic coexistence of two competing microbial populations in a periodically operated chemostat

It is well known that when two microbial populations competing for a single rate-limiting nutrient are grown in a chemostat with time-invariant inputs, with competition being the only interaction between them, they cannot coexist, but eventually one of the two populations prevails and the other becomes extinct. It has been suggested that periodic variation of one of the chemostat's operating parameters can stabilize the coexistence state of the two microbial populations. A systematic numerical study of the model equations describing microbial competition in a chemostat with periodically varying dilution rate is performed, and it is shown that coexistence of the competing microbial populations is obtained for a wide range of operating conditions. The coexistence state is usually in the form of limit cycle oscillations. However, cases of chaotic behavior resulting from successive period doublings and quasi-periodicity are also observed.

Interactions of bacteria and microflagellates in sequencing batch reactors exhibiting enhanced mineralization of toxic organic chemicals

Community level interactions were studied in non-axenic sequencing batch reactors (SBRs) being used to treat 2,4-dinitrophenol (DNP). Increasing the influent DNP concentrations from 1 to 10 µg ml(-1) eliminated large predatory organisms such as rotifers and ciliated protozoa from the SBRs. Under steady-state conditions at a DNP concentration of 10 µg ml(-1), supplemental additions of glucose enhanced DNP degradation and led to the establishment of a microbial community consisting of five species of bacteria and a variety of microflagellates. The bacteria and flagellates exhibited oscillating population dynamics in this system, possibly indicating predator-prey interactions between these two groups. Only two of the five bacteria isolated from this system could utilize glucose as a growth substrate, and one of these two species was the only organism that could mineralize DNP in the system. The other three bacteria could grow using metabolic by-products of one of the glucose-utilizing strains (Bacillus cereus) found in the reactors. Supplemental glucose additions increased the average size of bacterial floc particles to 172 µm, compared with 41 µm in SBRs not receiving glucose. It is theorized that the enhanced mineralization of DNP in this non-axenic system was attributable to increased community interactions resulting in increased bacterial flocculation in SBRs receiving supplemental glucose additions.

Microbial predation in a periodically operated chemostat: a global study of the interaction between natural and externally imposed frequencies

Predator-prey systems in continuously operated chemostats exhibit sustained oscillations over a wide range of operating conditions. When the chemostat is operated periodically, the interaction of the natural oscillation frequency with the external forcing gives rise to a wealth of dynamic behavior patterns. Using numerical bifurcation techniques, we perform a detailed computational study of these patterns and the transitions (local and especially global) between them as the amplitude and frequency of the forcing vary. The transition from low-forcing-amplitude quasiperiodicity to entrainment of the chemostat behavior by strong forcing (involving the concerted closing of resonance horns) is analyzed. We concentrate on certain strong resonance phenomena between the two frequencies and provide an extensive atlas of computed phase portraits for our model system. Our observations corroborate recent mathematical results and case studies of periodically forced chemical oscillators. In particular, the existence and relative succession of several distinct types of global bifurcations resulting in chaotic transients and multistability are studied in detail. The location in the operating diagram of several key codimension 2 local bifurcations of periodic solutions is computed, and their interaction with an interesting feature we name "real-eigenvalues horns" is examined.

Segregated, structured, distributed models and their role in microbial ecology: A case study based on work done on the filter-feeding ciliateTetrahymena pyriformis

Microbial populations are composed of individual organisms each of which, if environmental circumstances are favorable, is undergoing change of its internal state through the operation of the set of processes that we call the cell cycle. The rate of progression through the cycle is subject to internal controls as well as external influences, and exhibits random as well as deterministic features. Microorganisms of the same species in different stages of the cell cycle have different internal states, and thus, the operation of the cell cycle is by itself sufficient to produce a distribution of states among the individual organisms of a population. In turn, the distribution of states produces distributions of the rates at which the cells of a population carry on their activities. Mathematical models of microbial growth that take the operation of the cell cycle and its consequences into account are more complicated than the kinds of models that are often used in microbial ecology. This paper gives some account of the nature, formulation, and uses of complex growth models. The account is illustrated by work done by the author and his collaborators H. M. Tsuchiya and more recently F. Srienc, as well as by others, on the filter-feeding ciliateTetrahymena pyriformis.

Effects of spatial heterogeneity on the dynamics of a microbial feeding interaction

A mathematical model for an ideal chemostat in which one microbial population feeds on another and where Monod's model is used for the specific growth rates of both populations predicts a less stable behavior for the system than the one observed experimentally. Various factors have been proposed as being the reason for the increased stability of such systems. In this work, the effect of spatial heterogeneity on the dynamics of the microbial feeding interaction is studied. It is concluded that spatial heterogeneity has a stabilizing effect on the system. This effect combined with other factors could be the reason for the increased stability observed in systems where a microbial feeding interaction occurs.