Limit cycles in a chemostat with general variable yields and growth rates
The paper investigates a three-dimensional chemostat model whose variables are the concentrations of one nutrient \(S\) and two microorganisms \(x\) and \(y\) competing for this nutrient. The coefficients describing the yields and the growth rates of the microorganisms are assumed to be functions of \(S\). The authors are mainly interested in the existence and properties of periodic solutions. They show that the positive octant is positively invariant. And since, moreover, the boundary faces \(x=0\) and \(y=0\) are invariant, they may restrict the investigation to these two-dimensional problems where the Poincaré-Bendixson theorem can be applied to prove the existence of periodic orbits. Furthermore, Hopf bifurcation occurs on the two boundary faces. A numerical example illustrates the results. All proofs are rather elementary.
scientific article; zbMATH DE number 5125036