The Effect of Leachate Recycling on the Dynamics of Two Competing Bacteria with an Obligate One-Way Beneficial Relationship in a Chemostat

In the present work, we study a simple mathematical model that describes the competition of two bacterial species with an obligate one-way beneficial relationship for a limited substrate in a bioreactor associated with leachate recirculation. The substrate is present into two forms, insoluble and soluble substrates, where the latter is consumed by the two competing bacteria, which have two general nonlinear growth rates. The reduction of the model to a 2D one facilitates the study of the nature of the equilibrium points. The dynamic system admits multiple steady states. We provide necessary and sufficient conditions on the added insoluble and soluble substrates and the dilution rate to guarantee the existence, uniqueness, and local and global stability of such steady states. It is deduced that the coexistence of both bacteria is possible, which contradicts the competitive exclusion principle. In the second step, we propose an optimal control on the leachate recirculation rate that reduces the organic matter inside the reactor. Finally, we provide some numerical examples that corroborate and reinforce the theoretical findings.