Predator-prey power laws: trophic interactions give rise to scale-invariant ecosystems

Abstract

Scaling laws and power-law distributions are ubiquitous in ecological systems. However, it is not clear what factors give rise to such universal regularities. Here, I show scaling laws are a simple consequence of scale-invariant distributions, and both result from simple commonalities of diverse ecosystems. I introduce a simple model of predator-prey interactions in which predators and prey move on a two-dimensional space in search of resources that they use to survive and reproduce. As primary resources increase, the food web exhibits a series of transitions to phases with equilibrium dynamics and top-down control of the food web, non-equilibrium dynamics with bottom-up control, and unstable dynamics exhibiting the paradox of enrichment. The model shows resource heterogeneity can solve the paradox of enrichment and ensure the stability of ecosystems. Scale-invariant spatial distribution of prey and predators and a surprisingly rich set of scaling laws, including predator-prey and Taylor’s power laws, appear in the non-equilibrium phase. The model predicts both Taylor’s power law and predator-prey power law can be extended to a rich set of fluctuation scaling laws governing the fluctuation of predator’s and prey’s densities and growth. A mathematical theory suggests scaling laws result from the scale-invariance of the spatial distribution of prey and predators.