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

Abstract

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.

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

“…Every detail about the way of modeling and a complete analysis of the resulting random model can be found in [17,18] thus we will omit the details in this section. Instead we just give some remarks concerning the work in [17,18].…”

“…In order to show the relevance of this new way of modeling we have presented in the previous sections some examples which illustrate the effect of this bounded noise when perturbing some very well-known models such as the logistic growth or the Lotka-Volterra competition. In addition, in [17,18] the authors consider this noise to model random input flows in the chemostat model where some relevant improvements are also achieved. Finally, this way of modeling is full of advantages from the mathematical analysis point of view but also, which is essential, as a quite realistic modeling from the biological point of view.…”

Corrigendum to the paper: A way to model stochastic perturbations in population dynamics models with bounded realizations. Commun Nonlinear Sci Numer Simulat, 77 (2019), 239–257

“…This last property is completely unrealistic from the point of view of applications, since random perturbations in real life are bounded. In order to understand better the drawback caused by the standard Wiener process when modeling noise, we refer the readers to [4,5,7] , where the authors investigate chemostat models by means of the Brownian motion.…”

Understanding site choice and competition through the adaptive dynamics of microbial settling strategy in a chemostat

“…In a simple chemostat model, two physical operating conditions are under control: the concentration of nutrient input and the dilution rate. The effects of the two physical operating conditions on ecological dynamics have attracted much attention (see, e.g., Hale and Somolinos 1983; Butler et al 1985; Yang and Freedman 1991; Ruan 1993; Smith 1997; Ruan and He 1998; Caraballo et al 2015, 2018; Nguyen et al 2020). In this paper, we investigate how the physical operating conditions affect the evolution of microbial population with different settling mechanisms.…”