A COMPETITION MODEL IN THE CHEMOSTAT BETWEEN PLASMID-BEARING AND PLASMID-FREE ORGANISMS WITH AN INTERNAL INHIBITOR AND DIFFERENT REMOVAL RATES

In this work, we consider a competition model between plasmid-bearing and plasmid-free organisms in a chemostat, where plasmid-bearing organisms produce a toxin (allelopathic agent) that affects the other competitor. By using general monotonic growth rate functions and different removal rates of the species, we provide a complete analysis for the existence and local stability of all equilibria in the resulting four-dimensional model, which cannot be reduced to a two-dimensional one. With the same removal rates, it is well known that the model can exhibit two positive equilibria: one is stable as long as it exists, and the other is unstable. By incorporating different removal rates, we show that one of the positive equilibria destabilizes, leading to the emergence of a stable limit cycle through a supercritical Hopf bifurcation. Additionally, we present the operating diagram that illustrates the asymptotic behavior of the model according to the operating parameters. The bifurcation diagram, based on input concentration, illustrates the various types of bifurcations of equilibria and the coexistence either around a positive equilibrium or sustained oscillations.