Abstract
We consider a mathematical model that describes the competition of three species for a single nutrient in a chemostat in which the dilution rate is assumed to be controllable by means of state dependent feedback. We consider feedback schedules that are affine functions of the species concentrations. In case of two species, we show that the system may undergo a Hopf bifurcation and oscillatory behavior may be induced by appropriately choosing the coefficients of the feedback function. When the growth of the species obeys Michaelis-Menten kinetics, we show that the Hopf bifurcation is supercritical in the relevant parameter region, and the bifurcating periodic solutions for two species are always stable. Finally, we show that by adding a third species to the system, the two-species stable periodic solutions may bifurcate into the coexistence region via a transcritical bifurcation. We give conditions under which the bifurcating orbit is locally asymptotically stable.
Highlights
We study the mathematical model of a chemostat: nS = D(S0 − S) − xifi(S)/γi, i=1 xi = xi(fi(S) − D), i = 1, . . . , n, n ≥ 2 where S is the nutrient concentration and xi are the concentrations of the competing species
We have analyzed the dynamics of microbial competition in the chemostat which is controlled by means of state dependent feedback
We have considered the class of feedbacks where the dilution rate is a positive affine function of the microbial concentrations
Summary
By Theorem 4, for sufficiently small positive κ there exists an asymptotically stable Hopf orbit in the x, y plane. We determine a neutral stability condition for the Hopf orbit with respect to the third organism and establish a neutral stability curve by obtaining a relation between η and the Poincare-Lindstedt perturbation parameter ε.
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