Dynamics of some stochastic chemostat models with multiplicative noise

Abstract

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.

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]).…”

Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise

“…where λ > 0, β ∈ R and Q is a bounded domain in R n with arbitrary dimensions. The noise is multiplicative in the Stratonovich integrals sense, see [7,10,15,17,18,27]. The existence of a random attractor for 3D sine-Gordon equation had been discussed in the literature [11,29], for the 3D stochastic wave equation with other nonlinearity, see [26,28].…”

Some Aspects Concerning the Dynamics of Stochastic Chemostats

“…System (3)- (4) has been analyzed in [30] by using the classic techniques from stochastic analysis and some stability results are provided there. However, as in our opinion there are some unclear points in the analysis carried out there, our aim in this paper is to use an alternative approach to this problem, specifically the theory of random dynamical systems, which will allow us to partially improve the results in [30].…”