Stability and stabilization for models of chemostats with multiple limiting substrates

We study chemostat models in which multiple species compete for two or more limiting nutrients. First, we consider the case where the nutrient flow and species removal rates and input nutrient concentrations are all given as positive constants. In that case, we use Brouwer degree theory to give conditions guaranteeing that the models admit globally asymptotically stable componentwise positive equilibrium points, from all componentwise positive initial states. Then we use the results to develop stabilization theory for a class of controlled chemostats with two or more limiting nutrients. For cases where the dilution rate and input nutrient concentrations can be selected as controls, we prove that many different componentwise positive equilibria can be made globally asymptotically stable. This extends the existing control results for chemostats with one limiting nutrient. We demonstrate our methods in simulations.     

Asymptotic Behavior of a Chemostat Model with Constant Recycle Sludge Concentration

In this work, we study a several species aerobic chemostat model with constant recycle sludge concentration in continuous culture. We reduce the number of parameters by considering a dimensionless model. First, the existence of a global positive uniform attractor for the model with different removal rates is proved using the theory of dissipative dynamical systems. Hence, we investigate the asymptotic behavior of the model under small perturbations using methods of singular perturbation theory and we prove that, in the case of two species in competition, the unique equilibrium which is positive is globally asymptotically stable. Finally, we establish the link between the open problem of the chemostat with different removal rates and monotone functional responses, and our model when two species compete on the same nutrient. We give some numerical simulations to illustrate the results.     

Theoretical studies on extinction in the gradostat

A multistage continuous culture system in which nutrients (or substrate) are supplied in the form of gradients has been studied in the laboratory. Called a gradostat, it consists of several chemostats with adjacent vessels connected. Two mathematical models based on Michaelis-Menten kinetics in a gradostat with two culture vessels and two competing species of microorganisms for the cases where there is a gradient of one limiting substrate and there are opposing gradients of two limiting complementary substrates are investigated. Given the parameters of the system, we can answer the basic question as to which species survive and which do not and determine the limiting values.     

When the feasibility of an ecosystem is sufficient for global stability?

We show via a Liapunov function that in every model ecosystem governed by generalized Lotka-Volterra equations, a feasible steady state is globally asymptotically stable if the number of interaction branches equals n-1, where n is the number of species. This means that the representative graph for which the theorem holds is a 'tree' and not only an alimentary chain. Our result is valid also in the case of non-homogeneous systems, which model situations in which input fluxes are present.     

Do nutrient-competition models predict nutrient availabilities in limnetic ecosystems?

Recent theory on resource competition, predicated on the importance of hypothesized trade-offs between minimum requirements for nutrient resources, predicts that there should be negative correlations between the supply rate of major limiting nutrients and the availability of at least some secondary nutrients and/or among the availabilities of different limiting nutrients. However, an analysis of four data sets from large-scale surveys of lakes shows mostly positive correlations among the availabilities and supplies of nutrients. In contrast, a fifth data set, obtained in an area of high acidification, does show several important negative correlations that are consistent with the nutrient competition models. Further analyses suggest two possible explanations for the preponderance of positive correlation. Negative correlations between nutrients and light indicate that an important trade-off among species regulating phytoplankton may involve low light requirements versus low nutrient requirements. The existence of negative correlations in nutrient availabilities in acidic lakes (where herbivory appears less important than in buffered lakes) also suggests that another important trade-off may involve an ability to minimize loss rates (especially due to grazing) versus an overall ability to exploit nutrient resources. Received: 23 February 1996 / Accepted: 30 September 1996     

Dynamics of an HIV-1 therapy model of fighting a virus with another virus

In this paper, we rigorously analyse an ordinary differential equation system that models fighting the HIV-1 virus with a genetically modified virus. We show that when the basic reproduction ratio ?(0)<1, then the infection-free equilibrium E (0) is globally asymptotically stable; when ?(0)>1, E (0) loses its stability and there is the single-infection equilibrium E (s). If ?(0)∈(1, 1+δ) where δ is a positive constant explicitly depending on system parameters, then the single-infection equilibrium E (s) that is globally asymptotically stable, while when ?(0)>1+δ, E (s) becomes unstable and the double-infection equilibrium E (d) comes into existence. When ?(0) is slightly larger than 1+δ, E (d) is stable and it loses its stability via Hopf bifurcation when ?(0) is further increased in some ways. Through a numerical example and by applying a normal form theory, we demonstrate how to determine the bifurcation direction and stability, as well as the estimates of the amplitudes and the periods of the bifurcated periodic solutions. We also perform numerical simulations which agree with the theoretical results. The approaches we use here are a combination of analysis of characteristic equations, fluctuation lemma, Lyapunov function and normal form theory.     

Growth of Enterobacter aerogenes in a chemostat with double nutrient limitations.

The behavior of Enterobacter aerogenes during growth in chemostats limited by single and double nutrient restrictions was examined. On the assumption that different essential nutrients act to limit growth in different ways, we selected pairs of nutrients likely to affect different aspects of metabolism. Results show that macromolecular cell composition can be controlled by using more than one nutrient restriction. The polysaccharide content of the cells is readily manipulated by the ratio of carbon to nitrogen in the inlet nutrients. Also, at low dilution rates, ratios of protein to ribonucleic acid are dependent on the ratio of phosphate to nitrogen in the input nutrients. An examination of both acetic acid and metabolite production (as measured by ultraviolet absorbance of culture filtrates) showed that accumulation of these products was dependent on both dilution rate and type of nutrient limitation(s). These results were examined in terms of the problems of translation of batch to continuous culture processes and the use of selected nutrient limitations to control noncellular product formation.     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

A Computational Study of Alternate SELEX

Systematic evolution of ligands by exponential enrichment (SELEX) is a procedure for identifying nucleic acid (NA) molecules with affinities for specific target species, such as proteins, peptides, or small organic molecules. Here, we extend the work in Seo et al. (Bull Math Biol 72:1623–1665, 2010) (multiple-target SELEX or positive SELEX) and examine an alternate SELEX process with multiple targets by incorporating negative selection into a positive SELEX protocol. The alternate SELEX process is done iteratively by alternating several positive selection rounds with several negative selection rounds. At the end of each positive selection round, NAs are eluted from the bound product and amplified by polymerase chain reaction (PCR) to increase the size of the pool of NA species that bind preferentially to the given positive target vector. The enriched population of NAs is then exposed to the negative targets (undesired targets). The free NA species (instead of the bound NA species being eluted) are retained and amplified by PCR (negative selection). The goal is to minimize an enrichment of nonspecifically binding NAs against multiple targets. While positive selection alone results in a pool of NAs that bind tightly to a given target vector, negative selection results in the subset of the NAs that bind best to the nontarget vectors that are also present. By alternating the two processes, we eventually obtain a refined population of nucleic acids that bind to the desired target(s) with high “selectivity” and “specificity.” In the present paper, we give formulations of the negative and alternate selection processes and define their efficiencies in a meaningful way. We study the asymptotic behavior of alternate SELEX system as a discrete-time dynamical system. To do this, we use the chemical potential to examine how alternate SELEX leads to the selection of NAs with more specific interactions when the ratio of the number of positive selection rounds to the number of negative selection rounds is fixed. Alternate SELEX is said to be globally asymptotically stable if, given the initial target vector and a fixed ratio, the distribution of the limiting NA fractions does not depend on the relative concentrations of the NAs in the initial pool (provided that all of the NA species are initially present in the initial pool). We state conditions on the matrix of NA—target affinities that determine when the alternate SELEX process is globally asymptotically stable in this sense and illustrate these results computationally.     

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.     

Nutrient flows between ecosystems can destabilize simple food chains

Dispersal of organisms has large effects on the dynamics and stability of populations and communities. However, current metacommunity theory largely ignores how the flows of limiting nutrients across ecosystems can influence communities. We studied a meta-ecosystem model where two autotroph-consumer communities are spatially coupled through the diffusion of the limiting nutrient. We analyzed regional and local stability, as well as spatial and temporal synchrony to elucidate the impacts of nutrient recycling and diffusion on trophic dynamics. We show that nutrient diffusion is capable of inducing asynchronous local destabilization of biotic compartments through a diffusion-induced spatiotemporal bifurcation. Nutrient recycling interacts with nutrient diffusion and influences the susceptibility of the meta-ecosystem to diffusion-induced instabilities. This interaction between nutrient recycling and transport is further shown to depend on ecosystem enrichment. It more generally emphasizes the importance of meta-ecosystem theory for predicting species persistence and distribution in managed ecosystems.     

Independent colimitation for carbon dioxide and inorganic phosphorus

Simultaneous limitation of plant growth by two or more nutrients is increasingly acknowledged as a common phenomenon in nature, but its cellular mechanisms are far from understood. We investigated the uptake kinetics of CO(2) and phosphorus of the algae Chlamydomonas acidophila in response to growth at limiting conditions of CO(2) and phosphorus. In addition, we fitted the data to four different Monod-type models: one assuming Liebigs Law of the minimum, one assuming that the affinity for the uptake of one nutrient is not influenced by the supply of the other (independent colimitation) and two where the uptake affinity for one nutrient depends on the supply of the other (dependent colimitation). In addition we asked whether the physiological response under colimitation differs from that under single nutrient limitation.We found no negative correlation between the affinities for uptake of the two nutrients, thereby rejecting a dependent colimitation. Kinetic data were supported by a better model fit assuming independent uptake of colimiting nutrients than when assuming Liebigs Law of the minimum or a dependent colimitation. Results show that cell nutrient homeostasis regulated nutrient acquisition which resulted in a trade-off in the maximum uptake rates of CO(2) and phosphorus, possibly driven by space limitation on the cell membrane for porters for the different nutrients. Hence, the response to colimitation deviated from that to a single nutrient limitation. In conclusion, responses to single nutrient limitation cannot be extrapolated to situations where multiple nutrients are limiting, which calls for colimitation experiments and models to properly predict growth responses to a changing natural environment. These deviations from single nutrient limitation response under colimiting conditions and independent colimitation may also hold for other nutrients in algae and in higher plants.     

A small marine biosphere in the Proterozoic

The riverine supply of the globally limiting nutrient, phosphorus, to the ocean accounts for only a few percent of nutrient supply to photosynthetic organisms in surface waters. Recycling of marine organic matter by heterotrophic organisms provides almost all of the phosphorus that drives net primary production in the modern ocean. In the low‐oxygen environments of the Proterozoic, the lack of free oxygen would have limited rates of oxic respiration, slowing the recycling of nutrients and thus limiting global rates of photosynthesis. A series of steady‐state mass balance calculations suggest that the rate of net primary production in the ocean was no more than 10% of its modern value during the Proterozoic eon, and possibly less than 1%. The supply of nutrients in such a world would be dominated by river input, rather than recycling within the water column, leading to a small marine biosphere found primarily within estuarine environments.     

The distribution of soil nutrients with depth: Global patterns and the imprint of plants

To understand the importance of plants in structuring the vertical distributions of soil nutrients, we explored nutrient distributions in the top meter of soil for more than 10,000 profiles across a range of ecological conditions. Hypothesizing that vertical nutrient distributions are dominated by plant cycling relative to leaching, weathering dissolution, and atmospheric deposition, we examined three predictions: (1) that the nutrients that are most limiting for plants would have the shallowest average distributions across ecosystems, (2) that the vertical distribution of a limiting nutrient would be shallower as the nutrient became more scarce, and (3) that along a gradient of soil types with increasing weathering-leaching intensity, limiting nutrients would be relatively more abundant due to preferential cycling by plants. Globally, the ranking of vertical distributions among nutrients was shallowest to deepest in the following order: P > K > Ca > Mg > Na = Cl = SO₄. Nutrients strongly cycled by plants, such as P and K, were more concentrated in the topsoil (upper 20 cm) than were nutrients usually less limiting for plants such as Na and Cl. The topsoil concentrations of all nutrients except Na were higher in the soil profiles where the elements were more scarce. Along a gradient of weathering-leaching intensity (Aridisols to Mollisols to Ultisols), total base saturation decreased but the relative contribution of exchangeable K⁺ to base saturation increased. These patterns are difficult to explain without considering the upward transport of nutrients by plant uptake and cycling. Shallower distributions for P and K, together with negative associations between abundance and topsoil accumulation, support the idea that plant cycling exerts a dominant control on the vertical distribution of the most limiting elements for plants (those required in high amounts in relation to soil supply). Plant characteristics like tissue stoichiometry, biomass cycling rates, above- and belowground allocation, root distributions, and maximum rooting depth may all play an important role in shaping nutrient profiles. Such vertical patterns yield insight into the patterns and processes of nutrient cycling through time.     

Signatures of nutrient limitation and co‐limitation: responses of autotroph internal nutrient concentrations to nitrogen and phosphorus additions

Humans are modifying the availability of nutrients such as nitrogen (N) and phosphorus (P), and it is therefore important to understand how these nutrients, independently or in combination, influence the growth and nutrient content of primary producers. Using meta‐analysis of 118 field and laboratory experiments in freshwater, marine and terrestrial ecosystems, we tested hypotheses about co‐limitation of N and P by comparing the effects of adding N alone, P alone, and both N and P together on internal N (e.g. %N, C:N) and P (e.g. %P, C:P) concentrations in autotroph communities. In particular, we tested the following predictions. First, if only one nutrient was limiting, addition of that nutrient should decrease the concentration of the other nutrient, but addition of the non‐limiting nutrient would have no effect on the internal concentration of the limiting nutrient. If community co‐limitation was occurring then addition of either nutrient should result in a decrease in the internal concentration of the other nutrient. Community co‐limitation could also result in no change – or even an increase – in N concentrations in response to P addition if P stimulated growth of N fixers. Finally, if biochemically dependent co‐limitation was occurring, addition of a limiting nutrient would not decrease, and could even increase, the concentration of the other, co‐limited nutrient. We found no general evidence for the decrease in the internal concentration of one nutrient due to addition of another nutrient. The one exception to this overall pattern was marine systems, where N addition decreased internal P concentrations. In contrast, P addition increased internal N concentrations across all experiments, consistent with co‐limitation. These results have important implications for understanding the roles that N and P play in controlling producer growth and internal nutrient accumulation as well as for managing the effects of nutrient enrichment in ecosystems. Synthesis On a global scale, humans have doubled nitrogen (N) inputs and quadrupled phosphorus (P) inputs relative to pre‐industrial levels. N and P fertilization influences autotroph internal nutrient concentrations and ratios and thereby affects a variety of community and ecosystem processes, including decomposition and consumer population dynamics. It is therefore critical to understand the effects of nutrient additions on the growth and nutrient concentrations of primary producers. We used meta‐analysis to evaluate the responses of autotroph internal N and P concentrations to additions of N, P, and N+P and make inferences about limitation and co‐limitation of N and P across marine, terrestrial, and freshwater ecosystems. We found little evidence for single‐nutrient limitation, highlighting the fact that multiple nutrients generally limit primary production.     

Global stability of models of humoral immunity against multiple viral strains

We analyse, from a mathematical point of view, the global stability of equilibria for models describing the interaction between infectious agents and humoral immunity. We consider the models that contain the variables of pathogens explicitly. The first model considers the situation where only a single strain exists. For the single strain model, the disease steady state is globally asymptotically stable if the basic reproductive ratio is greater than one. The other models consider the situations where multiple strains exist. For the multi-strain models, the disease steady state is globally asymptotically stable. In the model that does not explicitly contain an immune variable, only one strain with the maximum basic reproductive ratio can survive at the steady state. However, in our models explicitly involving the immune system, multiple strains coexist at the steady state.     

A probabilistic framework for nutrient uptake length

The nutrient uptake length, the average displacement of a nutrient in a stream before being taken up by the biota, is an important quantity to characterize and compare streams and rivers, or to quantify certain aspects of their related ecosystems. This concept has been widely used for almost 30 years now, and uptake lengths have been estimated for several nutrients in many systems, but it also suffers from a number of limitations, one of them being the requirement of a spatially homogeneous stream or river. We combine recently advocated, transport-based models of stream processes with current concepts of dispersal theory into a novel framework for nutrient uptake length. The framework is based on the theory for dispersal kernels in terrestrial systems, where the entire distribution of dispersal distances is calculated and not only the average. Within this framework, we can re-derive all previous results and formulae for uptake length, and we can include spatially heterogeneous stream environments. In addition, we propose a number of new characteristic quantities that can complement nutrient uptake length when evaluating the health of a stream system or the impact of a source of nutrients. We illustrate our method with two examples of spatially non-homogeneous streams: point-source input of nutrients (e.g., wastewater treatment plant) and diffuse lateral input (e.g., agricultural run-off), and we show how to measure the relative contribution of the two sources to the uptake length and other characteristics.     

Do spatial effects appear at low dilution rate in chemostat?

The chemostat theory on two species competition has shown that the dilution rate where transition of dominance occurs – transition-dilution rate – is independent of limiting-nutrient concentration. However, we obtained the experimental data indicating that the transition-dilution rate changed with variations in limiting-ammonium concentrations, using the chemostat mixed-culture of the cyanobacterium Microcystis novacekii and the green alga Scenedesmus quadricauda. The transition-dilution rate was dependent on the concentration of limiting ammonium in the influx culture medium. We tried to simulate the experimental results. Though the dilution rate has been considered independent of nutrient concentration, we introduce the effective dilution rate that depends on nutrient concentration (ammonium concentration in this study). A hyperbolic Monod-type function is used to represent the effective dilution rate for each species. The maximum dilution rate of the function is set to be the mechanical dilution rate (nominal dilution rate) of the chemostat culture. The calculation shows that the nominal transition-dilution rate where transition of dominance occur decreases with increased concentration. This simulation is well consistent with our experimental data. These results may suggest that the species-specificity of limiting nutrients, here nitrogen. Or they may imply that the depreciation of nitrogen becomes critical when both dilution rate and concentration are very low, especially for the green algae. In the latter case, spatial effects are induced internally in the ecosystem.     

Almost periodic solution of non-autonomous Lotka-Volterra predator-prey dispersal system with delays

This paper studies a non-autonomous Lotka-Volterra almost periodic predator-prey dispersal system with discrete and continuous time delays which consists of n-patches, the prey species can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. By using comparison theorem and delay differential equation basic theory, we prove the system is uniformly persistent under some appropriate conditions. Further, by constructing suitable Lyapunov functional, we show that the system is globally asymptotically stable under some appropriate conditions. By using almost periodic functional hull theory, we show that the almost periodic system has a unique globally asymptotical stable strictly positive almost periodic solution. The conditions for the permanence, global stability of system and the existence, uniqueness of positive almost periodic solution depend on delays, so, time delays are "profitless". Finally, conclusions and two particular cases are given. These results are basically an extension of the known results for non-autonomous Lotka-Volterra systems.