Global dynamics of the chemostat with different removal rates and variable yields

How do seasonal temperature variations influence interplay between toxic and non-toxic cyanobacterial blooms? Evidence from modeling and experimental data

Summer cyanobacterial blooms exhibit a dynamic interplay between toxic and non-toxic genotypes, significantly influencing the cyanotoxin levels within a lake. The challenge lies in accurately predicting these toxin concentrations due to the significant temporal fluctuations in the proportions of toxic and non-toxic genotypes. Typically, the toxic genotypes dominate during the early and late summer periods, while the non-toxic variants prevail in mid-summer. To dissect this phenomenon, we propose a model that accounts for the competitive interaction between toxic and non-toxic genotypes, as well as seasonal temperature variations. Our numerical simulations suggest that the optimal temperature of the toxic genotypes is lower than that of the optimal temperatures of the non-toxic counterparts. This difference of optimal temperature may potentially contribute to explain the dominance of toxic genotypes at the early and late summer periods, situation often observed in the field. Experimental data from the laboratory align qualitatively with our simulation results, enabling a better understanding of complex interplays between toxic and non-toxic cyanobacteria.

Substrate inhibition can produce coexistence and limit cycles in the chemostat model with allelopathy

In this work, we consider a model of two microbial species in a chemostat in which one of the competitors can produce a toxin (allelopathic agent) against the other competitor, and is itself inhibited by the substrate. The existence and stability conditions of all steady states of the reduced model in the plane are determined according to the operating parameters. With Michaelis-Menten or Monod growth functions, it is well known that the model can have a unique positive equilibrium which is unstable as long as it exists. By including both monotone and non-monotone growth functions (which is the case when there is substrate inhibition), it is shown that a new positive equilibrium point exists which can be stable according to the operating parameters of the system. This general model exhibits a rich behavior with the coexistence of two microbial species, the multi-stability, the occurrence of stable limit cycles through super-critical Hopf bifurcations and the saddle-node bifurcation of limit cycles. Moreover, the operating diagram describes some asymptotic behavior of this model by varying the operating parameters and illustrates the effect of the inhibition on the emergence of the coexistence region of the species.

Global stability of a continuous bioreactor model under persistent variation of the dilution rate

In this work, the global stability of a continuous bioreactor model is studied, with the concentrations of biomass and substrate as state variables, a general non-monotonic function of substrate concentration for the specific growth rate, and constant inlet substrate concentration. Also, the dilution rate is time varying but bounded, thus leading to state convergence to a compact set instead of an equilibrium point. Based on the Lyapunov function theory with dead-zone modification, the convergence of the substrate and biomass concentrations is studied. The main contributions with respect to closely related studies are: i) The convergence regions of the substrate and biomass concentrations are determined as function of the variation region of the dilution rate (D) and the global convergence to these compact sets is proved, considering monotonic and non-monotonic growth functions separately; ii) several improvements are proposed in the stability analysis, including the definition of a new dead zone Lyapunov function and the properties of its gradient. These improvements allow proving convergence of substrate and biomass concentrations to their compact sets, while tackling the interwoven and nonlinear nature of the dynamics of biomass and substrate concentrations, the non-monotonic nature of the specific growth rate, and the time-varying nature of the dilution rate. The proposed modifications are a basis for further global stability analysis of bioreactor models exhibiting convergence to a compact set instead of an equilibrium point. Finally, the theoretical results are illustrated through numerical simulation, showing the convergence of the states under varying dilution rate.

An Improved Robust Adaptive Controller for a Fed-Batch Bioreactor with Input Saturation and Unknown Varying Control Gain via Dead-Zone Quadratic Forms

In this work, a new adaptive controller is designed for substrate control of a fed-batch bioreactor in the presence of input saturation and unknown varying control gain with unknown upper and lower bounds. The output measurement noise and the unknown varying nature of reaction rate and biomass concentration and water volume are also handled. The design is based on dead zone quadratic forms. The designed controller ensures the convergence of the modified tracking error and the boundedness of the updated parameters. As the first distinctive feature, a new robust adaptive auxiliary system is proposed in order to tackle input saturation and control gain uncertainty. As the second distinctive feature, the modified tracking error converges to a compact region whose bound is user-defined, in contrast to related studies where the convergence region depends on upper bounds of either external disturbances, system states, model parameters or terms and model parameter values. Simulations confirm the properties of the closed loop behavior.

Global Stability Analysis of the Model of Series/Parallel Connected CSTRs with Flow Exchange Subject to Persistent Perturbation on the Input Concentration

In this paper, we study the convergence properties of a network model comprising three continuously stirred tank reactors (CSTRs) with the following features: (i) the first and second CSTRs are connected in series, whereas the second and third CSTRs are connected in parallel with flow exchange; (ii) the pollutant concentration in the inflow to the first CSTR is time varying but bounded; (iii) the states converge to a compact set instead of an equilibrium point, due to the time varying inflow concentration. The practical applicability of the arrangement of CSTRs is to provide a simpler model of pollution removal from wastewater treatment via constructed wetlands, generating a satisfactory description of experimental pollution values with a satisfactory transport dead time. We determine the bounds of the convergence regions, considering these features, and also: (i) we prove the asymptotic convergence of the states; (ii) we determine the effect of the presence of the side tank (third tank) on the transient value of all the system states, and we prove that it has no effect on the convergence regions; (iii) we determine the invariance of the convergence regions. The stability analysis is based on dead zone Lyapunov functions, and comprises: (i) definition of the dead zone quadratic form for each state, and determination of its properties; (ii) determination of the time derivatives of the quadratic forms and its properties. Finally, we illustrate the results obtained by simulation, showing the asymptotic convergence to the compact set.

The operating diagram of a model of two competitors in a chemostat with an external inhibitor

Understanding and exploiting the inhibition phenomenon, which promotes the stable coexistence of species, is a major challenge in the mathematical theory of the chemostat. Here, we study a model of two microbial species in a chemostat competing for a single resource in the presence of an external inhibitor. The model is a four-dimensional system of ordinary differential equations. Using general monotonic growth rate functions of the species and absorption rate of the inhibitor, we give a complete analysis for the existence and local stability of all steady states. We focus on the behavior of the system with respect of the three operating parameters represented by the dilution rate and the input concentrations of the substrate and the inhibitor. The operating diagram has the operating parameters as its coordinates and the various regions defined in it correspond to qualitatively different asymptotic behavior: washout, competitive exclusion of one species, coexistence of the species around a stable steady state and coexistence around a stable cycle. This bifurcation diagram which determines the effect of the operating parameters, is very useful to understand the model from both the mathematical and biological points of view, and is often constructed in the mathematical and biological literature.

A density-dependent model of competition for one resource in the chemostat

This paper deals with a two-microbial species model in competition for a single-resource in the chemostat including general intra- and interspecific density-dependent growth rates with distinct removal rates for each species. In order to understand the effects of intra- and interspecific interference, this general model is first studied by determining the conditions of existence and local stability of steady states. With the same removal rate, the model can be reduced to a planar system and then the global stability results for each steady state are derived. The bifurcations of steady states according to interspecific interference parameters are analyzed in a particular case of density-dependent growth rates which are usually used in the literature. The operating diagrams show how the model behaves by varying the operating parameters and illustrate the effect of the intra- and interspecific interference on the disappearance of coexistence region and the occurrence of bi-stability region. Concerning the small enough interspecific interference terms, we would shed light on the global convergence towards the coexistence steady state for any positive initial condition. When the interspecific interference pressure is large enough this system exhibits bi-stability where the issue of the competition depends on the initial condition.

Competition for a single resource and coexistence of several species in the chemostat

We study a model of the chemostat with several species in competition for a single resource. We take into account the intra-specific interactions between individuals of the same population of micro-organisms and we assume that the growth rates are increasing and the dilution rates are distinct. Using the concept of steady-state characteristics, we present a geometric characterization of the existence and stability of all equilibria. Moreover, we provide necessary and sufficient conditions on the control parameters of the system to have a positive equilibrium. Using a Lyapunov function, we give a global asymptotic stability result for the competition model of several species. The operating diagram describes the asymptotic behavior of this model with respect to control parameters and illustrates the effect of the intra-specific competition on the coexistence region of the species.

Competition in the chemostat: A stochastic multi-species model and its asymptotic behavior

In this paper, a stochastic chemostat model in which n-species compete for a single growth-limiting substrate is considered. We first prove that the stochastic model has an unique global positive solution by using the comparison theorem for stochastic differential equations. Then we show that when the noise intensities are small, the competition outcome in the chemostat is completely determined by the species' stochastic break-even concentrations: the species with the lowest stochastic break-even concentration survives and all other species will go to extinction in the chemostat. In other words, the competitive exclusion principle holds for stochastic competition chemostat model when the noise intensities are small. Moreover, we find that noise may change the destiny of the species. Numerical simulations illustrate the obtained results.

Chemostats and epidemics: competition for nutrients/hosts

In a chemostat, several species compete for the same nutrient, while in an epidemic, several strains of the same pathogen may compete for the same susceptible hosts. As winner, chemostat models predict the species with the lowest break-even concentration, while epidemic models predict the strain with the largest basic reproduction number. We show that these predictions amount to the same if the per capita functional responses of consumer species to the nutrient concentration or of infective individuals to the density of susceptibles are proportional to each other but that they are different if the functional responses are nonproportional. In the second case, the correct prediction is given by the break-even concentrations. In the case of nonproportional functional responses, we add a class for which the prediction does not only rely on local stability and instability of one-species (strain) equilibria but on the global outcome of the competition. We also review some results for nonautonomous models.