Natural Selection as a Process That Increases Metabolic Entropy Production? A Regime-Dependent Analysis in Open Chemostat Systems with Michaelis–Menten Kinetics and Mutation–Selection Dynamics

We analyse whether Darwinian natural selection acts as a metabolic-entropy-production-increasing process in a model of an open chemostat ecosystem, where Michaelis–Menten uptake kinetics are grounded in a thermodynamically consistent mesoscopic chemical-reaction-network model, and the continuum limit of the discrete replicator equation is derived as a Crow–Kimura reaction–diffusion process. On the quasi-static ecological manifold, an exact Fisher-type identity yields a monotonic increase in the metabolic entropy-production rate, proportional to the uptake-rate variance. Solving the reduced moment equations in closed form, under an explicit quasi-static and zero-skewness closure, identifies a thermodynamic ceiling imposed by mass conservation, while phenotypic trade-offs produce a finite sub-ceiling. When ecological and evolutionary timescales become comparable, a frozen-parameter Routh–Hurwitz analysis identifies an instantaneous spectral instability boundary whose scope and self-consistency are assessed for both unbounded and bounded trait models. The results of this analysis delineate the precise dynamical and biochemical conditions under which selection increases metabolic entropy production and the regimes in which that tendency is reshaped.