Some Aspects Concerning the Dynamics of Stochastic Chemostats

Abstract

In this paper we study a simple chemostat model influenced by white noise which makes this kind of models more realistic. We use the theory of random attractors and, to that end, we first perform a change of variable using the OrnsteinUhlenbeck process, transforming our stochastic model into a system of differential equations with random coefficients. After proving that this random system possesses a unique solution for any initial value, we analyze the existence of random attractors. Finally we illustrate our results with some numerical simulations.

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

“…We would like to highlight that the O-U process is frequently used to transform stochastic models affected by the standard Wiener process into random ones (see [1,6,7,8]), which are much more tractable from the mathematical point of view, but both parameters β and γ are not taken into account or do not play any relevant role. Nevertheless, in the framework that we introduce in this paper we use directly this suitable O-U process depending on the parameters previously mentioned since they will be the key of the advantages provided by this way of modeling, as we will show in the rest of this work.…”

“…The most common stochastic process that is considered when modeling disturbances in real life is the well-known standard Wiener process, see for instance [1,2] where the authors study random and stochastic modeling for a SIR model, [3,4,5] where stochastic prey-predator Lotka-Volterra systems are analyzed or [6,7,8,9] where different ways of modeling stochasticity in the chemostat model are investigated. Nevertheless, this stochastic process has the property of having continuous but not bounded variation paths, which does not suit to the idea of modeling real situations since, in most of cases, the real life is subjected to fluctuations which are known to be bounded.…”

“…In some cases analyzed in the previous literature (see [6,8]), the use of a standard Wiener process to perturb some parameters in deterministic systems can lead to a non-realistic model; for instance, the positiveness of solutions are not necessarily preserved as a consequence of the arbitrary large values and the large fluctuations of this Wiener process. However, in other situations, one can modify the way to perturb the deterministic system, still using standard Wiener processes, and the positiveness of solutions is preserved too (see [7,9]).…”

Modeling and Analyzing the Long-Time Behavior of Random Chemostat Models

“…There are many different ways to introduce randomness and/or stochasticity in some deterministic model. The most common way is to use the standard Wiener process, ω, and replace D by D + αω(t), where α > 0 represents the intensity of the noise, as made in [1]. However, several drawbacks can be found when using this way to model disturbances on the input flow in the chemostat model.…”

“…As a consequence, we are interested in considering another approach which consists on replacing D by D+αz * β,ν (θ t ω), where z * β,ν (θ t ω) represents a suitable Ornstein-Uhlenbeck process. We refer the interested readers to [1,2] for more detailed explanations concerning this work.…”