Study of the dynamics of two chemostats connected by Fickian diffusion with bounded random fluctuations

Abstract

This paper investigates the dynamics of a model of two chemostats connected by Fickian diffusion with bounded random fluctuations. We prove the existence and uniqueness of non-negative global solution as well as the existence of compact absorbing and attracting sets for the solutions of the corresponding random system. After that, we study the internal structure of the attracting set to obtain more detailed information about the long-time behavior of the state variables. In such a way, we provide conditions under which the extinction of the species cannot be avoided and conditions to ensure the persistence of the species, which is often the main goal pursued by practitioners. In addition, we illustrate the theoretical results with several numerical simulations.

Effects of real random perturbations on Monod and Haldane consumption functions in the chemostat model

“…The chemostat model ( 1)-( 2) has been widely investigated in the literature, specially the case when considering the Monod consumption function. Nevertheless, it is a deterministic system and then it assumes restrictions that are very strong, particularly when taking into consideration that real life is often subject to suffer random disturbances (see, for instance, [17,18,19,20,21,22,23]).…”

“…, where ϕ(ξ * (θ t ω)) denotes a bounded noise (see details in Section 2) which has been proved to be a good tool when modeling real random fluctuations not only in the chemostat model but also in other models arising in population dynamics (see [17,18,19,20,21,22,23,24]).…”

Extinction and strong persistence in the Beddington–DeAngelis predator–prey random model

“…On the other hand, the framework of a random dynamical system (RDS) allows us to discuss the pullback behavior of stochastic or random biological systems. As for forwarding dynamics, there are a few shots with the random chemostats models [11][12][13][14][15], the eco-epidemiological model [16], and the generalized logistic random differential equation model [17]. However, little seems to be known about the forward dynamics of the stochastic or random predator-prey model.…”

“…So it implies that the attracting set(15) reduces to a point (K, 0), which means the extinction of the predator.…”

Coupling of Bio-Reactors to Increase Maximum Sustainable Yield

“…This is by analogy the principle of exclusion in ecology, a single ecological niche allows only one species to exploit it. The gradostat was studied in particular to find the conditions for the coexistence of two competitors in a system of two coupled chemostats [18][19][20]. The method was extended to the case of N > 2 coupled chemostats to search for the coexistence of several competitors from a single resource [16,17].…”