The general chemostat model with multiple nutrients and flocculating agent: From deterministic behavior to stochastic forcing transition

Abstract

Quantifying the effects of environmental perturbations on ecosystems is essential to understanding many rules in nature, such as the survival and extinction of populations. In this paper, a class of deterministic chemostat models with perfectly complementary nutrients, flocculating agent and general functional response functions is investigated in details as well as its stochastic version. For the deterministic model, the well-posedness of the solutions, the dissipativeness and the existence of the equilibria (forward/backward bifurcation) are discussed. Sufficient conditions are given for the global stability of the boundary equilibrium (the washout equilibrium). If the basic reproduction number is greater than one, the model is uniformly persistent, and an explicit expression for the estimation of the ultimately lower bound of microorganism population is given by using different techniques from the standard persistence theory. For the stochastic model, the existence of an invariant probability measure is analyzed by establishing two thresholds which automatically become the basic reproduction number when stochastic perturbations are not considered. Furthermore, sufficient conditions are given for the extinction of microorganism populations. The numerical simulations reveal that noise excitation may make the prediction and control of the evolutionary trend of microorganism population more difficult and complex.