Global Dynamics of a Multispecies Chemostat Model Driven by Degenerate Fractional Brownian Motion

Abstract.

Understanding how environmental noise with long-range dependence affects ecosystem dynamics is fundamental to the maintenance of stability and biodiversity. This paper investigates a chemostat model driven by degenerate fractional Brownian motion (fBm) to explore multispecies competition dynamics. Since the solution is neither a Markov process nor a semimartingale, classical stochastic analysis is inapplicable, which poses fundamental challenges for the dynamical analysis of the system. A logarithmic transformation is first employed to reformulate the system into an additive-noise form, and global well-posedness is then established via a Lyapunov function method. Asymptotic pathwise estimates are obtained by applying the majorizing measure technique, the Borell–TIS inequality and stochastic comparison theorem. Subsequently, the asymptotic behavior of the boundary system is analyzed, and a stochastic average break-even concentration [Formula: see text] for each species [Formula: see text] is defined for each species. Then the global dynamics of the system is analyzed in terms of [Formula: see text] and the substrate input concentration [Formula: see text]. The analysis shows that species [Formula: see text] becomes extinct whenever [Formula: see text] or [Formula: see text]. More importantly, in the case where [Formula: see text], species [Formula: see text] goes extinct, whereas species [Formula: see text] persists. These results rigorously demonstrate that the competitive exclusion principle holds for stochastic chemostat models driven by fBm. Finally, numerical simulations not only validate the theoretical results but also show that outcome of species competition, whether exclusion or coexistence, depends on the range of the Hurst index [Formula: see text].