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.     

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.     

Algal defenses,population stability,and the risk of herbivore extinctions: a chemostat model and experiment

The effects of inducible defenses and constitutive defenses on population dynamics were investigated in a freshwater plankton system with rotifers as predators and different algal strains as prey. We made predictions for these systems using a chemostat predator–prey model and focused on population stability and predator persistence as a function of flow-through rate. The model exhibits three major types of behavior at a high nutrient concentration: (1) at high dilution rates, only algae exist; (2) at intermediate dilution rates, algae and rotifers show stable coexistence; (3) at low dilution rates, large population fluctuations occur, with low minimum densities entailing a risk of stochastic rotifer extinctions. The size and location of the corresponding areas in parameter space critically depend on the type of algal defense strategy. In an 83-day high-nutrient chemostat experiment we changed the dilution rate every 3 weeks, from 0.7 to 0.5 to 0.3 to 0.1 per day. Within this range of dilution rates, rotifers and algae coexisted, and population fluctuations of algae clearly increased as dilution rates decreased. The CV of herbivore densities was highest at the end of the experiment, when the dilution rate was low. On day 80, herbivorous rotifers had become undetectable in all three chemostats with permanently defended algae (where rotifer densities had already been low) and in two out of three chemostats where rotifers had been feeding on algae with inducible defenses (that represented more edible food). We interpret our results in relation to the paradox of enrichment.     

Dynamics of chemostat in which one microbial population grows on multiple complementary nutrients

The equations of a chemostat in which one microbial population grows on multiple rate-limiting nutrients are formulated. The dynamics of a chemostat involving growth on complementary nutrients is studied through stability analysis of the system of equations. Some conditions are derived that relate the dynamic behavior of the chemostat to its operating conditions and can be applied to any model for the specific growth rate of the population. It is shown that, if maintenance of the population is neglected, the system exhibits no sustained or damped oscillations. If maintenance of the population is considered, damped oscillations are observed for some operating conditions.     

Monod's bacterial growth model revisited

An attempt to justify Monod's bacterial growth model is presented. The justification is based on a mechanistic approach to growth which leads to a differential equation with delay and then to Monod's model. An unexpected increase of parameterK _s with μ_m is predicted by the theory. A survey of literature shows that this effect is present in a large majority of published data.     

The ideal free distribution and bacterial growth on two substrates

A population dynamical model describing growth of bacteria on two substrates is analyzed. The model assumes that bacteria choose substrates in order to maximize their per capita population growth rate. For batch bacterial growth, the model predicts that as the concentration of the preferred substrate decreases there will be a time at which both substrates provide bacteria with the same fitness and both substrates will be used simultaneously thereafter. Preferences for either substrate are computed as a function of substrate concentrations. The predicted time of switching is calculated for some experimental data given in the literature and it is shown that the fit between predicted and observed values is good. For bacterial growth in the chemostat, the model predicts that at low dilution rates bacteria should feed on both substrates while at higher dilution rates bacteria should feed on the preferred substrate only. Adaptive use of substrates permits bacteria to survive in the chemostat at higher dilution rates when compared with non-adaptive bacteria.     

Limit cycles in a chemostat model for a single species with age structure

We model an age-structured population feeding on an abiotic resource by combining the Gurtin-MacCamy [Math. Biosci. 43 (1979) 199] approach with a standard chemostat model. Limit cycles arise by Hopf bifurcations at low values of the chemostat dilution rate, even for simple maternity functions for which the original Gurtin-MacCamy model has no oscillatory solutions. We find the exact location in parameter space of the Hopf bifurcations for special cases of our model. The onset of cycling is largely independent of both the form of the resource uptake function and the shape of the maternity function.     

Failure of antibiotic treatment in microbial populations

The tolerance of bacterial populations to biocidal or antibiotic treatment has been well documented in both biofilm and planktonic settings. However, there is still very little known about the mechanisms that produce this tolerance. Evidence that small, non-mutant subpopulations of bacteria are not affected by an antibiotic challenge has been accumulating and provides an attractive explanation for the failure of typical dosing protocols. Although a dosing challenge can kill the susceptible bacteria, the remaining persister cells can serve as a source of population regrowth. We give a condition for the failure of a periodic dosing protocol for a general chemostat model, which supports the simulations of an earlier, more specialized batch model. Our condition implies that the treatment protocol fails globally, in the sense that a mixed bacterial population will ultimately persist above a level that is independent of the initial composition of the population. We also give a sufficient condition for treatment success, at least for initial population compositions near the steady state of interest, corresponding to bacterial washout. Finally, we investigate how the speed at which the bacteria are wiped out depends on the duration of administration of the antibiotic. We find that this dependence is not necessarily monotone, implying that optimal dosing does not necessarily correspond to continuous administration of the antibiotic. Thus, genuine periodic protocols can be more advantageous in treating a wide variety of bacterial infections.      

A size-structured model of bacterial growth and reproduction

We consider a size-structured bacterial population model in which the rate of cell growth is both size- and time-dependent and the average per capita reproduction rate is specified as a model parameter. It is shown that the model admits classical solutions. The population-level and distribution-level behaviours of these solutions are then determined in terms of the model parameters. The distribution-level behaviour is found to be different from that found in similar models of bacterial population dynamics. Rather than convergence to a stable size distribution, we find that size distributions repeat in cycles. This phenomenon is observed in similar models only under special assumptions on the functional form of the size-dependent growth rate factor. Our main results are illustrated with examples, and we also provide an introductory study of the bacterial growth in a chemostat within the framework of our model.     

Postprandial changes in methanogenic and acidogenic bacteria in the rumens of steers fed high- or low-forage diets once daily

Four ruminally fistulated Hereford steers (400 kg) were fed two isocaloric diets at 1.5 x maintenance once daily in a repeated measurement crossover experiment. Postprandial changes in hydrogen-oxidizing, carbon dioxide-reducing bacterial groups were monitored. The methanogenic bacterial populations were present at densities of 4 x 10(8) to 8 x 10(8)/g of ruminal contents on either the high- or low-forage diet. Numbers remained constant postprandially on the high-forage diet but showed a distinct rise and fall with the once-daily feeding of the low-forage diet. Presumed hydrogen- and carbon dioxide-utilizing, acid-producing (acidogenic) bacteria were present between 2 x 10(8) and 12 x 10(8)/g of ruminal contents, with the density of the low-forage population being twofold higher than that of the high-forage population. Acidogenic bacteria exhibited similar postprandial changes on both diets, with the predominant shift being associated with the feeding event. This is the first study which documents the postfeeding trends in ruminal methanogenic bacteria on specified, production-level diets. It is also the first study to suggest that other hydrogen-oxidizing, carbon dioxide-reducing bacteria which produce acid instead of methane are present at high population densities in the normally fed adult ruminant.     

Postprandial changes in methanogenic and acidogenic bacteria in the rumens of steers fed high- or low-forage diets once daily.

Four ruminally fistulated Hereford steers (400 kg) were fed two isocaloric diets at 1.5 x maintenance once daily in a repeated measurement crossover experiment. Postprandial changes in hydrogen-oxidizing, carbon dioxide-reducing bacterial groups were monitored. The methanogenic bacterial populations were present at densities of 4 x 10(8) to 8 x 10(8)/g of ruminal contents on either the high- or low-forage diet. Numbers remained constant postprandially on the high-forage diet but showed a distinct rise and fall with the once-daily feeding of the low-forage diet. Presumed hydrogen- and carbon dioxide-utilizing, acid-producing (acidogenic) bacteria were present between 2 x 10(8) and 12 x 10(8)/g of ruminal contents, with the density of the low-forage population being twofold higher than that of the high-forage population. Acidogenic bacteria exhibited similar postprandial changes on both diets, with the predominant shift being associated with the feeding event. This is the first study which documents the postfeeding trends in ruminal methanogenic bacteria on specified, production-level diets. It is also the first study to suggest that other hydrogen-oxidizing, carbon dioxide-reducing bacteria which produce acid instead of methane are present at high population densities in the normally fed adult ruminant.     

Mathematical modeling of population dynamics of unstable plasmid-containing bacteria during continuous cultivation in a chemostat

A structural approach to studying the regularities of the population dynamics of unstable recombinant bacterial strains in a chemostat was elaborated. The approach is based on the mathematical modeling of cell distribution in a population with different numbers of plasmid copies. The effect of decreased selective preference of plasmidless variants of the recombinant strain in the chemostat, which is related to a decrease in the number of plasmid copies in cells upon long-term incubation was analyzed. It is shown that the time of half-elimination of plasmids from the bacterial population in the steady state in the chemostat T1/2 does not depend on the maximum number of plasmid copies in cells N but is determined only by the mean time of generation g and the probability of the loss of one plasmid copy tau. The dependence of the preference of bacterial plasmidless variants on the efficiency of expression of genes cloned into plasmids in chemostat was analyzed using the recombinant strain E. coli Z905, whose plasmids pPHL-7 contain cloned genes for the luminescence system of marine luminescing bacteria Photobacterium leiognathi.     

A model approach to the population dynamics of the rotifer Brachionus rubens in two-stage chemostat culture

Summary A model, based on energy-flow considerations, is presented which describes the population dynamics of Brachionus rubens in the second stage of a two-stage algalrotifer chemostat. The rotifers are foodlimited with substrate-inhibition occurring at high algal densities. The model shows two stable states: steady state with constant density of rotifers and washout of the animals. Which one of the stable states is reached depends on the initial conditions.Empirical data are in general agreement with the model. Deviations may be explained by the fact that the data underlying the model calculations are based on a different food alga (Chlorella vulgaris) than the one used in the experiments (Monoraphidium minutum).The observed population growth rate reaches a maximum value of 0.84 (1/day) at algal densities of 3–4. 10⁶ cells/ml. It decreases at higher algal densities. The egg ratio is related linearily to algal density without being reduced at high algal densities.This study is dedicated to the memory of Prof. Dr. Udo Halbach     

Periodic coexistence in the chemostat with three species competing for three essential resources

A chemostat model of three species of microorganisms competing for three essential, growth-limiting nutrients is considered. J. Husiman and F.J. Weissing [Nature 402 (1999) 407] show numerically that this model can generate periodic oscillations. The present contribution is concerned with rigorous analysis regarding the existence of periodic oscillations in this model. Our analysis is based on the following observation made by Huisman and Weissing: there is a cyclic replacement of species, if each species becomes limited by the resource for which it is the intermediate competitor. Using a permanence theory, an index theory, and a Poincaré-Bendixson theory for three-dimensional competitive systems, we analytically succeed to give sufficient conditions for the existence of periodic orbits in the limit sets in this model. The results in this paper suggest that with a wide range of parameter values, sustained periodic oscillations of species abundances for the model are possible, without involving external disturbances. Our results also suggest that competition is not necessarily destructive, i.e., in the case of existence of sustained periodic oscillations, if one of three competitors is absent, one of the other two rivals cannot survive.     

Intra- and interspecific density dependence mediates weather effects on the population dynamics of a plant–insect herbivore system

Temperature and precipitation are two major factors determining arthropod population densities, but the effects from these climate variables are seldom evaluated in the same study system and in combination with inter- and intraspecific density dependence. In this study, I used a 19 year time series on plant variables (shoot height and flowering incidence) and insect density in order to understand direct and indirect effects of climatic fluctuations on insect population densities. The study system includes two closely related leaf beetle species (Galerucella spp.) and a flower feeding weevil Nanophyes marmoratus attacking the plant purple loosestrife Lythrum salicaria. Results suggest that both intraspecific density dependence and weather variables affected Galerucella population densities, with interactive effects of rain and temperature on insect densities that depended on the timing relative to insect life cycles. In spring, high temperatures increased Galerucella densities only when combined with high rain, as low rain implies a high drought risk. Low temperatures are only beneficial if combined with little rain, as high rain cause chilly and wet conditions that are bad for insects. In summer, interactive effects of rain and temperature are different because high temperatures and little rain cause drought that induce wilting in plants, thus reducing food availability for the leaf feeding larvae. In contrast, the density of the flower feeding weevil was less affected by temperature and precipitation directly, and more indirectly interspecific density dependent effects through reduced resource availability caused by previous Galerucella damage.     

Senescence and antibiotic resistance in an age-structured population model

Different theories have been proposed to understand the growing problem of antibiotic resistance of microbial populations. Here we investigate a model that is based on the hypothesis that senescence is a possible explanation for the existence of so-called persister cells which are resistant to antibiotic treatment. We study a chemostat model with a microbial population which is age-structured and show that if the growth rates of cells in different age classes are sufficiently close to a scalar multiple of a common growth rate, then the population will globally stabilize at a coexistence steady state. This steady state persists under an antibiotic treatment if the level of antibiotics is below a certain threshold; if the level exceeds this threshold, the washout state becomes a globally attracting equilibrium.     

Dynamics of a chemostat in which one microbial population feeds on another

Under certain simplifying hypotheses, a chemostat in which one microbial population feeds on another can be described by a system of three ordinary differential equations. A study is conducted to find which features of the equations are the most important to the dynamics of the system; it is found that the main influence comes from the from the particular form of the specific growth rate of the feeding population. Some conditions are derived that relate the form of the specific growth rate of the feeding population to the dynamic behaviour of the system.     

Contest and Scramble Competition and the Carry-Over Effect in Globodera spp. in Potato-Based Crop Rotations Using an Extended Ricker Model

The Ricker model extended with a linear term was used to model the dynamics of a potato cyst nematode population on different potato cultivars over a wide range of population densities. The model accounts for contest and scramble competition and between-year carryover of unhatched eggs. Contest competition occurs due to the restricted amount of available root sites that are the feeding source of the female nematode. Nematodes not reaching such a feeding site turn into males and do not contribute to a new generation. Scramble competition results in a decrease of the number of eggs per cyst at high densities due to the decrease in the food supply per feeding site. At still higher densities, the size of the root system declines; then dynamics are mostly governed by carryover of cysts between subsequent years. The restricted number of three parameters in the proposed model made it possible to calculate the equilibrium densities and to obtain analytical expressions of the model''s sensitivity to parameter change. The population dynamics model was combined with a yield-loss assessment model and, using empirical Bayesian methods, was fitted to data from a 3-year experiment carried out in the Netherlands. The experiment was set up around the location of a primary infestation of Globodera pallida in reclaimed polder soil. Due to a wide range of population densities at short distances from the center of the infestation, optimal conditions existed for studying population response and damage in different cultivars. By using the empirical Bayesian methods it is possible to estimate all parameters of the dynamic system, in contrast to earlier studies with realistic biological models where convergence of parameter estimation algorithms was a problem. Applying the model to the outcome of the experiment, we calculated the minimum gross margin that a fourth crop needs to reach in order to be taken up in a 3-year rotation with potato. An equation was derived that accounted for both gross margin changes and nematode-related yield loss. The new model with its three parameters has the right level of complexity for the amount and type of collected data. Two other important models from the literature, containing five and 10 parameters respectively, may at this point turn out to be less appropriate. Consequences for research priorities are discussed and prediction schemes are taken in consideration.     

Some stochastic features of bacterial constant growth apparatus

Stochastic models for bacterial constant growth apparatus such as the chemostat are posed and studied. Approximations are given for the mean and variance of the size of the bacterial population when the population is in steady state. Procedures for stimulating a chemostat are developed and the approximate moments are compared with simulated values. The distribution is derived for the waiting time until the occurrence of a population change-over to a faster growing strain. Research supported by National Institutes of Health Grant 5-R01-GM21214.