Aggregation in the Schelling model and inverted biomass pyramids in ecosystems

When do integrated cities segregate spontaneously and how do we measure segregation? When do inverted biomass pyramids occur in ecosystems and what effect do refuges have on biomass pyramids? We study these questions in this dissertation.;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. This agent based lattice model has become quite influential amongst social scientists, demographers, and economists. Aggregation relates to individuals coming together to form groups and Schelling equated global aggregation with segregation. 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. We identify distinct scales of global aggregation. 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. We also discover several scaling laws for the aggregation measures. 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. We extend our analysis to cities where one category of agents is in a minority. Overall, we find that agents in a minority have fewer opportunities to move to locations where they can be happy and this leads to a rise in the number of isolated unhappy agents, although in special circumstances we observe the formation of compact segregated minority clusters.;Coral reefs around the world have experienced a dramatic decline during the past 25 years. Overfishing is believed to play a major role. There is significant interest in stabilizing and restoring damaged reefs, and first steps include understanding the functioning of reefs in their natural state and examining the effects of fishing.;The isolated Kingman and Palmyra reefs are believed to provide a baseline for the natural state of coral reefs. At 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 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. Since the prey hide from predators in the coral, predators do not have access to all the prey and interactions between predators and prey are rare. Therefore, the fundamental assumption of Lotka-Volterra model that predators and prey are well-mixed does not approximate the situation at Kingman. 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. We also show that prey fishing alone has the same effect.;Refuges protect prey from predators and diminish the food supply of predators in coral reefs, but affect the feeding habits of predators in other ecosystems in more complex ways. Current models in the ecological literature incorporate the role of refuges in an ad-hoc manner and are not rooted in the refuge mediated behavior of predators and prey. Based on evidence from previous field studies, we generalize the classical predator-prey models to incorporate three possible effects of refuges on the feeding habits of predators. We show that refuges can facilitate inverted biomass pyramids but can also discourage them. We also show that immigrating prey unequivocally support the existence of inverted biomass pyramids.