Source – sciencedirect.com
– “…The new dynamics of this model can also reproduce the irreversible collapses found in history. Collapse can be avoided, and population can reach a steady state at maximum carrying capacity if the rate of depletion of nature is reduced to a sustainable level and if resources are distributed equitably”:
(NASA Claims: Over 30 ADVANCED civilizations have COLLAPSED before us – Are we next?)
A NASA-funded study has found that ADVANCED ancient civilizations that existed on Earth thousands of years ago were as susceptible to collapse as WE are today.
Is there a mysterious, chaotic pattern within civilization?
How many times have you heard in history books that a superpower in the distant past became so advanced and uncontrollably powerful that it eventually collapsed?
You may think you haven’t heard of that in history books at all, but you may be wrong.
If we look back at our history books –which by the way are totally incomplete— you will notice that in the distant past, looking at the 18th century, for example, there were countries like France who were considered a world superpower.
Their mighty culture developed at such rate that they eventually felt supremacy and acted irresponsibly in turn.
With vast territories overseas, spending money uncontrollably, eventually lead to catastrophic budget deficits which in turn caused them to enter into debts and their currency was devaluated, chaos followed.
Even though France did not disappear entirely from history, it is evidence of a small chaotic pattern present in modern-day society. This chaotic way researchers believe has been present in civilization for thousands of years.
The truth is that even before that, there were numerous cultures and civilizations that just as France and other countries, were the makers of their own collapse.
If we look back in history 3000–5000 years, we will find a historical record that clearly shows us how advanced and sophisticated civilizations were just as susceptible to collapse as we are today.
While lifestyles were much different in the distant past than they are today, it turns out that there is an ongoing pattern encrusted into humanity that raises numerous questions about our sustainability as a civilization which ultimately challenges our ability to stay ‘afloat’ as a complex society and culture.
If we look back in time over 10,000 years, we will discover that numerous advanced civilizations that predate the Inca, Olmec, and Egyptian civilization existed on Earth which mysteriously vanished without a trace.
Look back at the Ancient Maya for example. The Ancient Maya were one of the most advanced and sophisticated ancient civilizations ever to rule on Earth.
This ancient Civilization, which flourished in the jungles of present-day Mexico(south), Guatemala and Belize is noted for the Maya script, the only known developed writing system of the Pre-Columbian Americas.
In fact, the ancient Maya were so ahead of other societies even in Europe, that they had perfected the use of rubber creating various products from it 3000 years before people in the Old World even knew what it was.
The Maya created fascinating monuments, pyramids, ancient cities, writing techniques and strangely, like many other great civilizations, they too disappeared under mysterious circumstances.
But in addition to the Ancient Maya, researchers around the globe have found similar patterns in other civilizations that just as the Maya, collapsed and vanished.
Many scientists argue that it is difficult to overlook the sheer number of repeating patterns in most of these civilizations, and a study partially funded by NASA provides us with evidence that there are certain (chaotic) characteristics present in ancient cultures that lived on Earth thousands of years ago and eventually disappeared.
This is considered by many people as a sign that clearly states that ancient civilizations have reset a number of times.
In the report, applied mathematician Safa Motesharri and his “Human and Nature Dynamical” model claims that “the process of rise-and-collapse is actually a recurrent cycle found throughout history.”
“The fall of the Roman Empire and the equally (if not more) advanced Han, Mauryan, and Gupta Empires, as well as so many advanced Mesopotamian Empires, are all testimony to the fact that advanced, sophisticated, complex, and creative civilizations can be both fragile and impermanent.”
In the study, we find that“Technological change can raise the efficiency of resource use, but it also tends to raise both per capita resource consumption and the scale of resource extraction, so that, absent policy effects, the increases in consumption often compensate for the increased efficiency of resource use.”
There are widespread concerns that current trends in resource-use are unsustainable, but possibilities of overshoot/collapse remain controversial. Collapses have occurred frequently in history, often followed by centuries of economic, intellectual, and population decline. Many different natural and social phenomena have been invoked to explain specific collapses, but a general explanation remains elusive.
In this paper, we build a human population dynamics model by adding accumulated wealth and economic inequality to a predator–prey model of humans and nature. The model structure, and simulated scenarios that offer significant implications, are explained. Four equations describe the evolution of Elites, Commoners, Nature, and Wealth. The model shows Economic Stratification or Ecological Strain can independently lead to collapse, in agreement with the historical record.
The measure “Carrying Capacity” is developed and its estimation is shown to be a practical means for early detection of a collapse. Mechanisms leading to two types of collapses are discussed. The new dynamics of this model can also reproduce the irreversible collapses found in history. Collapse can be avoided, and population can reach a steady state at maximum carrying capacity if the rate of depletion of nature is reduced to a sustainable level and if resources are distributed equitably.
- Human–nature dynamics;
- Societal collapse;
- Carrying capacity;
- Overshoot vs. sustainability;
- Economic inequality;
- Ecological strain
There are widespread concerns that current trends in population and resource-use are unsustainable, but the possibilities of an overshoot and collapse remain unclear and controversial. How real is the possibility of a societal collapse? Can complex, advanced civilizations really collapse? It is common to portray human history as a relentless and inevitable trend toward greater levels of social complexity, political organization, and economic specialization, with the development of more complex and capable technologies supporting ever-growing population, all sustained by the mobilization of ever-increasing quantities of material, energy, and information. Yet this is not inevitable. In fact, cases where this seemingly near-universal, long-term trend has been severely disrupted by a precipitous collapse – often lasting centuries – have been quite common. A brief review of some examples of collapses suggests that the process of rise-and-collapse is actually a recurrent cycle found throughout history, making it important to establish a general explanation of this process (Chase-Dunn and Hall, 1997, Goldstein, 1988, Meadows et al., 1972, Modelski, 1987, Tainter, 1988, Turchin and Nefedov, 2009 and Yoffee and Cowgill, 1988).
The Roman Empire’s dramatic collapse (followed by many centuries of population decline, economic deterioration, intellectual regression, and the disappearance of literacy) is well known, but it was not the first rise-and-collapse cycle in Europe. Prior to the rise of Classical Greco-Roman civilization, both the Minoan and Mycenaean Civilizations had each risen, reached very advanced levels of civilization, and then collapsed virtually completely (Morris, 2006 and Redman, 1999). The history of Mesopotamia – the very cradle of civilization, agriculture, complex society, and urban life – presents a series of rise-and-declines including the Sumerians, the Akkadian, Assyrian, Babylonian, Achaemenid, Seleucid, Parthian, Sassanid, Umayyad, and Abbasid Empires (Redman et al., 2004 and Yoffee, 1979). In neighboring Egypt, this cycle also appeared repeatedly. In both Anatolia and in the Indus Valley, the very large and long-lasting Hittite and Harrapan civilizations both collapsed so completely that their very existence was unknown until modern archeology rediscovered them. Similar cycles of rise and collapse occurred repeatedly in India, most notably with the Mauryan and the Gupta Empires (Edwards et al., 1971, Edwards et al., 1973, Jansen et al., 1991, Kenoyer, 1998 and Thapar, 2004). Southeast Asia similarly experienced “multiple and overlapping histories of collapse and regeneration” over 15 centuries, culminating in the Khmer Empire based in Angkor, which itself was depopulated and swallowed by the forest during the 15th Century (Stark, 2006). Chinese history is, very much like Egypt’s, full of repeated cycles of rises and collapses, with each of the Zhou, Han, Tang, and Song Empires followed by a very serious collapse of political authority and socioeconomic progress (Chu and Lee, 1994, Lee, 1931 and Needham and Wang, 1956).
Collapses are not restricted to the “Old World”. The collapse of Maya Civilization is well known and evokes widespread fascination, both because of the advanced nature of Mayan society and because of the depth of the collapse (Demerest et al., 2004 and Webster, 2002). As Diamond (2005) puts it, it is difficult to ignore “the disappearance of between 90 and 99% of the Maya population after A.D. 800 …and the disappearance of kings, Long Count calendars, and other complex political and cultural institutions.” In the nearby central highlands of Mexico, a number of powerful states also rose to high levels of power and prosperity and then rapidly collapsed, Teotihuacan (the sixth largest city in the world in the 7th C) and Monte Alban being just the largest of these to experience dramatic collapse, with their populations declining to about 20–25% of their peak within just a few generations (Tainter, 1988).
We know of many other collapses including Mississippian Cultures such as Cahokia, South West US cultures such as the Pueblo and Hohokam, Andean civilizations such as Tiwanaku, Sub-Saharan civilizations such as Great Zimbabwe, and many collapses across the Pacific Islands, such as Easter Island. It is also likely other collapses have also occurred in societies that were not at a sufficient level of complexity to produce written records or archeological evidence. Indeed, a recent study (Shennan et al., 2013) of the Neolithic period in Europe has shown that “in contrast to the steady population growth usually assumed, the introduction of agriculture into Europe was followed by a boom-and-bust pattern in the density of regional populations”. Furthermore “most regions show more than one boom–bust pattern”, and in most regions, population declines “of the order of the 30–60%” can be found. The authors also argue that, rather than climate change or diseases, the timing and evidence point to endogenous causes for these collapses in 19 out of 23 cases studied, suggesting the possibility of “rapid population growth driven by farming to unsustainable levels”. Moreover, through wavelet analysis of the archeological data, S. Downey [personal communication] has shown that the average length of such boom-and-bust cycles is about 300–500 years.
In summary, despite the common impression that societal collapse is rare, or even largely fictional, “the picture that emerges is of a process recurrent in history, and global in its distribution” (Tainter, 1988). See also Yoffee and Cowgill (1988), Goldstein (1988), Ibn Khaldun (1958), Kondratieff (1984), and Parsons (1991). As Turchin and Nefedov (2009) contend, there is a great deal of support for “the hypothesis that secular cycles — demographic–social–political oscillations of a very long period (centuries long) are the rule, rather than an exception in the large agrarian states and empires.”
This brings up the question of whether modern civilization is similarly susceptible. It may seem reasonable to believe that modern civilization, armed with its greater technological capacity, scientific knowledge, and energy resources, will be able to survive and endure whatever crises historical societies succumbed to. But the brief overview of collapses demonstrates not only the ubiquity of the phenomenon, but also the extent to which advanced, complex, and powerful societies are susceptible to collapse. The fall of the Roman Empire, and the equally (if not more) advanced Han, Mauryan, and Gupta Empires, as well as so many advanced Mesopotamian Empires, are all testimony to the fact that advanced, sophisticated, complex, and creative civilizations can be both fragile and impermanent.
A large number of explanations have been proposed for each specific case of collapse, including one or more of the following: volcanoes, earthquakes, droughts, floods, changes in the courses of rivers, soil degradation (erosion, exhaustion, salinization, etc.), deforestation, climate change, tribal migrations, foreign invasions, changes in technology (such as the introduction of ironworking), changes in the methods or weapons of warfare (such as the introduction of horse cavalry, armored infantry, or long swords), changes in trade patterns, depletion of particular mineral resources (e.g., silver mines), cultural decline and social decadence, popular uprisings, and civil wars. However, these explanations are specific to each particular case of collapse rather than general. Moreover, even for the specific case where the explanation applies, the society in question usually had already experienced the phenomenon identified as the cause without collapsing. For example, the Minoan society had repeatedly experienced earthquakes that destroyed palaces, and they simply rebuilt them more splendidly than before. Indeed, many societies experience droughts, floods, volcanoes, soil erosion, and deforestation with no major social disruption (Tainter, 1988).
The same applies to migrations, invasions, and civil wars. The Roman, Han, Assyrian, and Mauryan Empires were, for centuries, completely militarily hegemonic, successfully defeating the neighboring “barbarian” peoples who eventually did overrun them. So external military pressure alone hardly constitutes an explanation for their collapses. With both natural disasters and external threats, identifying a specific cause compels one to ask, “yes, but why did this particular instance of this factor produce the collapse?” Other processes must be involved, and, in fact, the political, economic, ecological, and technological conditions under which civilizations have collapsed have varied widely. Individual collapses may have involved an array of specific factors, with particular triggers, but a general explanation remains elusive. Individual explanations may seem appropriate in their particular case, but the very universal nature of the phenomenon implies a mechanism that is not specific to a particular time period of human history, nor a particular culture, technology, or natural disaster (Tainter, 1988, Turchin, 2003 and Yoffee and Cowgill, 1988).
In this paper we attempt to model collapse mathematically in a more general way. We propose a simple model, not intended to describe actual individual cases, but rather to provide a general framework that allows carrying out “thought experiments” for the phenomenon of collapse and to test changes that would avoid it. This model (called HANDY, for Human and Nature DYnamics) advances beyond existing biological dynamic population models by simultaneously modeling two separate important features which seem to appear across so many societies that have collapsed: (1) the stretching of resources due to the strain placed on the ecological carrying capacity (Abel, 1980, Catton, 1980, Kammen, 1994, Ladurie, 1987, Ponting, 1991, Postan, 1966, Redman, 1999, Redman et al., 2004, Wood, 1998 and Wright, 2004), and (2) the economic stratification of society into Elites and Masses (or “Commoners”) (Brenner, 1985, Parsons, 1991, Turchin, 2005, Turchin, 2006, Turchin and Nefedov, 2009, Diamond, 2005 and Goldstone, 1991; Ibn Khaldun, 1958). In many of these historical cases, we have direct evidence of Ecological Strain and Economic Stratification playing a central role in the character or in the process of the collapse (Culbert, 1973, Diamond, 2005, Goldstone, 1991, Lentz, 2000 and Mitchell, 1990). For these empirical reasons, and the theoretical ones explained in Section 3, our model incorporates both of these two features. Although similar to the Brander and Taylor (1998) model (hereafter referred to as “BT”) in that HANDY is based on the classical predator–prey model, the inclusion of two societal classes introduces a much richer set of dynamical solutions, including cycles of societal and ecological collapse, as well as the possibility of smoothly reaching equilibrium (the ecological carrying capacity). We use Carrying Capacity in its biological definition: the population level that the resources of a particular environment can sustain over the long term (Catton, 1980, Cohen, 1995 and Daly and Farley, 2003). In this paper, we call these environment resources “Nature”.
The paper is organized as follows: Section 2 gives a brief review of the predator–prey model; Section 3 includes the mathematical description of HANDY; Section 4 covers a theoretical analysis of the model equilibrium and possible solutions; Section 5 presents examples of scenarios within three distinct types of societies; Section 6 gives an overall discussion of the scenarios from 5 and 7 offers a short summary of the paper and a discussion of future work.
2. Predator–Prey Model
The predator–prey model, the original inspiration behind HANDY, was derived independently by two mathematicians, Alfred Lotka and Vitto Volterra, in the early 20th century (Lotka, 1925 and Volterra, 1926). This model describes the dynamics of competition between two species, say, wolves and rabbits. The governing system of equations is
In the above system, x represents the predator (wolf) population; y represents the prey (rabbit) population; a determines the predator’s birth rate, i.e., the faster growth of wolf population due to availability of rabbits; b is the predator’s death rate; c is the prey’s birth rate; d determines the predation rate, i.e., the rate at which rabbits are hunted by wolves.
Rather than reaching a stable equilibrium, the predator and prey populations show periodic, out-of-phase variations about the equilibrium values
As indicated above, Human And Nature DYnamics (HANDY) was originally built based on the predator–prey model. We can think of the human population as the “predator”, while nature (the natural resources of the surrounding environment) can be taken as the “prey”, depleted by humans. In animal models, carrying capacity is an upper ceiling on long-term population. When the population surpasses the carrying capacity, mechanisms such as starvation or migration bring the population back down. However, in the context of human societies, the population does not necessarily begin to decline upon passing the threshold of carrying capacity, because, unlike animals, humans can accumulate large surpluses (i.e., wealth) and then draw down those resources when production can no longer meet the needs of consumption. This introduces a different kind of delay that allows for much more complex dynamics, fundamentally altering the behavior and output of the model. Thus, our model adds the element of accumulated surplus not required in animal models, but which we feel is necessary for human models. We call this accumulated surplus “wealth”.
Empirically, however, this accumulated surplus is not evenly distributed throughout society, but rather has been controlled by an elite. The mass of the population, while producing the wealth, is only allocated a small portion of it by elites, usually at or just above subsistence levels. Based on this, and on the historical cases discussed in the introduction, we separated the population into “Elites” and “Commoners”, and introduced a variable for accumulated wealth. For an analysis of this two-class structure of modern society, see Drăgulescu and Yakovenko (2001) and Banerjee and Yakovenko (2010). This adds a different dimension of predation whereby Elites “prey” on the production of wealth by Commoners. As a result, HANDY consists of four prediction equations: two for the two classes of population, Elites and Commoners, denoted by xE and xC, respectively; one for the natural resources or Nature, y; and one for the accumulated Wealth, w, referred to hereafter as “Wealth”. This minimal set of four equations seems to capture essential features of the human–nature interaction and is capable of producing major potential scenarios of collapse or transition to steady state.
A similar model of population and renewable resource dynamics based on the predator–prey model was developed in the pioneering work of Brander and Taylor (1998) demonstrating that reasonable parameter values can produce cyclical “feast and famine” patterns of population and resources. Their model showed that a system with a slow-growing resource base will exhibit overshooting and collapse, whereas a more rapidly growing resource base will produce an adjustment of population and resources toward equilibrium values. They then applied this model to the historical case of Easter Island, finding that the model provides a plausible explanation of the population dynamics known about Easter Island from the archeological and scientific record. They thus argue that the Polynesian cases where population did collapse were due to smaller maximum resource bases (which they call “carrying capacity”) that grew more slowly, whereas those cases which did not experience such a collapse were due to having a larger resource base (i.e., a larger carrying capacity). They then speculate that their model might be consistent with other historical cases of collapse, such as the ancient Mesopotamian and Maya civilizations or modern Rwanda.
However, the BT approach only models Population and Nature and does not include a central component of these historical cases: economic stratification and the accumulation of wealth. Thus, despite clear evidence for a stratified class structure in Easter Island’s history prior to the collapse (as well as for Mesopotamia, the ancient Maya, and modern Rwanda), the BT model does not include class stratification as a factor. In their model, society produces and consumes as a single homogeneous unit. We feel that a historically realistic modeling of the evolution of human–nature dynamics in these stratified complex societies cannot be achieved without including this class stratification in the model. Brander and Taylor recognize that their model is simple, and that application to more complex scenarios may require further development of the structure of the model. We have found that including economic stratification, in the form of the introduction of Elites and Commoners, as well as accumulated Wealth, results in a much richer variety of solutions, which may have a wider application across different types of societies. HANDY’s structure also allows for “irreversible” collapses, without the need to introduce an explicit critical depensation mechanism into the model as other models need to do. Thus while the Brander–Taylor model has only two equations, HANDY has four equations to predict the evolution of the rich and poor populations (Elites and Commoners), Nature, and accumulated Wealth (we examine other differences in Section 6.4 of the paper) The HANDY equations are given by:
3.1. Model Description
The total population is divided between the two variables, xC and xE, representing the population of commoners and of elites. The population grows through a birth rate β and decreases through a death rate α. β is assumed to be constant for both Elites and Commoners but α depends on Wealth as explained below.
In reality, natural resources exist in three forms: nonrenewable stocks (fossil fuels, mineral deposits, etc.), regenerating stocks (forests, soils, animal herds, wild fish stocks, game animals, aquifers, etc.), and renewable flows (wind, solar radiation, precipitation, rivers, etc.). Future generations of the model will disaggregate these forms. We have adopted a single formulation intended to represent an amalgamation of the three forms, allowing for a clear understanding of the role that natural resources play in collapse or sustainability of human societies.
Thus, the equation for Nature includes a regeneration term, γy(λ − y), and a depletion term, − δxCy. The regeneration term has been written in the form of a logistic equation, with a regeneration factor, γ, exponential regrowth for low values of y, and saturation when y approaches λ, Nature’s capacity — maximum size of Nature in absence of depletion. As a result, the maximum rate of regeneration takes place when y = λ / 2. Production is understood according to the standard Ecological Economics formulations as involving both inputs from, and outputs to, Nature (i.e., depletion of natural sources and pollution of natural sinks) ( Daly, 1996 and Daly and Farley, 2003). This first generation of HANDY models the depletion side of the equation as if it includes the reduction in Nature due to pollution.
The depletion term includes a rate of depletion per worker, δ, and is proportional to both Nature and the number of workers. However, the economic activity of Elites is modeled to represent executive, management, and supervisory functions, but not engagement in the direct extraction of resources, which is done by Commoners. Thus, only Commoners produce.
It is frequently claimed that technological change can reduce resource depletion and therefore increase carrying capacity. However, the effects of technological change on resource use are not unidirectional. Technological change can raise the efficiency of resource use, but it also tends to raise both per capita resource consumption and the scale of resource extraction, so that, absent policy effects, the increases in consumption often compensate for the increased efficiency of resource use. These are associated with the phenomena referred to as the Jevons Paradox, and the “Rebound Effect” ( Greening et al., 2000, Polimeni et al., 2008 and Ruth, 2009). For example, an increase in vehicle fuel efficiency tends to enable increased per capita vehicle miles driven, heavier cars, and higher average speeds, which then negate the gains from the increased fuel-efficiency. In addition, technological advances can enable greater resource extraction and throughput, which then appears as increases in the productivity of other factors of production. As Daly points out, much of the increase in productivity in both agriculture and industry in the last two centuries has actually come from increased (rather than decreased) resource throughput (Daly, 1991). A decline in the price of a resource is usually thought to reflect an increase in the abundance of that resource, but in fact, it often reflects that the resource is simply being extracted more rapidly. Rather than extend carrying capacity, this reduces it. Over the long-term, per capita resource-use has tended to rise over time despite dramatic technological advances in resource efficiency. Thus, the sign and magnitude of the effect of technological change on resource use varies and the overall effect is difficult to predict. Therefore, in this generation of HANDY, we assume that the effects of these trends cancel each other out. The model will be developed further to allow the rates of these technology-induced trends to be adjusted in either direction.
Finally, there is an equation for accumulated Wealth, which increases with production, δxCy, and decreases with the consumption of the Elites and the Commoners, CC and CE, respectively. The consumption of the Commoners (as long as there is enough wealth to pay them) is sxC, a subsistence salary per capita, s, multiplied by the working population. The Elites pay themselves a salary κ times larger, so that the consumption of the Elites is κsxE. However, when the wealth becomes too small to pay for this consumption, i.e., when w < wth, the payment is reduced and eventually stopped, and famine takes place, with a much higher rate of death. κ is meant to represent here the factors that determine the division of the output of the total production of society between elites and masses, such as the balance of class power between elites and masses, and the capacity of each group to organize and pursue their economic interests. We recognize the inherent limitations, in this initial generation of our model, of holding that balance (κ) constant in each scenario, but we expect to develop κ further in later generations of HANDY so that it can be endogenously determined by other factors in the model.
CC and CE, the consumption rates for the Commoner and the Elite respectively, are given by the following equations:
Wealth threshold, wth, is a threshold value for wealth below which famine starts. It depends on the “minimum required consumption per capita”, ρ:
Even when Commoners start experiencing famine, i.e., when w ≤ wth, the Elites continue consuming unequally as indicated by the factor κ in the second term on the right hand side of Eq. (5). A graphical representation of the consumption rates are given in Fig. 2a.
The death rates for the Commoner and the Elite, αC and αE, are functions of consumption rates:
The death rates vary between a normal (healthy) value, αm, observed when there is enough food for subsistence, and a maximum (famine) value, αM that prevails when the accumulated wealth has been used up and the population starves. There are a variety of mechanisms which can reduce population when it exceeds carrying capacity, including everything from emigration, increased disease susceptibility, and outright starvation to breakdowns in social order and increased social violence, such as banditry, riots, rebellions, revolutions, and wars. These mechanisms are described in detail in Turchin (2003) but the net effect of all of them is a reduction in population, and that is what the dynamics of our model is meant to represent when we say “population decline” or “famine”. Note also that an increase in the death rates (α) is equivalent to an equal decrease in the birth rates (β). The death rates αC and αE can be expressed in terms of , a graphical representation of which is given Fig. 2b.
3.2. A Note on Units and Dimensions
There are three dimensions for quantities in HANDY:
Population (either Commoner or Elite), in units of people.
Nature/Wealth, in units of “eco–Dollars”.
Time, in units of years.
The structure of the model requires Nature and Wealth to be measured with the same units, therefore we created the unit eco-dollar. Other parameters and functions in the model carry units that are compatible with the abovementioned dimensions following Eq. (3). For example, Carrying Capacity, χ, and the Maximum Carrying Capacity, χM, defined in Section 4.1, are both expressed in units of people.
4. Equilibrium Values and Carrying Capacity
We can use the model to find a sustainable equilibrium and maximum carrying capacity in different types of societies. In order for population to reach an equilibrium, we must have αm ≤ βE ≤ βC ≤ αM. We define a dimensionless parameter, η:
Since we assume αm ≤ βC ≤ αM, η will always be bounded by 0 ≤ η ≤ 1.
4.1. Equilibrium when xE = 0 (No Elites): Egalitarian Society
Assuming xE ≡ 0, we can find the equilibrium values of the system (subscript “e” denotes the equilibrium values):
We define χ, the Carrying Capacity for the population, to be equal to xC,e in Eq. (8), i.e., the equilibrium value of the population in the absence of Elites:
Carrying Capacity can be maximized if Nature’s regeneration rate is maximal, i.e., if . This requires δ to be set equal to a value δ* that can result in a steady state with the maximum (sustainable) Population, which in this paper we call the “optimal” value of δ. From the second equation in Eq. (8), it can be seen that δ* is given by:
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