An optimal control problem for resource utilisation by microorganisms

Abstract

AbstractDecomposition of organic matter controls the flow of carbon and nutrients in terrestrial and aquatic ecosystems. Several kinetic laws have been proposed to describe decomposition rates, but they neglect adaptation of the microbial decomposer to environmental conditions. Here we formalise decomposition as an optimal control problem by assuming that microorganisms regulate the uptake rate of a substrate to maximise their growth over the period of decomposition. The result is an optimal control problem consisting of two differential equations and auxiliary conditions that determine the optimal value of the control variable (the uptake rate), the remaining substrate at any given time, and the optimal completion time. This problem serves as a case study to illustrate the solution of differential equations and optimal control problems for students in undergraduate courses. The mathematical analysis of the problem requires rewriting the differential equations in reverse time along with the solution of a nonhomogeneous linear first order differential equation. We then return to modelling with some biologically motivated questions about how the parameters of the model representing environmental conditions and microbial functional traits affect the outcome. Finally, we discuss alternative ways to use the material with students.Keywords: Optimal controlresource utilisationmicrobial decomposition Disclosure statementNo potential conflict of interest was reported by the authors.Notes1 We are using capital T for time to reserve the usual symbol t for the dimensionless version of the problem.2 In a different context, with G a function of X rather than U, this is the Monod function, which can be derived from first principles (Liu, Citation2007).3 See (Ledder, Citation2023), for example, for a derivation of this function from first principles (where it is presented as the Holling type 2 function for a consumer-resource system). Note that the function is approximately αU/β when U is small.4 With a change of sign from the usual statement, the Lagrange multiplier rule can be recast as a necessary condition that the vector u that maximises a function g(u) subject to a constraint f(u)=0 must maximise the combined function H(u)=g(u)+λf(u) for some constant λ.Additional informationFundingThis project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 101001608).