The Logistic Sigmoid: A Narrative Review of Population Ecology, from Verhulst's Curve to the Allee Effects

Abstract

Population ecology is growth's regulation science: the logistic curve that Pierre-François Verhulst fitted in 1838, the carrying capacity's ceiling, the r- and K-selection continuum that Pianka named in 1970, and the Allee effects whose inverse density dependence—cooperation's ecology—makes small populations smaller. This article presents a narrative review of the primary literature of that science, from Verhulst's 1838 logistic law and Pearl and Reed's 1920 rediscovery, through Allee's 1931 Animal Aggregations, Gause's 1934 The Struggle for Existence, Lack's 1954 natural regulation, Pianka's 1970 r- and K-selection, May's 1976 chaos in the logistic map, Hutchinson's 1978 Population Ecology, Courchamp, Clutton-Brock, and Grenfell's 1999 inverse density dependence, Stephens, Sutherland, and Freckleton's 1999 Allee definition, Turchin's 2003 Complex Population Dynamics, and Sibly and colleagues' 2005 regulation across taxa. The synthesis is organized around three themes: the logistic settlement, in which growth's ceiling became ecology's first law and its map's chaos; the selection's continuum, in which density's two strategies organized the life histories; and the Allee's recovery, in which cooperation's ecology made smallness a risk—the conservation's arithmetic. It is concluded that population ecology's history is the logistic curve's elaboration—one sigmoid whose parameters, chaos, and inverse densities contain the field.