A method for the reconstruction of unknown non-monotonic growth functions in the chemostat

We propose an adaptive control law that allows one to identify unstable steady states of the open-loop system in the single-species chemostat model without the knowledge of the growth function. We then show how one can use this control law to trace out (reconstruct) the whole graph of the growth function. The process of tracing out the graph can be performed either continuously or step-wise. We present and compare both approaches. Even in the case of two species in competition, which is not directly accessible with our approach due to lack of controllability, feedback control improves identifiability of the non-dominant growth rate.Comment: expansion of ideas from proceedings paper (17 pages, 8 figures), proceedings paper is version v

Global stability for a model of competition in the chemostat with microbial inputs

International audienceWe propose a model of competition of nn species in a chemostat, with constant input of some species. We mainly emphasize the case that can lead to coexistence in the chemostat in a non-trivial way, i.e., where the n−1n−1 less competitive species are in the input. We prove that if the inputs satisfy a constraint, the coexistence between the species is obtained in the form of a globally asymptotically stable (GAS) positive equilibrium, while a GAS equilibrium without the dominant species is achieved if the constraint is not satisfied. This work is round up with a thorough study of all the situations that can arise when having an arbitrary number of species in the chemostat inputs; this always results in a GAS equilibrium that either does or does not encompass one of the species that is not present in the input

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

Global stabilization of the chemostat with delayed and sampled measurements and control

International audienceThe classical model of the chemostat with one substrate, one species and a Haldane type growth rate function is considered. The input substrate concentration is supposed to be constant and the dilution rate is considered as the control. The problem of asymptotically stabilizing an equilibrium point of this system in the case where the measured concentrations are delayed and piecewise constant with a piecewise constant control is addressed

Influence of temperature on the complex dynamic behaviour of a microbial food web

Ecosystems are set to extrinsic drivers like climate parameters. These are known to show non-linear dynamics and potential chaotic behaviour. One of the most important drivers is temperature; it influences a large variety of ecological processes (e.g. growth rate and other metabolic rates). On the other hand, populations show density dependent, intrinsic, non-linear dynamics including complex, irregular patterns. The interactions between extrinsic and intrinsic dynamic behaviour are difficult to determine in natural ecosystems and have been discussed in literature. The assessment of the consequences derived from climate change represents a great challenge for ecologist. Deeper knowledge on the mechanisms driving temperature effects on natural food webs is needed. In this work I investigated a well defined simplified microbial food web consisting of two prey bacteria (Pedobacter sp. and Acinetobacter johnsonii) and one predator ciliate (Tetrahymena pyriformis). This simple food web permits the study of intrinsic dynamics as well as the influence of extrinsic disturbances. The experimental setup developed by my colleagues and me, consisted of chemostats where parameters like the flow rate were computer controlled, so external noise was reduced to the minimum. Experimental parameters could be determined with great precision, and therefore the dynamic behaviour showed by the experiments is considered to be purely intrinsic. A mathematical model based on experimental data was developed with the aim of analyzing the temperature scenario investigated experimentally. The model included temperature dependent growth rate functions that were fitted to experimental data. Although the model did not capture the whole complexity of the food web, reflected some qualitative temperature effects observed experimentally. The bacterium Acinetobacter johnsonii showed grazing-resistant morphologies. I hypothesised that this morphological plasticity was responsible for part of the irregular fluctuations of the abundances observed in the chemostat experiments. In cooperation with David Heckman I developed a mathematical model with the aim of testing this hypothesis at a theoretical level. Numerical analysis showed a stabilization of the food web represented by two characteristics: the possibility of coexistence for a wider range of external parameters, and the absence of chaotic fluctuations. I was able to show for the first time experimentally, that changes on extrinsic temperatures may shift population dynamics to different attractors depending on the specific temperature response of populations. I analyzed the impact of a temperature increase from 20°C to 25°C. The results presented here suggest that the ecological responses to temperature can affect the dynamic behaviour in food webs

Further results on stabilization of periodic trajectories for a chemostat with two species,

Abstract We discuss an important class of problems involving the tracking of prescribed trajectories in the chemostat model. We provide new tracking results for chemostats with two species and one limiting substrate, based on Lyapunov function methods. In particular, we use a linear feedback control of the dilution rate and an appropriate time-varying substrate input concentration to produce a locally exponentially stable oscillatory behavior. This means that all trajectories for the nutrient and corresponding species concentrations in the closed loop chemostat that stay near the oscillatory reference trajectory are attracted to the reference trajectory exponentially fast. We also obtain a globally stable oscillatory reference trajectory for the species concentrations, using a nonlinear feedback control depending on the dilution rate and the substrate input concentration. This guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory. Finally, we construct an explicit Lyapunov function for the corresponding global error dynamics. We demonstrate the efficacy of our method in a simulation

Productivity analysis and non-linear gain scheduling approach for multi-species bioprocesses with product inhibition

International audienceBioprocesses with product inhibition are known to allow species coexistence. In this work, we first study the productivity of the different possible equilibria, depending on the operating conditions, and show that single species offers the best performances. Then, we propose a control strategy to stabilize the dynamics about the desired equilibria, in presence of instability. Based on output feedback linearization, we propose a family of controllers and a gain-scheduling approach to adapt the controller. Finally, we illustrate our approach on numerical simulations, showing that the attraction basin of the closed-loop system is improved by considering the gain-scheduling approach