Aggregation in the Schelling model and inverted biomass pyramids in ecosystems
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Thomas Schelling proposed a simple spatial model to illustrate how, even with relatively mild assumptions on each individual's nearest neighbor preferences, an integrated city would likely unravel to a segregated city, even if all individuals prefer integration. Many authors assumed that the segregation which Schelling observed in simulations on very small cities persists for larger, realistic size cities. We describe how different measures can be used to quantify the segregation and unlock its dependence on city size, disparate neighbor comfortability threshold, and population density. We develop highly efficient simulation algorithms and quantify aggregation in large cities based on thousands of trials. In particular, we show that for the values of disparate neighbor comfortability threshold used by Schelling, the striking global aggregation Schelling observed is strictly a small city phenomenon. Along the way we prove that in the Schelling model, in the process of evolution, the total perimeter of the interface between the different agents always decreases, which provides a useful analytical tool to study the evolution. At the isolated reef Kingman, it was recently discovered that apex predators constitute 85% of the total fish biomass. This is in sharp contrast to most reefs, where the prey biomass substantially dominates the total fish biomass. The recent study at the two pristine reefs, Kingman and Palmyra also indicates that the predator:prey fish biomass ratio is an increasing function of reef cover. Based on these field observations, we model the fish biomass structure at a pristine coral reef. We introduce a new refuge based mechanism for predator-prey interaction with an explicit dependence on refuge size. Our refuge based model does not assume mass action interaction between predators and prey and may provide a new mechanism in ecology to produce inverted biomass pyramids. Our model yields both the inverted biomass pyramid and the increasing dependence of the predator:prey biomass ratio on reef cover. We add various forms of fishing to our model, and show that sufficiently high fishing pressure with quite general types of fishing transforms the inverted biomass pyramid to be bottom heavy.