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

Javier López-de-la-Cruz

Abstract

In this paper we analyze a chemostat model with wall growth where the input flow is affected by two different stochastic processes: the well-known standard Wiener process, which leads into several drawbacks from the biological point of view, and a suitable Orsntein-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 which, moreover, does not provide much information about the long-time behavior of the systems in many situations.

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

Abstract

In this paper we analyze a chemostat model with wall growth where the input flow is affected by two different stochastic processes: the well-known standard Wiener process, which leads into several drawbacks from the biological point of view, and a suitable Orsntein-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 which, moreover, does not provide much information about the long-time behavior of the systems in many situations.
Paper Structure (12 sections, 5 theorems, 97 equations, 15 figures)

This paper contains 12 sections, 5 theorems, 97 equations, 15 figures.

Table of Contents

  1. Introduction
  2. The Ornstein-Uhlenbeck process
  3. Random chemostat model with wall growth
  4. Existence and uniqueness of global solution.
  5. Existence of a deterministic attracting set.
  6. Internal structure of the deterministic attracting set.
  7. Numerical simulations.
  8. Stochastic chemostat model with wall growth
  9. Stochastic chemostat becomes a random chemostat
  10. Numerical simulations and final comments
  11. Comparison between random and stochastic models
  12. Final comments

Key Result

Proposition 2.1

There exists a $\theta _t$-invariant set $\widetilde{\Omega }\in \mathcal{F}$ of $\Omega$ of full $\mathbb{P}-$measure such that for $\omega \in \widetilde{\Omega }$ and $\beta,\gamma>0$, we have

Figures (15)

  • Figure 1: The chemostat model
  • Figure 2: Effects of the mean reverting constant on the O-U process
  • Figure 3: Effects of the volatility constant on the O-U process
  • Figure 4: Attracting set $B^{(s,x)}_0$
  • Figure 5: Attracting set $\widetilde{B}^{(s,x)}_0$
  • ...and 10 more figures

Theorems & Definitions (11)

  • Proposition 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.3
  • proof
  • Remark 1
  • ...and 1 more