Predator-Prey Interactions Give Rise to Scale-Invariant Ecosystems

Abstract

AbstractA large body of empirical evidence has shown that scaling laws and power-law distributions are ubiquitous features of ecological systems. However, it is not clear what factors give rise to such universal regularities and how scaling laws and scale-invariant distributions relate to each other. Here, I show scaling laws can be a simple consequence of scale-invariant distributions, and both result from simple commonalities of apparently different 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. The model predicts predator-prey systems can be found in different phases: as primary resources increase, the system exhibits a series of consecutive transitions to different phases with equilibrium dynamics and top-down control of the food web, non-equilibrium dynamics with bottom-up control, or unstable dynamics. While unstable predator-prey dynamics can result in a homogeneous environment by enrichment of resources, resource heterogeneity restores the stability of the system. Scale-invariant group size distribution and a rich set of scaling laws result in the non-equilibrium phase. By developing a general theory, I argue scaling laws can result from the scale-invariance of group size distributions under broad conditions. While some of the scaling laws predicted by the model await empirical examination, consistently with a recently discovered empirical pattern, the model shows predator abundance and prey production show a sublinear scaling with prey abundance. The model links the nature of the control in the food web, prey’s and predators’ behavioral responses to each other, and their life histories. While in small densities, mass-action law holds, and a weak density dependence and top-down control follows, in large densities, a shelter effect - the benefit of living in groups for preys - plays a role and makes prey production non-invariant to density fluctuations. Due to higher density fluctuations in higher densities, prey per-capita production in large densities decreases. This leads to a longer lifespan and lower population turnover of prey, and a scale-invariant ecosystem with bottom-up control in large densities.