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Decomposition 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.

Decomposition kinetics are fundamental for quantifying carbon and nutrient cycling in terrestrial and aquatic ecosystems. Several theories have been proposed to construct process-based kinetics laws, but most of these theories do not consider that microbial decomposers can adapt to environmental conditions, thereby modulating decomposition. Starting from the assumption that a homogeneous microbial community maximizes its growth rate over the period of decomposition, we formalize decomposition as an optimal control problem where the decomposition rate is a control variable. When maintenance respiration is negligible, we find that the optimal decomposition kinetics scale as the square root of the substrate concentration, resulting in growth kinetics following a Hill function with exponent 1/2 (rather than the Monod growth function). When maintenance respiration is important, optimal decomposition is a more complex function of substrate concentration, which does not decrease to zero as the substrate is depleted. With this optimality-based formulation, a trade-off emerges between microbial carbon-use efficiency (ratio of growth rate over substrate uptake rate) and decomposition rate at the beginning of decomposition. In environments where carbon substrates are easily lost due to abiotic or biotic factors, microbes with higher uptake capacity and lower efficiency are selected, compared to environments where substrates remain available. The proposed optimization framework provides an alternative to purely empirical or process-based formulations for decomposition, allowing exploration of the effects of microbial adaptation on element cycling.