Random and stochastic disturbances on the input flow in chemostat models with wall growth

In this paper, we analyze a chemostat model with wall growth where the input flow is perturbed by two different stochastic processes: the well-known standard Wiener process, which leads into several draw- backs from the biological point of view, and a suitable Ornstein- Uhlenbeck process depending on some parameters which allow us to control the noise to be bounded inside some interval that can be fixed previously by practitioners. Thanks to this last approach, which has already proved to be very realistic when modeling other simplest chemostat models, it will be possible to prove the persistence and coexistence of the species in the model without needing the theory of random dynamical systems and pullback attractors needed when dealing with the Wiener process. This is an advantage since the theoretical framework in this paper is much less complicated and provides us much more information than the other

Studying the long time dynamics of fermentation models: production of dry and sweet wine

In this talk we will consider two classical mathematical models of wine fermentation. The first model will describe the wine-making process that is used to produce dry wine. The second model will be obtained by introducing a term in the equation of the dynamics of the yeast. Thanks to this change it will be possible to inhibit the fermentation of the sugar and as a consequence a sweet wine will be obtained. We will first prove the existence, uniqueness, positiveness and boundedness of solutions for both models. Then we will pass to analyze the long-time dynamics. For the second model we will also provide estimates for the concentration of ethanol, nitrogen and sugar at the end of the process. Moreover, several numerical simulations will be provided to support the theoretical results

Modeling and analysis of random and stochastic input flows in the chemostat model

In this paper we study a new way to model noisy input flows in the chemostat model, based on the Ornstein-Uhlenbeck process. We introduce a parameter β as drift in the Langevin equation, that allows to bridge a gap between a pure Wiener process, which is a common way to model random disturbances, and no noise at all. The value of the parameter β is related to the amplitude of the deviations observed on the realizations. We show that this modeling approach is well suited to represent noise on an input variable that has to take non-negative values for almost any time.European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Ministerio de Economía y Competitividad (MINECO). EspañaJunta de Andalucí

Bounded random fluctuations on the input flow in chemostat models with wall growth and non-monotonic kinetics

Dedicated to the memory of María José Garrido Atienza.This paper investigates a chemostat model with wall growth and Haldane consumption kinetics. In addition, bounded random fluctuations on the input flow, which are modeled by means of the well-known Ornstein-Uhlenbeck process, are considered to obtain a much more realistic model fitting in a better way the phenomena observed by practitioners in real devices. Once the existence and uniqueness of global positive solution has been proved, as well as the existence of deterministic absorbing and attracting sets, the random dynamics inside the attracting set is studied in detail to provide conditions under which persistence of species is ensured, the main goal pursued from the practical point of view. Finally, we support the theoretical results with several numerical simulations.Junta de Andalucía P12-FQM-1492Ministerio de Ciencia, Innovación y Universidades (MCIU). España PGC2018-096540-B-I00Junta de Andalucía (Consejería de Economía y Conocimiento) FEDER US-1254251Junta de Andalucía (Consejería de Economía y Conocimiento) P18-FR-450

Study of the dynamics of two chemostats connected by Fickian diffusion with bounded random fluctuations

This paper investigates the dynamics of a model of two chemostats connected by Fickian diffusion with bounded random fluctuations. We prove the existence and uniqueness of non-negative global solution as well as the existence of compact absorbing and attracting sets for the solutions of the corresponding random system. After that, we study the internal structure of the attracting set to obtain more detailed information about the long-time behavior of the state variables. In such a way, we provide conditions under which the extinction of the species cannot be avoided and conditions to ensure the persistence of the species, which is often the main goal pursued by practitioners. In addition, we illustrate the theoretical results with several numerical simulations

A way to model stochastic perturbations in population dynamics models with bounded realizations

International audienceIn this paper, we analyze the use of the Ornstein-Uhlenbeck process to model dynamical systems subjected to bounded noisy perturbations. In order to discuss the main characteristics of this new approach we consider some basic models in population dynamics such as the logistic equations and competitive Lotka-Volterra systems. The key is the fact that these perturbations can be ensured to keep inside some interval that can be previously fixed, for instance, by practitioners, even though the resulting model does not generate a random dynamical system. However, one can still analyze the forwards asymptotic behavior of these random differential systems. Moreover, to illustrate the advantages of this type of modeling, we exhibit an example testing the theoretical results with real data, and consequently one can see this method as a realistic one, which can be very useful and helpful for scientists

Dynamics of some stochastic chemostat models with multiplicative noise

In this paper we study two stochastic chemostat models, with and without wall growth, driven by a white noise. Specifically, we analyze the existence and uniqueness of solutions for these models, as well as the existence of the random attractor associated to the random dynamical system generated by the solution. The analysis will be carried out by means of the well-known Ornstein-Uhlenbeck process, that allows us to transform our stochastic chemostat models into random ones.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadJunta de Andalucí