Extensions of the chemostat model with flocculation

Abstract

a b s t r a c tIn this work, we study a model of the chemostat where the species are present in two forms, isolated and aggregated individuals, such as attached bacteria or bacteria in flocks. We show that our general model contains a lot of models that were previously considered in the literature. Assuming that flocculation and deflocculation dynamics is fast compared to the growth of the species, we construct a reduced chemostat-like model in which both the growth functions and the apparent dilution rate depend on the density of the species. We also show that such a model involving monotonic growth rates may exhibit bi-stability, while it may occur in the classical chemostat model, but when the growth rate is nonmonotonic.

Properties of the chemostat model with aggregated biomass

“…According to the study conducted in Section 3.1, a positive steady-state exists as soon as the condition (3.8) is verified and is unique. By particularizing the attachment and detachment functions as we did in Section 3.1, the following stability result is obtained (the case in which D u and D v are different from D is addressed in [8]). Proof As mentioned previously, it is enough to study the local stability for the reduced dynamics (3.3).…”

“…In this Section, we consider that the removal rates of planktonic and attached bacteria are distinct, and accordingly to Assumptions (2.1) one has D v < D u D. This Section follows part of the work [7,8]. The reduction technique we use in Section 4 gives the following reduced model:…”

Competition for a single resource and coexistence of several species in the chemostat

“…there exists a unique solution S * > λ 2 of equation H(S) = 0 if and only if H(λ 2 ) > 0, that is, S in >λ 2 . If a 2 = 0 then the third equation of system (4) implies that S * = λ 2 and from the second one we have…”

“…All other equilibria corresponding to the intersections of ∆ with the blue steady-state characteristics are unstable. In order to do this, we calculate the Jacobian matrix and we use mainly Lemma 6.3 of [4], which we recall here:…”

“…The consideration of density-dependent growth functions in the chemostat model has been also introduced in the literature in the field of mathematical ecology [1] or wastewater process engineering [9]. For instance, it has been proven that flocculation systems can be reduced, under certain assumptions, to systems with density-dependent growth rates where coexistence may occur through this mechanism, [3,4,7].…”