Operating diagram of a flocculation model in the chemostat

Abstract

International audience The objective of this study is to analyze a model of the chemostat involving the attachment and detachment dynamics of planktonic and aggregated biomass in the presence of a single resource. Considering the mortality of species, we give a complete analysis for the existence and local stability of all steady states for general monotonic growth rates. The model exhibits a rich set of behaviors with a multiplicity of coexistence steady states, bi-stability, and occurrence of stable limit cycles. Moreover, we determine the operating diagram which depicts the asymptotic behavior of the system with respect to control parameters. It shows the emergence of a bi-stability region through a saddle-node bifurcation and the occurrence of coexistence region through a transcritical bifurcation. Finally, we illustrate the importance of the mortality on the destabilization of the microbial ecosystem by promoting the washout of species. L'objectif de cette étude est d'analyser un modèle du chémostat impliquant la dynamique d'attachement et de détachement de la biomasse planctonique et agrégée en présence d'une seule ressource. En considérant la mortalité des espèces, nous donnons une analyse complète de l'existence et de la stabilité locale de tous les équilibres pour des taux de croissance monotones. Le modèle pré-sente un ensemble riche de comportements avec multiplicité d'équilibres de coexistence, bi-stabilité et apparition des cycles limites stables. De plus, nous déterminons le diagramme opératoire qui dé-crit le comportement asymptotique du système par rapport aux paramètres de contrôle. Il montre l'émergence d'une région de bi-stabilité via une bifurcation noeud col et l'occurrence d'une région de coexistence via une bifurcation transcritique. Enfin, nous illustrons l'importance de la mortalité sur la déstabilisation de l'écosystème microbien en favorisant le lessivage des espèces.

Mortality can produce limit cycles in density-dependent models with a predator-prey relationship

“…Table 1. Necessary and sufficient conditions of existence and stability of steady states of system (1) where c 4 is defined by (20).…”

“…Several suggestions to address the disparity in predictions of models involving multiple microbial species with observations from the laboratory can be found in the literature. Note these various mechanisms to promote coexistence: the intraspecific and interspecific competition [1,10], the flocculation [16,18,19,20,21], the densitydependence of the growth functions [7,17,25,30,31,32,33,36], the predator-prey interactions [6,29], the complex food webs [2,26,55], the presence of inhibitors that affects the strongest competitor [3,11,12], the commensalistic relationship [4,5,45], and the syntrophic relationship [9,15,43,56].…”

“…This diagram is very important to understand the model from the mathematical and biological point of view. It is often built in the mathematical literature (see [1,3,8,9,11,12,17,20,24,36,42,43,44,54]) and the biological literature (see [27,45,53,56]). In [37], we have studied the existence and the local stability of model (1) in the particular case where α i = 1.…”

“…where the function c 4 can be written as its expression (20). Using the expressions of F 1 in (13) and of F 2 in (15), we obtain…”

“…The advantages of this method is that in some cases, it can be applied for growth rates which are not specified and must simply verify the general assumptions on the model. Therefore, it is a useful tool to predict the behavior of the system [1,3,8,9,11,12,17,20,22,36,43].…”