A general chemostat model with second-order Poisson jumps: asymptotic properties and application to industrial waste-water treatment

Abstract

A chemostat is a laboratory device (of the bioreactor type) in which organisms (bacteria, phytoplankton) develop in a controlled manner. This paper studies the asymptotic properties of a chemostat model with generalized interference function and Poisson noise. Due to the complexity of abrupt and erratic fluctuations, we consider the effect of the second order Itô-Lévy processes. The dynamics of our perturbed system are determined by the value of the threshold parameter $ \mathfrak{C}^{\star}_0 $. If $ \mathfrak {C}^{\star}_0 $ is strictly positive, the stationarity and ergodicity properties of our model are verified (practical scenario). If $ \mathfrak {C}^{\star}_0 $ is strictly negative, the considered and modeled microorganism will disappear in an exponential manner. This research provides a comprehensive overview of the chemostat interaction under general assumptions that can be applied to various models in biology and ecology. In order to verify the reliability of our results, we probe the case of industrial waste-water treatment. It is concluded that higher order jumps possess a negative influence on the long-term behavior of microorganisms in the sense that they lead to complete extinction.

Keywords

• chemostat
• poisson jumps
• dynamics
• ergodicity
• waste-water treatment