Abstract
Competition among plants of the same species often results in power-law relations between measures of crowding, such as plant density, and average size, such as individual biomass. Yoda’s self-thinning rule, the constant final yield rule, and metabolic scaling, all link individual plant biomass to plant density and are widely applied in crop, forest, and ecosystem management. These dictate how plant biomass increases with decreasing plant density following a given power-law exponent and a constant of proportionality. While the exponent has been proposed to be universal and thus independent of species, age, and environmental, and edaphic conditions, different theoretical mechanisms yield absolute values ranging from less than 1 to nearly 2. Here, eight hypothetical mechanisms linking the exponent to constraints imposed on plant competition are featured and contrasted. Using dimensional considerations applied to plants growing isometrically, the predicted exponent is -3/2 (Yoda's rule). Other theories based on metabolic arguments and network transport predict an exponent of -4/3. These rules, which describe stand dynamics over time, differ from the 'rule of constant final yield' that predicts an exponent of -1 between the initial planting density and the final yield attained across stands. The latter can be recovered from statistical arguments applied at the time scale in which the site carrying capacity is approached. Numerical models of plant competition produce plant biomass-density scaling relations with an exponent between -0.9 and -1.8 depending on the mechanism and strength of plant-plant interaction. These different mechanisms are framed here as a generic dynamical system describing the scaled-up carbon economy of all plants in an ecosystem subject to differing constraints. The implications of these mechanisms for forest management under a changing climate are discussed and recent research on the effects of changing aridity and site 'quality' on self-thinning are highlighted.
Highlights
Power-law relations in ecology remain a subject of fascination and research interest given their simultaneous ubiquity and practical significance (Thompson, 1942; Vogel, 1988; Niklas, 1994; Brown and West, 2000; Farrior et al, 2016; West, 2017)
The constant final yield rule applies when stands sown at different initial densities p0 all achieve the same biomass per unit area or yield yc at a fixed time after sowing (i.e., yc = f (p0); Figures 1B,C, 2)
When density-driven mortality or self-thinning is absent (i.e., p(t) = p0), this rule leads to an exponent −1 between w(p0|t) and p0, as it can be shown by multiplying both sides of Equation (3) by p0, and recalling that y(t) = w(t)p(t)
Summary
Power-law relations in ecology remain a subject of fascination and research interest given their simultaneous ubiquity and practical significance (Thompson, 1942; Vogel, 1988; Niklas, 1994; Brown and West, 2000; Farrior et al, 2016; West, 2017). The usage of the term “rule” reflects the extensive experimental evidence supporting the universal character of the exponents of the size-density relations. The significance of these power-law relations to crop production, forestry and ecosystem management is rarely in dispute and has been reviewed elsewhere (Willey and Heath, 1969; Drew and Flewelling, 1977, 1979; White, 1981; Westoby, 1984; Peet and Christensen, 1987; Friedman, 2016). The ecological mechanisms responsible for their apparent universal character remains a subject of inquiry and debate since their inception in 1864 (Spencer, 1864) This debate frames the scope of this review
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