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Human and nature dynamics (HANDY): Modeling inequality and use of resources in the collapse or sustainability of societies /Safa Motesharrei a,⁎, Jorge Rivas b, Eugenia Kalnay c

Summary
Collapses of even advanced civilizations have occurred many times
in the past five thousand years, and they were frequently followed by
centuries of population and cultural decline and economic regression.
Although many different causes have been offered to explain individual
collapses, it is still necessary to develop a more general explanation. In
this paper we attempt to build a simplemathematicalmodel to explore
the essential dynamics of interaction between population and natural
resources. It allows for the two features that seemto appear across societies
that have collapsed: the stretching of resources due to strain
placed on the ecological carrying capacity, and the division of society
into Elites (rich) and Commoners (poor).
The Human And Nature DYnamical model (HANDY)was inspired by
the predator and prey model, with the human population acting as the
predator and nature being the prey. When small, Nature grows exponentially
with a regeneration coefficient γ, but it saturates at a maximum
value λ. As a result, the maximum regeneration of nature takes
place at λ / 2, not at the saturation level λ. The Commoners produce
wealth at a per capita depletion rate δ, and the depletion is also proportional
to the amount of nature available. This production is saved as accumulated
wealth, which is used by the Elites to pay the Commoners a
subsistence salary, s, and pay themselves κs, where κ is the inequality
coefficient. The populations of Elites and Commoners grow with a
birth rate β and die with a death rate α which remains at a healthy
low level when there is enough accumulated food (wealth). However,
when the population increases and the wealth declines, the death rate
increases up to a famine level, leading to population decline.
We show how the carrying capacity – the population that can be indefinitely
supported by a given environment (Catton, 1980) – can be
defined within HANDY, as the population whose total consumption is
at a level that equals what nature can regenerate. Since the regrowth
of Nature is maximum when y = λ / 2, we can find the optimal level
of depletion (production) per capita, δ* in an egalitarian society where
xE ≡ 0, δ∗∗(≥δ∗) in an equitable society where κ ≡ 1, and δ* in an unequal
society where xE ≥ 0 and κ N 1.
In sum, the results of our experiments, discussed in Section 6, indicate
that either one of the two features apparent in historical societal
collapses – over-exploitation of natural resources and strong economic
stratification – can independently result in a complete collapse. Given
economic stratification, collapse is very difficult to avoid and requires
major policy changes, includingmajor reductions in inequality and population
growth rates. Even in the absence of economic stratification, collapse
can still occur if depletion per capita is too high.However, collapse
can be avoided and population can reach equilibrium if the per capita
rate of depletion of nature is reduced to a sustainable level, and if resources
are distributed in a reasonably equitable fashion.