From Logistic to Gompertz: A Microscopic Theory of Coherent Growth in Biological Systems

Abstract

Logistic and Gompertz growth have traditionally been connected by augmenting the logistic model with an extra parameter, giving $θ$-logistic or Richards growth. In spite of this bridge, the biological foundation of Gompertz growth remains only vaguely understood. We propose a novel microscopic version of the Richards model where a coherence parameter sets the coupling between nodes on a network. This model reveals Gompertz growth ($θ\to 0$) as the coherent limit within this family: the system is asymptotically stable with a spectral gap protecting the macroscopic state, and each entity contributes linearly to the collective growth rate regardless of network topology. In contrast, Richards ($θ> 0$) and logistic ($θ= 1$) growth require synchronization as a precondition for macroscopic validity, a condition that Gompertz growth instead imposes. The coherence parameter $θ$ thus acts as a symmetry-breaking parameter: at $θ= 0$ the aggregate dynamics depend only on the collective mean and are invariant to how fluctuations are distributed among entities, an invariance broken at first order in $θ$, where the macroscopic drift acquires a dependence on the microscopic variance. These observations support interpreting Gompertz growth in biological systems as driven by a source external to the individual entities: a field that stimulates a response simultaneously across all entities, such as an environmental stressor, an electromagnetic field, or time over longer horizons. Our results also admit a phenomenological classification of well-known growth models: uncorrelated growth without time dependence yields the exponential function, uncorrelated growth with linear time dependence yields the Gaussian, pairwise correlated growth yields the logistic, and correlated growth (or independent growth with a common driver) yields the Gompertz.