Modeling and Optimal Control of a Two-Species Bioproducing Microbial Consortium

Motivated by recent laboratory experiments, we study microbial populations with light-inducible genetic differentiation that generates a two-species microbial consortium relevant for bioproduction. First, we derive a hierarchy of models describing the evolution of the microbial populations, each with decreasing complexity. This sequential order reduction reveals the connections between several popular classes of models used in this context. Second, we demonstrate the analytical insight the order reduction provides by studying the optimal control of such a reduced-order system of nonlinear ordinary differential equations. Appealing to Pontryagin's maximum principle, we find different optimal control structures within different regions of the parameter space. Explicit solutions are obtained in a subset of parameter space, while, for the remainder of parameter space, closed-form solutions are obtained that depend on a scalar value that solves a particular transcendental equation. We show that a unique solution of the scalar equation exists and lies in a known compact interval, making its numerical approximation particularly easy. The analytical results are verified against direct numerical calculations.