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

Abstract

We study a model of the chemostat with several species in competition for a single resource. We take into account the intra-specific interactions between individuals of the same population of micro-organisms and we assume that the growth rates are increasing and the dilution rates are distinct. Using the concept of steady-state characteristics, we present a geometric characterization of the existence and stability of all equilibria. Moreover, we provide necessary and sufficient conditions on the control parameters of the system to have a positive equilibrium. Using a Lyapunov function, we give a global asymptotic stability result for the competition model of several species. The operating diagram describes the asymptotic behavior of this model with respect to control parameters and illustrates the effect of the intra-specific competition on the coexistence region of the species.

Interspecific density-dependent model of predator–prey relationship in the chemostat

“…Thus, the classical chemostat model is unable to reproduce reality even qualitatively and new hypotheses should be considered in order to reconcile the theory and the experimental results. Several mechanisms of coexistence in the chemostat were considered in the literature, such as the intra-and interspecific competition between the species [1,3,5,13,27], the flocculation of the species [9,11,12,14], and the densitydependence of the growth functions [6-8, 10, 21]. Mathematical study of several chemostat-like models can be found in the monograph [25], see also [16].…”

A density-dependent model of competition for one resource in the chemostat

“…The biodiversity is also found in biological reactors with a mixture including at least two competitors for one resource, see [19,45]. This has triggered a lot of mathematical research aimed to extend model (1) to bring theory and observations in better accordance. Different mechanisms of coexistence which were proposed in the literature are the intra-and interspecific competition [1,11,28,51], the flocculation [8,9,14,15] and the density-dependence [32][33][34][35].…”

“…This has triggered a lot of mathematical research aimed to extend model (1) to bring theory and observations in better accordance. Different mechanisms of coexistence which were proposed in the literature are the intra-and interspecific competition [1,11,28,51], the flocculation [8,9,14,15] and the density-dependence [32][33][34][35]. Several mathematical models [4,12,13,[23][24][25]30] have attempted to understand the effects of an inhibitor on the competition and the coexistence of species in the chemostat.…”

“…In [8] it is shown that the method of steady-state characteristic is still applicable in the case where both the growth rate µ i (S , x i ) and the removal rate D i = d i (x i ) of each species depend on the density of the same species. In [1] it is shown that the method of steady-state characteristic permits a quite comprehensive analysis of the model considered in [28], where the growth rate µ i (S ) depends only on S but the removal rate of x i is of the form D i + a i x i . The term a i x i corresponds to the so-called crowding effect.…”

“…On the other hand, our study describes the operating diagram which shows the stability regions, in dependence of the operating parameters D and S in , when all biological parameters are fixed. This bifurcation diagram is an important tool for the experimentation as discussed in [1,12,42,48,49].…”