Only a few years ago, humanity reached a new demographic milestone, the birth of the Earth's seventh-billion resident. Demographers have not been alone in expressing concern that with the population expected to reach 9 billion by the middle of this century, Earth's ability to feed and provide other necessities for human life will soon be surpassed. The collapse of historical societies has attracted the attention of both ecologists and economists in relation to current issues of population growth, environmental degradation, and sustainability. A large body of literature, popularized by Jared Diamond in his bestselling book Collapse (Diamond, 2005), argues that ancient civilizations such as the Maya of Mesoamerica, the Anasazi of the southwest United States, the Norse of Greenland, and the Rapanui of Easter Island, collapsed because they degraded the environment or overexploited the ecological resources sustaining them.

This field of research has generated an emerging modeling literature focusing on human-resource interactions. Most of the modeling studies are grounded around the Malthusian theory of population growth (Malthus, 1798) and designed as Lotka-Volterra predator-prey systems, in which the human population is the predator and the resource is the prey. The Malthusian theory postulates that over time population growth tends to outrun food supply locking humans into a trap characterized by environmental degradation and overexploitation and leading to a collapse of the human-resource system. Among all potential cases of historical collapse, Easter Island is probably the most attractive because it constitutes a sort of natural laboratory of the past, due to its small size and isolation (Figure 1).

Modeling studies on Easter Island, which started to emerge approximately two decades ago with the pioneering work of Brander and Taylor (1998), are not very numerous and could be covered, at least briefly, by a short review such as the present one. Therefore, my aims are to survey the existing literature on models of Easter Island (comprising some 30 articles), critically review the advances made over the last two decades, point out the existing gaps, and provide an outlook on where we could direct our attention in the future. The rest of this article covers these aspects as follows. The next two sections briefly describe modeling studies focused, respectively, on socioeconomic aspects and on ecological aspects. Although, as we shall see, the line separating these two kind of studies may be a thin one. After these descriptions, I will highlight gaps in our approaches and current knowledge and will suggest possible areas for future research efforts. I will conclude in the final section by summarizing the main points.

Some years ago, Reuveny (2012) conducted a very insightful analysis of the literature on the collapse of historical civilizations from a resource economics modeling perspective. Considering Easter Island the best example for studying collapse, due to its isolation and its potential for responding with innovations, institutions, and conflict, Reuveny (2012) centred his attention on models applied to the Rapanui society. To avoid unnecessary repetitions here I will present these studies only very briefly and qualitatively (i.e., without describing equations), and I will refer the reader to Reuveny's (2012) review for further details. I will complement the list of articles discussed by Reuveny (2012) with the latest publications in the field.

Reuveny (2012) subdivided the literature into two major types of studies: (1) those assuming a myopic resource extraction (i.e., agents do not maximize utility across multiple time periods but instead focus on the given period) and (2) those based on farsighted decision making. The earliest and most representative modeling study based on a myopic resource extraction assumption (one that satisfies many proponents of the ecocidal view of the ancient Rapanui people, e.g., Flenley and Bahn, 2003; Diamond, 2005) is that of Brander and Taylor (1998). Brander and Taylor (1998) considered a resource stock, represented by a forest-soil complex, increasing logistically and decreasing linearly due to harvest. The economy produced and consumed two goods, the resource harvested and some manufactured good. Labor was the only other factor of production, which was assumed equal to population size. Harvest was simulated with a Schaefer function depending on resource and population levels and manufacturing on population levels only. This model could reproduce a boom and bust cycle in human population, peaking at about 10,000 individuals around 1250 AD and falling to 3,800 by the year 1850, in line with the understanding of the island's history at that time. Given the model assumptions, they concluded that the ancient Rapanui collapsed because they were myopic, lacked understanding of the forest-soil complex, and experienced increase growth with resource consumption. They also argued that resource depletion might have gone unnoticed because the forest disappeared relatively slowly compared to the typical 30-year life span in prehistoric Polynesia, although later studies suggested that about 40 % of the Rapanui lived between 40 and 55 years (Shaw, 2000; Hunt and Lipo, 2001).

The model of Brander and Taylor (1998) has been refined in a number of ways. Anderies (2000) added a minimum resource consumption per capita to capture the end of monument carving at around 1500 AD. Erickson and Gowdy (2000) included a capital stock, accumulating by a full conversion of the extracted resource into manufacturing of tools and boats and depreciating at a constant rate. Also Roman et al. (2017) included a capital stock produced from extracting natural resources but in addition explored the effects of movements of people and goods between two hypothetical societies sharing the characteristics of Easter Island. Matsumoto (2002) considered a discretized version of the time-continuous Brander and Taylor's (1998) model to investigate the sensitivity of the model dynamics to changes in time-dependent parameters. D'Alessandro (2007) considered two resources, forest and land area, and included a threshold in resource level below which exhaustion became inevitable. Taylor (2009) also included a threshold in resource stock. Other models have considered the possibility for social adaptation either through negative changes, for example via conflict (Reuveny and Maxwell, 2001; Maxwell and Reuveny, 2005; Reuveny et al., 2011) or rivalry (de la Croix and Dottori, 2008), or through positive changes, for example by technical progress (Reuveny and Decker, 2000; Decker and Reuveny, 2005; Dalton et al., 2005), by forming resource management institutions (Dalton and Coats, 2000; Erickson and Gowdy, 2000; Pezzey and Anderies, 2003; Taylor, 2009; Chu et al., 2014), or by introducing economic stratification in the form of elites, commoners, and wealth accumulation (Motesharrei et al., 2014). All these modeling studies showed that a collapse was inevitable unless the population growth was sufficiently slow or resource renewal rate was sufficiently fast.

Models based on foresight, (i.e., models simulating a population who cares about its future, a premise consistent with the genocidal view on the fate of the Rapanui, e.g., Peiser, 2005; Hunt, 2007; Hunt and Lipo, 2011), are only a few. Among them, the work of Good and Reuveny (2006) is probably the most prominent and the first applied to Easter Island. They showed that even if the islanders had complete assignment of property rights and implemented optimal resource management with infinite horizon, or equivalently had a social planner that implemented such an optimal plan, a similar boom and bust cycle obtained by Brander and Taylor's (1998) model would have occurred. This work was expanded a few years later by the same authors (Good and Reuveny, 2009) with the inclusion of social welfare functions to confirm that these best-case institutions would have done little to prevent the social collapse of the Rapanui.

As for studies focused on ecological aspects, I will consider those based on models that do not entirely follow the orthodox resource economics approach, which typically structures models to capture a small, closed economy represented by a renewable resource and human labor producing different types of goods (e.g., harvested and manufactured goods). I will therefore focus on models in which a population of assumed identical agents harvests from a resource assumed to follow a logistic growth function, without much considerations of economical nature. This distinction may appear minimal, but it can lead to substantial differences in the way these two classes of models can be further developed and applied. A model with such an ecologically-centred flavor is that of Basener and Ross (2005). Much in line with the results of models embedded into resource economics principles, Basener and Ross (2005) found that when the harvest rate was faster relative to the rate at which the resource renewed itself, a boom and bust cycle occurred. Alternatively, population and resource coexisted, when the harvest rate was slower than the resource regeneration rate. In a subsequent study, Basener et al. (2008) further developed their model to include a rat population that ate palm seeds, in accordance with the hypothesis advanced by Hunt (2007) asserting that the rats could have been the primary cause of the forest decline. Their modeling work confirmed Hunt's (2007) hypothesis. Bologna and Flores (2008) developed a model very similar to the one of Basener and Ross (2005) with the difference that the original harvest rate is now called deforestation rate and is defined by the ratio between the capability of exploiting the resource and the capability of the resource to regenerate itself. They thus reproduced the conditions that lead to a collapse. Wolcott and Conrad (2011), developed two models: a two-state variable model of human population and resource, and a three-state variable model of human population, resource, and agriculture. They showed that the three-state model reduced to the two-state model and produced dynamics that were consistent with the history of Easter Island. Basener et al. (2011) created a spatially-explicit version of their previous model (Basener et al., 2008) by including a reaction-diffusion scheme to capture movements of people and rats within the island. They showed that by increasing the mobility of the rats the model shifted from a stable equilibrium, characterized by the coexistence of people, rats and forest, to an unstable equilibrium, driving the populations to a collapse. Takács et al. (2016) took Basener et al. (2011)'s spatial model, introduced a diffusion parameter for tree, and showed that this had a stabilizing effect on the model system. Finally, Brandt and Merico (2015) developed a modified version of (Basener et al., 2008)'s model and included an epidemic component comprising susceptible, infected, and recovered individuals to capture the effects of the diseases introduced by Europeans. Rather than trying to predict the classical collapse, Brandt and Merico (2015) used the model to test the plausibility of the two main competing hypothesis, collapse (ecocide) and long-term resilience with a fall at the time of the European contacts due to diseases (genocide). In addition, Brandt and Merico (2015) suggested and tested a third scenario, a long and slow demise. While none of the three scenarios could be completely ruled out, the slow demise appeared to be the most plausible when compared to the temporal pattern of deforestation inferred from the latest compilations of radiocarbon dates on charcoal remains (Hunt and Lipo, 2006; Mulrooney, 2013). The model could reproduce the three scenarios fairly easily. This was surprising especially because only a few parameters (seven, excluding those relevant to the epidemic module, which was activated after 1722 AD) needed to be adjusted within reasonably small ranges to reproduce the different scenarios. In a way, this model result mirrors the controversy between scholars and demonstrates how uncertainties in the assumptions (i.e., the parameter values from a modeling perspective) can lead to very conflicting views. With new archaeological evidence coming to light, casting doubts on the classical narrative of a human-induced collapse, some of these models have begun to incorporate the new pieces of evidence and started to describe a more complex historical ecology.

Scholars are divided into two opposing factions. A larger group supporting the notion of a collapse based on the early, classical archaeological and anthropological studies and a smaller group supporting the notion of a steady population size with the involvement of rats in the deforestation process based on most recent archaeological evidence. Modeling studies have been following these two different school of thoughts in a chronological manner. Early modeling applications were embedded into the notion of collapse. As recent studies proposing a later colonization and a resilient response of the Rapanui society to declining resources emerged, models have started to include elements of this new narrative, contributing to the description of a more complex ecological history for the island. Agent-Based Models offer appealing possibilities for overcoming the limits of both dynamic systems models and available archaeological data, for example, through the systematic analyses of alternative scenarios. Experiments with different agents' characteristics, their interactions, and features of the landscape would produce a variety of behavioral responses to social relationships and environmental forcing. Such investigations could provide valuable insights into arguably some the most intriguing and debated events of recent human history.

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