A hybrid control against species invasion in the chemostat

In this work, we consider the classical chemostat model with the objective to limit the invasion of a new species having negative effect on the resident one, playing with the removal rate. We study the resilience of the system to the apparition of an invader, and propose a new concept of weak resilience when the system cannot return and stay at its original state (whatever is the removal rate). The weak resilience guarantees that the measure of the time for which the density of the resident species is above a given threshold is infinite. We show that this can be achieved by a hybrid controller with very few knowledge on the dynamics of the system. As it is not possible to eradicate totally the invasive species, the controller makes the resident species return indefinitely many times above the desired threshold, and the solutions converge asymptotically to periodic solutions. We illustrate the effectiveness of the proposed controller on numerical simulations.