Substrate inhibition can produce coexistence and limit cycles in the chemostat model with allelopathy

摘要

Abstract In this work, we consider a model of two microbial species in a chemostat in which one of the competitors can produce a toxin (allelopathic agent) against the other competitor, and is itself inhibited by the substrate. The existence and stability conditions of all steady states of the reduced model in the plane are determined according to the operating parameters. 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. By including both monotone and non-monotone growth functions (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. This general model exhibits a rich behavior with the coexistence of two microbial species, the multi-stability, the occurrence of stable limit cycles through super-critical Hopf bifurcations and the saddle-node bifurcation of limit cycles. Moreover, the operating diagram describes some asymptotic behavior of this model by varying the operating parameters and illustrates the effect of the inhibition on the emergence of the coexistence region of the species.