Population Modeling with Delay Differential Equations

We investigate a delay differential equation system version of a model designed to describe finite time population collapse. The most commonly utilized population models are presented, including their strengths, weaknesses and limitations. We introduce the Basener-Ross model, and implement the Hopf bifurcation test to identify whether there is a Hopf bifurcation in this system. We attempt to improve upon the Basener-Ross model (which uses ordinary differential equations) by introducing delay differential equations to account for the gestational period of humans. We utilize the singularity-removing transformation of the original Basener-Ross system for the delay differential equation system as well. The new system is shown to have a Hopf bifurcation. We also investigate how the bifurcation diagram of the original ODE model changes with the introduction of delays

Easter Island’s collapse: A tale of a population race

We study different extended formulations for the set (formule) in order to tackle the feasibility problem for the set (formule). Here the goal is not to find an improved polyhedral relaxation of conv(X+), but rather to reformulate in such a way that the new variables introduced provide good branching directions, and in certain circumstances permit one to deduce rapidly that the instance is infeasible. For the case that A has one row a we analyze the reformulations in more detail. In particular, we determine the integer width of the extended formulations in the direction of the last coordinate, and derive a lower bound on the Frobenius number of a. We also suggest how a decomposition of the vector a can be obtained that will provide a useful extended formulation. Our theoretical results are accompanied by a small computational study.fertility, war, bargaining power, collapse, natural resources.

Easter Island’s Collapse : A Tale of a Population Race

The Easter Island tragedy has become an allegory for ecological catastrophe and a warning for the future. In the economic literature the collapse is usually attributed to irrational or myopic behaviors in the context of a fragile ecosystem. In this paper we propose an alternative story involving non-cooperative bargaining between clans to share the crop. Each clan’s bargaining power depends on its threat level when fighting a war. The biggest group has the highest probability of winning. A clan’s fertility is determined ex ante by each group. In the quest for greater bargaining power, each clan’s optimal size depends on that of the other clan, and a population race follows. This race may exhaust the natural resources and lead to the ultimate collapse of the society. In addition to well-known natural factors, the likelihood of a collapse turns out to be greater when the cost of war is low, the probability of succeeding in war is highly responsive to the number of fighters, and the marginal return to labor is not too low. We analyze whether these factors can account for the difference between Easter and Tikopia Islands. The paper also makes a methodological contribution in that it is the first fertility model to include strategic complementarities between groups’ fertility decisionsFertility, War, Bargaining Power, Collapse, Natural Resources

Darboux integrability and dynamics of the Basener-Ross population model

We deal with the Basener and Ross model for the evolution of human population in Easter island. We study the Darboux integrability of this model and characterize all its global dynamics in the Poincaré disc, obtaining 15 different topological phase portraits

Natural and anthropogenic drivers of cultural change on Easter Island: Review and new insights

Easter Island (Rapa Nui) is a remote Pacific island known for its megalithic statues, the moai, built by an ancient culture which disappearance is still debated. Theories claiming for either self-destruction (ecocide) of this ancient culture or an eventual genocide after the European contact have been the most popular. Anthropogenic drivers have been traditionally preferred as causes of this major cultural shift, whereas climatic changes have been dismissed or underrated. However, the latest findings suggest that the topic is more complex than formerly thought and demand a more holistic perspective. This paper reviews the main paleoclimatic, paleoecological, archaeological and historical evidence of the major Rapanui cultural shift leading to the end of the moai-building civilization and uses an integrated approach to analyze its timing and potential causes. The disappearance of the ancient Eastern Island culture that erected the moai was a dramatic cultural shift with significant changes in lifestyle, socio-political organization, religious performance, art and also in the geographical settlement of the cultural core of the Rapanui society. The ancient society, represented by the so called Ancient Cult (or moai cult) was centered on the Rano Raraku crater, to the east of the island, whose soft volcanic rocks (tuff) where suitable for moai carving. This society was replaced by the Birdman-Cult society, based on Rano Kao, to the westernmost end of the island. The assumed date for such shift is uncertain ranging between mid-16th and late-18th centuries. It is suggested that such geographical change, as well as the associated societal transformations, may have been the result of a combination of climatic, ecological and cultural drivers and events. The latest paleoecological reconstructions show that the Rano Raraku catchment was deforested by AD 1450 and the lake inside dried out by AD 1550 owing to an intense climatic drought. This would have caused a landscape deterioration transforming the Raraku catchment into a wasteland devoid of freshwater and unsuitable for human life and the cultural flourishment that characterized the Ancient-Cult society. The drought lasted for about a century and a half and would have forced the Rapanuis to look for alternative freshwater sources. The only feasible option was the freshwater lake inside the then forested Rano Kao crater, where the ceremonial village of Orongo, the center of the Birdman Cult, was funded by AD 1600. The Kao crater is made of hard volcanic rocks (basalts) unsuitable for moai carving by the Neolithic Rapanui culture, unaware of metals, which would have contributed to the end of the moai-building phase. Deforestation and drought would have led to a general demographic decline. The shift from the rigid socio-political organization of the Ancient Cult to the more flexible system characteristic of the Birdman Cult could be viewed as a cultural adaptation to changing environmental conditions. The occurrence of a further, rather catastrophic, genocide caused by slave trading and epidemic diseases, occurred shortly after the European contact (AD 1722), has been documented historically. Therefore, the Rapanui civilization has undergone at least two cultural crises caused by natural and anthropogenic drivers. A complex synergistic scenario like that proposed here can conciliate multidisciplinary lines of evidence formerly used to defend more simplistic and apparently contradictory hypotheses of cultural change. © 2016 Elsevier LtdPeer reviewe

Coupled Societies are More Robust Against Collapse:A Hypothetical Look at Easter Island

Inspired by the challenges of environmental change and the resource limitations experienced by modern society, recent decades have seen an increased interest in understanding the collapse of past societies. Modelling efforts so far have focused on single, isolated societies, while multi-patch dynamical models representing networks of coupled socio-environmental systems have received limited attention. We propose a model of societal evolution that describes the dynamics of a population that harvests renewable resources and manufactures products that have positive effects on population growth. Collapse is driven by a critical transition that occurs when the rate of natural resource extraction passes beyond a certain point, for which we present numerical and analytical results. Applying the model to Easter Island gives a good fit to the archaeological record. Subsequently, we investigate what effects emerge from the movement of people, goods, and resources between two societies that share the characteristics of Easter Island. We analyse how diffusive coupling and wealth-driven coupling change the population levels and their distribution across the two societies compared to non-interacting societies. We find that the region of parameter space in which societies can stably survive in the long-term is significantly enlarged when coupling occurs in both social and environmental variables

Interconnections accelerate collapse in a socio-ecological metapopulation

Resource over-exploitation can have profound effects on both ecosystems and the human populations residing in them. Models of population growth based on a depletable resources have been studied previously, but relatively few consider metapopulation effects. Here we analyze a socio-ecological metapopulation model where resources grow logistically on each patch. Each population harvests resources on its own patch to support population growth, but can also harvest resources from other patches when their own patch resources become scarce. We find that allowing populations to harvest from other patches significantly accelerates collapse and also increases the parameter regime for which collapse occurs, compared to a model where populations are not able to harvest resources from other patches. As the number of patches in the metapopulation increases, collapse is more sudden, more severe, and occurs sooner. These effects also persist under scenarios of asymmetry and inequality between patches. We conclude that metapopulation effects in socio-ecological systems can be both helpful and harmful and therefore require urgent study.NSERC Discovery Gran