Scaling of plant size and age emerges from linked aboveground and belowground transport network properties

Vegetation growth modulates cycling of water, carbon, and nutrients at local-to-global scales. It is therefore critical to quantify plant growth rates and how they are constrained by environmental conditions (especially limited resource availability). Various theoretical approaches have been proposed to this aim. Specifically, allometric theory provides a powerful tool to describe plant growth form and function, but it is focused on the properties of plant xylem networks, neglecting any role played by soils in supplying water to plants. On the other hand, percolation theory addresses physical constraints imposed by the soil pore network to water and nutrient transport, neglecting roles of root networks and vegetation taking up soil resources. In this contribution, we merge these two perspectives to derive scaling relations between plant size (namely height) and age. Our guiding hypothesis is that the root network expands in the soil at a rate sufficient to match the rate of transport of water and nutrients in an idealized optimal fractal pore network, as predicted by percolation theory; with nutrient transport distance vs. time scaling exponent 0.82, and water transport (saturated conditions) distance vs. time scaling exponent 1. The root expansion rate is mirrored by growth aboveground, as in allometric theory, which predicts an isometric relation between root extension and plant height. Building on these results, we predict that the scaling of plant height and age should also have exponent 0.82 in natural systems where nutrients are heterogeneously distributed, and 1 in fertilized systems where nutrients are homogeneously distributed. These predictions are successfully tested with extensive datasets covering major plant functional types worldwide, showing that soil and root network properties constrain vegetation growth by setting limits to the rates of water and nutrient supply to plants.