A competition model in the chemostat with allelopathy and substrate inhibition

Abstract

A model of two microbial species in a chemostat competing for a single resource is considered, where one of the competitors that produces a toxin, which is lethal to the other competitor (allelopathic inhibition), is itself inhibited by the substrate. Using general growth rate functions of the species, necessary and sufficient conditions of existence and local stability of all equilibria of the four-dimensional system are determined according to the operating parameters represented by the dilution rate and the input concentration of the substrate. With Michaelis-Menten or Monod growth functions, it is well known that the model can have a unique positive equilibrium which is unstable as long as it exists. If a non monotonic growth rate is considered (which is the case when there is substrate inhibition), it is shown that a new positive equilibrium point exists which can be stable according to the operating parameters of the system. We describe its operating diagram, which is the bifurcation diagram giving the behavior of the system with respect to the operating parameters. By means of this bifurcation diagram, we show that the general model presents a set of fifteen possible behaviors: washout, competitive exclusion of one species, coexistence, multi-stability, occurrence of stable limit cycles through a supercritical Hopf bifurcations, homoclinic bifurcations and flip bifurcation. This diagram is very useful to understand the model from both the mathematical and biological points of view.

Operating diagrams for a three-tiered microbial food web in the chemostat

“…11(b). Hence, all stability conditions of SS6 in (9) hold in the region bounded by the curve Γ 4 and located at the right of this curve, which corresponds to r 5 S in ch , D > 0. Therefore, SS6 is stable (resp.…”

“…As it was claimed in [28], the operating diagram remains the most useful answer for the analysis of the behavior of the model according to the parameters. This diagram is often performed both in the biological literature [22,27,30,33] and the mathematical literature [1,2,7,8,9,10,13,14,15,16,17,19,23,25,26,32,31].…”

Operating diagram of a flocculation model in the chemostat

“…In [35], the constant input of some species in a chemostat of n species competing on a single nutrient can lead to coexistence. In the literature, we can cite these various mechanisms of coexistence: flocculation [14,[16][17][18][19], intra-and interspecific interference [1,8], densitydependence [15, 25-27, 31, 32], presence of internal or external inhibitors [2,[9][10][11]23], predator-prey interaction [3,4], simple or complex food web [5,22,45], and the references therein.…”