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

Tomás Caraballo, Renato Colucci, Javier López-de-la-Cruz, Alain Rapaport

TL;DR

This paper analyzes the use of the Ornstein–Uhlenbeck process to model dynamical systems subjected to bounded noisy perturbations and sees this method as a realistic one, which can be very useful and helpful for scientists.

Abstract

In 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.

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

TL;DR

This paper analyzes the use of the Ornstein–Uhlenbeck process to model dynamical systems subjected to bounded noisy perturbations and sees this method as a realistic one, which can be very useful and helpful for scientists.

Abstract

In 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.
Paper Structure (8 sections, 1 theorem, 56 equations, 18 figures)

This paper contains 8 sections, 1 theorem, 56 equations, 18 figures.

Table of Contents

  1. Introduction
  2. The Ornstein-Uhlenbeck process.
  3. Environmental perturbations in the logistic model.
  4. Perturbations on the growth rate in the logistic equation.
  5. Parameter estimation in the logistic model.
  6. Random competitive Lotka-Volterra models
  7. Random chemostat model
  8. Conclusions and 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 (18)

  • Figure 1: Effects of the mean reverting constant on the O-U process
  • Figure 2: Effects of the volatility constant on the O-U process
  • Figure 3: Real data: input flow in a bioreactor
  • Figure 4: Realizations of the O-U process generated with parameters from the real data with $\beta$, $\mu$, $\gamma$ as above
  • Figure 5: Examples of non-realistic realizations of the perturbed parameter
  • ...and 13 more figures

Theorems & Definitions (1)

  • Proposition 2.1