Growth revisited: revealing the conditions of sigmoidal growth and the limited scope of the logistic function

Sigmoidal functions are widely used in ecology to represent bounded growth, saturation, and asymptotic responses, yet curve form alone does not specify the ecological state relations that generate such trajectories. This paper revisits Klem’s 1933 yeast experiments, which historically contributed to the introduction of sigmoidal reasoning into fisheries surplus-production theory. The experiments are reconstructed through a minimal algebraic and resource-explicit approach that distinguishes yeast biomass, remaining nutrient resource, a biomass-equivalent growth basis, a restriction factor, a realized carrying structure, and the accessible basis remaining for further growth. From these relations, a constrained growth recursion is derived and compared with the discrete logistic form. The reconstruction shows that sigmoid-like bounded growth is reproduced when the biomass-equivalent basis is fixed, the restriction regime remains effectively stationary, and the accessible growth basis declines as biomass approaches the realized upper bound. Under these conditions, the recursion can be written in fixed-K logistic notation. Under recurrent nutrient supply, the biomass-equivalent basis and realized carrying structure change through time, and the same fixed-K interpretation no longer applies. The analysis further distinguishes the state-defined structure of growth from its interval-specific realization: α scales the transition between observed states without prescribing an explicit temporal growth trajectory. The contribution of the reconstruction is therefore a clarification of scope. Logistic notation is most directly ecologically interpretable when the state relations compressed into K remain sufficiently stationary, a condition that is relevant to both ecological growth models and fisheries surplus-production reasoning.