The debate on the causes of conflict in human societies has deep roots. In particular, the extent of conflict in hunter-gatherer groups remains unclear. Some authors suggest that large-scale violence only arose with the spreading of agriculture and the building of complex societies. To shed light on this issue, we developed a model based on operatorial techniques simulating population-resource dynamics within a two-dimensional lattice, with humans and natural resources interacting in each cell of the lattice. The model outcomes under different conditions were compared with recently available demographic data for prehistoric South America. Only under conditions that include migration among cells and conflict was the model able to consistently reproduce the empirical data at a continental scale. We argue that the interplay between resource competition, migration, and conflict drove the population dynamics of South America after the colonization phase and before the introduction of agriculture. The relation between population and resources indeed emerged as a key factor leading to migration and conflict once the carrying capacity of the environment has been reached.

Copyright: © 2017 Gargano et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

As a further step, we proposed a possible coordinating mechanism and implemented it into the model. More specifically, we argued that the interplay between resource competition, migration and conflict drove the population dynamics in South America after the colonization phase and before the introduction of agriculture. Unlike the no-migration case, under this second scenario, the model was able to fit remarkably well the reconstructed curve, strongly supporting the correctness of our interpretation. The rest of the paper is organized as follows. The next section illustrates our model, the subsequent one presents the results, and the last one discusses them and derives their general implications.

This intuition was further confirmed by our model, where, consistently with the mosaic concept, South America is represented as a two-dimensional lattice, with humans and resources interacting in each cell of the lattice. The model produced demographic curves that indeed were oscillating at the local level (namely the single cell) but smooth at the global one (namely the whole lattice). The reconstructed curve for the population density in South America instead exhibits high peaks and dramatic drops, from a density of SCPDs less than 0.13 to almost 0.29 around a mean carrying capacity K ≈ 0.185, which cannot be entirely attributed to the calibration process [ 13 ]. This suggests that some coordinating mechanism must have been at work to keep in phase the local oscillations.

Our research focuses on the pre-agriculture period and more specifically on the second phase, when the carrying capacity was reached: a condition which is simulated in our model. The demographic curve on that phase is characterized by sharp oscillations of large amplitude (see figure 3 in [ 13 ]), a behavior that resembles the dynamics of invasive species. Similar dynamics have indeed been observed in the field. However, the spatial scale of such observations is generally much smaller than the continental one [ 15 ]. Consequently, considering the whole South America as a single invaded site does not seem appropriate. From an ecological point of view, it could be better conceptualized as a mosaic of different ecosystems, in some cases separated by geographic barriers. Its population can hence be regarded as a meta-population split into different sub-populations, each of them occupying a patch of the mosaic [ 16 ]. This kind of representation appears especially suitable considering hunter-gatherers groups of the mid-Holocene, who normally lived within well defined territories whose size depended on the local ecological conditions [ 9 , 12 ]. While in each patch of the mosaic we can expect oscillating dynamics between human and resource populations, there is no reason why these oscillations should be in phase on a larger scale, and empirical research confirmed that they often are not [ 17 ]. As a consequence, local oscillations should compensate each other, leading to a relatively smooth curve for the global population density.

In order to shed light on this issue, we developed a model based on operatorial techniques which simulates population-resource dynamics in a pre-agricultural context at the continental scale. The model outcomes under different conditions were compared with recently available demographic data for prehistoric South America [ 13 ]. Using as a proxy for population size the number of occupied archaeological sites and the probability density of summed calibrated radiocarbon dates (SCPDs), the authors reconstructed the spatio-temporal patterns of human population growth in South America, from 14 k to 2 k years ago. This period encompassed three phases: a first phase of rapid expansion from 14 k to 9 k years ago, which represents the continent colonization, followed by approximatively 4 k years of a dynamical equilibrium around a carrying capacity, with no net growth. Finally a new growth phase occurred, due to the rising of agriculture which allowed to sustain a larger population [ 13 , 14 ].

The debate on the causes of conflict in human societies has deep philosophical roots [ 1 ], with recent works even proposing it as one of the keys to understand human evolution [ 2 ]. However, the actual extent of the conflict in small hunter-gatherer groups remains unclear [ 3 ], with some authors suggesting that large-scale warfare and mass-killing only arose with the spreading of agriculture and the building of complex societies, as before resources were too sparse to make their conquest or defense meaningful [ 3 – 6 ]. Evidence supporting this view points out to the fact that, at least in some regions, the Neolithic revolution was accompanied by “unprecedented levels” of violence [ 7 ]. Even if evidence of widespread warfare also exists for previous periods [ 8 – 11 ], the question of whether the rate of conflict in pre-agricultural societies was sufficient to significantly affect the evolution of human populations on scales larger than the single group remains open [ 12 ].

Methods

Model overview We modeled South America as a two-dimensional lattice formed by L2 different cells. Each cell includes two interacting populations and , respectively humans and natural resources. Being interested in a pre-agricultural situation, the latter population represents the only sustenance for humans. Starting from some initial conditions, populations evolve following a dynamics similar to the one of a predator-prey system, which from an ecological point of view is the most appropriate to represent a user-resource system with users depending on renewable biogenic resources [18, 19]. We explored two scenarios. In the first, no migration is allowed; in the second, humans are allowed to migrate to neighboring cells, while resources cannot diffuse across cells. Moreover, migration from a cell α to a cell β is assumed to depend on a variable K α , which reflects the local (i.e., in α) ratio between the densities of resources and humans. More specifically, humans migrate at a rate that increases for decreasing K α , simulating the fact that migration from α increases when local resources are low. After humans migrate from α to β, resources in α tend to recover, while human density in β, after an initial obvious increment, can subsequently decline. This produces a reduction of the total human populations in α and β, which we interpret as the outcome of increased mortality due to competition and/or conflict over local resources. In order to reproduce population dynamics, we employed operatorial techniques commonly used in quantum mechanics and already adopted in the analysis of population dynamics and migration [20–22]. The details are presented in the following subsections.

One-cell case Let be a system composed by two populations: the users (i.e. the humans), and the natural resources . The system can assume four basic states, while any other possible state can be obtained as a combination of these: (i) φ 0,0 , i.e. the vacuum of the system, corresponding to an almost complete absence of both humans and resources; (ii) φ 0,1 (resp. φ 1,0 ) corresponding to a low (resp. high) density of humans and abundance (resp. lack) of resources; (iii) φ 1,1 abundance of both human and resources. We associate the four basic states of the system with four independent mutually orthogonal vectors in a four-dimensional Hilbert space endowed with the scalar product 〈⋅, ⋅〉, a procedure that follows the general framework described in details in Bagarello [20] and used in several contexts. A convenient way to construct the vectors φ j, k with j, k = 0,1 makes use of two fermionic operators, a and b, which together with their adjoints, a† and b†, satisfy the canonical anticommutation relation (CAR): (1) where {x, y} = xy + yx is the anticommutator of x and y, can be both a or a†, and is the identity operator in the Hilbert space . We first introduce the vacuum vector φ 0,0 which satisfies aφ 0,0 = bφ 0,0 = 0. The existence of a non zero vector φ 0,0 with this property is a standard result in quantum mechanics [23]. The other vectors φ j, k can be constructed from φ 0,0 in the following way: The set is an orthonormal basis for , so that a general vector Ψ of can be expanded as , with . By using a standard argument, we can interpret Ψ as a state where the probability to find in the state φ j, k is given by |c j, k |2. We now introduce the number operators and of and . Then: (2) where n(a), n(b) can be either 0 or 1. It holds: and the mean values, or densities, n(a), (b) of the operators over the state Ψ are defined as , leading to the following expressions: (3) From a biological point of view, n(a) (resp. n(b)) is interpreted as the density for the population (resp. ) associated with the state Ψ. Both n(a), n(b) are real numbers between 0 and 1. For instance, if at t = 0 n(a) = 0, it means that Ψ is only a combination of φ 0,0 and φ 0,1 corresponding to the (almost complete) absence of humans (zero density for ). Conversely, if n(a) = 1 then Ψ is a combination of φ 1,0 and φ 1,1 corresponding to the maximum density for humans.

2D case A square lattice divided in L2 cells is introduced in the 2D case. In each cell the two populations and interact as in the previous case. In each cell α = 1, ⋯, L2, we introduce the fermionic operators a α , and the related number operators for and b α , and for . As before, we suppose that these operators satisfy the standard CAR anticommutation rules: (4) (5) Moreover holds for all α. In Eq (4) is the identity operator on the Hilbert space , with , endowed with the scalar product 〈⋅, ⋅〉. Extending what we have done before, we first introduce the vacuum vector of the system , where are two L2-dimensional vectors. The vacuum is annihilated by all the operators a α , b α , i.e., We then construct the states of the basis of by acting with the operators over , (6) where , are all the possible L2-dimensional vectors whose entries are only 0 or 1. In fact, , any other choice would destroy the state. Notice that we can construct at most 2L2 vectors and 2L2 vectors , so that we can build 4L2 possible pairs . The set of all the vectors obtained by this construction forms an orthonormal basis of . Moreover: (7) (8) for all α = 1, ⋯, L2. For more details on CAR, we refer to Roman [23]. The vectors can now be interpreted similarly to φ j, k in the previous section. The vacuum , for instance, describes a situation where all the lattice cells hold few of both humans and resources. The vector with and describes the same situation, except that there is a large amount of humans in the first cell and of resources in the last one. A generic state Ψ(0) of the system can be written as a linear combination of the elements in : (9) where are complex scalars such that . From Eqs (7) and (8), the are eigenstates of the number operators and the densities of and in α are simply their related eigenvalues. This is true at the initial time t = 0. At a later time, we need to compute the mean values over the state Ψ(t) describing the system at time t: (10) (11) As already stated (see also [20]), these mean values are phenomenologically interpreted as densities of humans and resources in the cell α.