Toward a theory of punctuated subsistence changeIsaac I. T. Ullaha,1, Ian Kuijtb, and Jacob Freemanc

aSchool of Human Evolution and Social Change, Arizona State University, Tempe, AZ 85287-2402; bDepartment of Anthropology, University of Notre Dame,Notre Dame, IN 46556; and cDepartment of Sociology, Social Work and Anthropology, Utah State University, Logan, UT 84322-0730

Edited by Dolores R. Piperno, Smithsonian Institution, Fairfax, VA, and approved June 12, 2015 (received for review February 20, 2015)

Discourse on the origins and spread of domesticated species focuseson universal causal explanations or unique regional or temporaltrajectories. Despite new data as to the context and physicalprocesses of early domestication, researchers still do not understandthe types of system-level reorganizations required to transition fromforaging to farming. Drawing upon dynamical systems theory andthe concepts of attractors and repellors, we develop an understand-ing of subsistence transition and a description of variation in, andemergence of, human subsistence systems. The overlooked role ofattractors and repellors in these systems helps explain why the ori-gins of agriculture occurred quickly in some times and places, butslowly in others. A deeper understanding of the interactions of alimited set of variables that control the size of attractors (a proxyfor resilience), such as population size, number of dry months, netprimary productivity, and settlement fixity, provides new insights intothe origin and spread of domesticated species in human economies.

complex adaptive systems | subsistence change | origins of agriculture |social-ecological systems

Recent work highlights that the transition from foraging tofarming was nonlinear and heterogeneous (e.g., refs. 17). Thatis, rather than an inevitability, early shifts to food production wereonly one of many possible outcomes that could have been reachedfor a given set of dynamically interacting social and ecologicalvariables. Although the foragerfarmer transition is one of themost fundamental changes in human evolution, our understandingof the foragerfarmer transition is theoretically fractious (1, 36,836), with scholarly discourse dominated by the assumption thatthe forager adoption of domesticates was driven either by sub-sistence necessity or because domesticates provided a desirableopportunity or assurance. Given our adaptive flexibility, however, itis clear that both options are possible, depending on the situation.The challenge is distinguishing the contextual settings in whichadoptions were linked to necessity versus opportunity. Simply put,there is currently no sufficient theory to explain the nonlinear andcontingent worldwide transitions from foraging to farming.In this paper, we use concepts from Dynamical Systems Theory

(SI Text S1) to model subsistence variation among contemporaryethnographic groups from an evolutionary perspective. We focuson a critical question: How can researchers use the concepts ofattractors and repellorsso integral to understanding many non-linear dynamical systemsto describe variation in the subsistencestrategies of human societies? This framing provides general in-sights into why transitions from foraging to farming, at a globalscale, exhibit nonlinearity and heterogeneity, and why the shift wassometimes gradual, and other times punctuated. Drawing uponcomparative ethnographic case studies, we formalize the use ofcross-cultural data in a theory-backed methodology to ascertainhow the attractor/repellor concept can be use to describe sub-sistence variation in human societies. Our analysis elucidatesbroad, multidimensional trends across the breadth of human sub-sistence practices and is a step toward developing a theory ofnonlinear subsistence change in human societies.

Dynamical Systems and Human EcologyWe start from the premise that human societies are complexadaptive systems, with heterogeneous agents at several levels oforganization, who interact with each other and the biophysical

environment (SI Text S1, SI Text S2, and Figs. S1 and S2).Complex adaptive systems are a special subset of dynamicalsystems in which the independent decisions and actions of in-dividual components of the system (i) self organize, (ii) changeover time, (iii) interact to derive novel emergent properties ofthe system, and (iv) may adapt to work in the interest of thesystem as a whole (37, 38). Viewing human societies as complexsystems helps generate hypotheses for the way in which humannatural system components should interact and evolve over time(Figs. S3 and S4). Here, we propose that (i) the feedback be-tween humans and their resources may lead to attractors andrepellors that describe the variation in human subsistencesystems and that (ii) a small number of controlling variablesmay disproportionately affect change within subsistence systems.If these propositions have merit, then, using a few importantvariables, we should be able to identify clusters of societies in theethnographic record that occupy similar, although not iso-morphic, attractors. These clusters will be separated by economicvoids where subsistence strategies are unlikely to persist due tothe presence of repellors.In dynamical systems theory, attractors are system states toward

which (all else being equal) a complex adaptive system will tend toevolve, and, conversely, repellors are system states that the systemwill tend to avoid (39, 40). In other words, an attractor is a con-figuration of system subcomponents that are relatively stable overtime whereas a repellor is a configuration that is not. Thus, at anygiven time, a complex adaptive system is always evolving toward anattractor, but rarely reaches it (i.e., the system is never in equi-librium). Importantly, the size of an attractor determines itsresilience, which is how much environmental change a system cancope with before feedbacks between variables change (SI Text S2and Fig. S5) (41). Attractors emerge from these kinds of feedbacks,and our working supposition is that human subsistence attractorsemerge from cross-scale feedbacks between human and naturalresources in socio-natural systems.

Significance

The questions of how, when, and why humans transitionedfrom hunting and gathering to food production are importantto understand the evolution and sustainability of agriculturaleconomies. We explore cross-cultural data on human subsis-tence with multivariate techniques and interpret the resultsfrom the perspective of human societies as complex adaptivesystems. We gain insight into several controlling variables thatmay inordinately influence the possibilities for subsistencechange and into why the foragerfarmer transition occurredquickly in some cases and more gradually in others.

Author contributions: I.I.T.U. designed research; I.I.T.U. performed research; I.I.T.U. contrib-uted new reagents/analytic tools; I.I.T.U., I.K., and J.F. analyzed data; and I.I.T.U., I.K., andJ.F. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Data deposition: The R code used to create all of the figures in the main text and theimputed SCCS datasets used in the analyses are made available as a free download at thefollowing persistent URL: figshare.com/articles/Cross_cultural_data_for_multivariate_analysis_of_subsistence_strategies/1404233.1To whom correspondence should be addressed. Email: iullah@asu.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1503628112/-/DCSupplemental.

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Complex systems are usually in a dynamic state that is largelydominated by the force of a local attractor, but not solely con-trolled by it. Because these systems are dynamical, evolving, andopen, their attractors and repellors also change in strength andconfiguration as system subcomponents and emergent, macrolevelconditions change (Fig. S5). This property holds major implica-tions for how complex adaptive systems change over time. Aninteresting set of possibilities for change occurs when a system ispositioned between two or more attractors. Systems operating insuch intermediary locations are available to move from the sphereof influence of one attractor to another. These transitions canoccur either gradually or as very rapid-phase changes, dependingon the resilience of each of the attractors (SI Text S2 and Figs. S3and S4) (39). These properties also control whether the transitionis immediately reversible (i.e., related to choices of opportunity) orrepresents a true bifurcation (i.e., related to choices of necessity).In dynamical systems, it is often the case that a few variables

control the dynamics of the system, including the emergence ofattractors and repellors (42). Controlling variables are those thathave a disproportionate impact on the feedback between indiv-iduals and their use of resources and thus strongly affect thestructure of a system. For example, Holling (43) found that, fromhostparasite, to lakes, to the boreal forest, only three to fourvariables control the structure of dynamical systems models ofecosystems. Similar observations have been made of dynamicalsystems models that describe socio-natural systems (e.g., refs. 4449). In one of these examples, Freeman and Anderies (50) arguethat the ratio of population to resources controls a regime changefrom a mobile foraging to an intensive, property-based foragingattractor. The meta-insight that we pull from these models is thatreframing human subsistence systems as emergent outcomes ofcomplex adaptive processes in a nonlinear system suggests that it isuseful to identify potential controlling variables. Such variableswould point the way forward for further theoretical development.

Materials and MethodsAll explanations for subsistence change in archaeology are built on an un-derstanding of ethnographically documented societies (e.g., refs. 17, 23,4549, 51, and 52). Much of this knowledge derives from traditional eth-noarchaeological research from the last century that provided case studiesof human groups living traditional, or near-traditional, lifeways (SI Text S3).These studies often resulted in models designed to address specific archae-ological problems (e.g., refs. 53 and 54) or to relate archaeological phe-nomena to theories of human behavior (e.g., ref. 55). Comparativeethnoarchaeology, on the other hand, attempts to identify global patternsof human behavior to either generate new hypotheses or to evaluate eco-logical models (e.g., refs. 47, 50, 51, and 5658).

In the tradition of comparative approaches, in this study, we used auto-mated multidimensional techniques to identify patterns in human sub-sistence variability with which to assess the attractor/repellor hypothesis. Weexamined subsistence, mobility, economic, demographic, and environmentaldata for the 186 societies of the Standard Cross-Cultural Sample (SCCS) (SIText S3 and Tables S1 and S2) (59). We supplemented these data with in-formation about Net Primary Production (NPP) from the Atlas of the Bio-sphere (60, 61) (SI Text S3 and Table S2). We address issues of missing data,potential autocorrelation (i.e., Galtons problem), and alternative expla-nations of patterning within the SCCS data in SI Text S4 and SI Text S5. Ourworkflow followed Le Roux and Rouanets (62) geometric data analysis (SIText S6) and was designed to graphically identify and intuit natural divisionsin cross-cultural data by combining the result of multivariate clustering withdimensional reduction analyses. Specifically, we used K-medoids clusteringpaired with nonmetric multidimensional scaling (NMMDS) or canonicalcorrespondence analysis (CCA). We used the workflow to plot societies in aphase space created by the ordination routine, which condenses many di-mensions of subsistence activities, mobility characteristics, settlement types,and so on, into a biplot. We used plot symbology to thematically display therelationship between the spatial patterning of societies within that phasespace to the input variables, other variables, or other data analyses. We usedthese techniques to ascertain whether the global SCCS sample of subsis-tence systems is characterized by clusters and gaps that might be analogousto attractors and repellors. All analyses were undertaken in R, using thecluster and vegan multidimensional analysis modules. We include ourimputed datasets and R code as supplemental data (SI Text S7).

ResultsOur multivariate analysis of human subsistence identified fourdiscrete clusters consistent with attractors separated by gapsthat may be repellors (Fig. 1A). These clusters are as follows:intensive agriculture, extensive agriculture, pastoralism, andhunting or gathering terrestrial or marine resources. Data depthanalysis (via hierarchical convex hulls) of these clusters (Fig. 1B)delimits a potential range of influence and resilience of each hy-pothesized subsistence attractor (i.e., society clusters) and revealsthe potential location of repellors (i.e., gaps) that may separatethem. There is weak cluster separation between intensive andextensive agricultural groups, but stronger separation of huntergatherers and extensive agriculturalists, huntergatherers andherders, and herders and intensive agriculturalists. Herders clusterquite distantly from extensive agriculturalists, as do huntergath-erers from intensive agriculturalists. Comparison of the clusterresults with traditional ethnographic subsistence classification ofeach SCCS society (Fig. 1A) suggests that there exists smaller scalevariation within the clusters surrounding each macrolevel attrac-tor, which larger samples may help illuminate. Fig. 1B illustratesthe variables that influence the positioning and internal configu-ration of the clusters. Different configurations of subsistence ac-tivities, settlement fixity, and population density seem to be majorfactors pulling the clusters apart from one another.CCA allows us to constrain the axes to be linear combinations

of a subset of the analyzed variables and evaluate the importanceof particular variables in the clustering results. Mobility anddemographic variables (Fig. 2A) account for 27.6% of the totalvariability, with the variation roughly split along two axes: mo-bility (settlement pattern and settlement fixity) and demography(total population, population density, and community size). En-vironmental variables (Fig. 2B) account for 18.3% of the totalvariability, with the variability again split between two axes: tem-perature seasonality (absolute latitude, average temperature, andnumber of frost months) and water availability (number of drymonths, average precipitation, coefficient of variation in pre-cipitation, and NPP). Subsistence variables (Fig. 2C) account for18% of the total variability, with three major axes aligned to de-gree of reliance on agriculture, herding and trade, and huntinggatheringfishing, respectively.Finally, we can gain insight into how clustering is reconfigured

as variables change by dividing and analyzing subsets of theoriginal dataset partitioned by cutoffs in NPP, absolute latitude,residential mobility, and total population. Viewing the resultstogether (Fig. 3) facilitates several unintuitive observations: (i) Notall of the macrolevel clusters are present in each partition, sug-gesting potential clinal variation in attractors. (ii) Our analysisseparates huntergatherers from fishers in some partitions, sug-gesting that the two may be weakly distinct attractors. (iii) Fishingshares some of the same tensions of extensive agriculture, implyingthat people intensively using an abundant wild resource (such asfish) may face pressures similar to those relying on extensivemanagement of cultivated resources. (iv) Cluster separation (i.e.,the resilience of attractors) in the partitioned phase spaces changesbetween partitions. This separation illustrates the interplay be-tween a set of highly influential constraints of mutually in-compatible variables. For example, a high degree of residentialmobility is largely incompatible with a very large population, anda high degree of residential sedentism conflicts with a high re-liance on animal products. These incompatibilities seem to shapethe resilience of subsistence attractors in different environments.Following societies across the partitioned datasets provides

insight into the conditions in which societies transition betweensubsistence systems. For example, the Ainu, Nama, and Chiri-cahua graph variably near two or more different clusters as thedataset is parsed. These groups can be thought of as in a systemstate that is far from any one local attractor, or, perhaps, haveindividuals who are capable of moving between attractors. TheNamaKhoisan-speaking hunterherders in the Western Caperegion of South Africabegan to adopt some agriculture whenunder pressure from European settlers in the late 19th and early

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20th centuries (63, 64). At the time of study, the Ainu werehunterfishers who had traded with feudal Japanese merchants forat least 200 y. The trade had severely depleted important gameanimals, and the Ainu had begun small-scale swidden horticultureto supplement wild foods (65). In these two cases, the transitionphase seems to have been rapid and not very stable. The Chiri-cahua, although under similar pressures, may be a case in whichindividuals opportunistically moved between foraging and agri-culture subsistence attractors. Although the central and westernChiricahua bands were foragers, Oplers informant from theeastern band was clear that maize was a favorite food and that itwas obtained for planting whenever possible (66). He relates,Only about six or seven families out of the hundred in a bigencampment might plant corn....The seeds came from the Mexi-cans, and many didnt plant because they didnt have seeds (ref.66, p. 374). The presence of a stable agricultural system on thelandscape made the opportunity available for individuals to ex-periment in some years without abandoning a foraging economy.

Discussion and ConclusionA Dynamical Systems Approach to Human Subsistence. The transi-tion from foraging to farming seems nonlinear and heteroge-neous at a global scale. Why? Our goal in this paper has beento widen our perspective from this particular transition andoutline how concepts from nonlinear dynamical systemstheoryattractors and repellorshelp us describe variation andchange in human subsistence systems. The spatial clustering and

separation apparent in multidimensional plots are consistentwith the presence of attractors and repellors in cross-culturaldata (Figs. 13), and the nature of these attractors and repellorsmay relate to zones of stability and instability within the totalspectrum of food procurement strategies in coupled human andnatural systems. Detailed dynamic models will help us to un-derstand the feedback processes that are likely to control theemergence and resilience of subsistence attractors and repellors.We must also advance by investigating alternative explanationsfor the clustered patterning in the SCCS. We have determinedthat some of these potential alternatives, such as randomness,sociocultural autocorrelation, or observer biases are unlikely toproduce the observed variation (SI Text S5 and Fig. S6). Twoother alternativesthe effect of competitive exclusion andsampling biaswarrant further consideration. The former re-quires investigation as an important controlling variable in itsown right, and the second requires an expanded set of inputsocieties that also includes prehistoric case studies (SI Text S5).Nonetheless, we suggest that the interaction of socio-naturalforces keeps the subsistence practices of human societies near anattractor. Populations may make a transition to a new sub-sistence mode when system conditions change enough to erodethe resilience of their former attractor. But under many systemconfigurations, societies may remain near an attractor even in theface of increasing pressures that might otherwise induce gradualchange. In these cases, dynamical systems theory suggests thatcritical thresholds may exist that, once surpassed, propagate quick

Fig. 1. The results of nonmetric multidimensionalscaling (NMMDS) and subsequent K-medoids clusteranalysis of subsistence, mobility, and demographicSCCS variables. (A) Biplot showing four macrolevelclusters. Clusters are represented by point color andtwo levels of hierarchical convex hulls. They mayreflect subsistence attractors for huntergathererfishers, herders, extensive agriculture, and intensiveagriculture. Point symbology represents SCCS sub-sistence labels (v858). Selected societies are labeled.(B) The same biplot, but showing the weightings ofinput variables instead of input cases, indicatingtheir importance in determining clusters.

Fig. 2. The results of canonical correspondence analysis (CCA) and subsequent K-medoids cluster analysis. In all biplots, colors and hierarchical convex hulls rep-resent clusters, symbology represents SCCS v858, and select societies are labeled. (A) This biplot shows the results of a CCA conducted where the axes were con-strained to be linear combinations of variables related to subsistence economy. (B) This biplot shows the results of a CCA conducted where the axes were constrainedto environmental variables. (C) This biplot shows the results of a CCA conducted where the axes were constrained to variables related to mobility and demography.

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phase changes from one attractor (and period of stability) to an-other, which are difficult to recoup (39). This possibility allows forboth gradual and rapid transition, and explains why a transitioncould occur quickly or slowly.The four potential macroscale subsistence attractors identified

by our analysis (pastoralism, hunting/gathering/fishing, extensiveagriculture, and intensive agriculture) (Fig. 1) are influenced by asmall group of controlling variables, including temperature andprecipitation seasonality, environmental productivity, degree ofresidential mobility, and population size. Mobility and popula-tion size may be the most influential (Fig. 2). Other variablesincluding social oneslikely become more important in themanifestation of smaller scale patterning, but, at either scale, so-cieties seem to coalesce around general ways of doing things dueto the inherent incompatibilities between important limiting var-iables. For example, our analysis shows that relatively high pop-ulations can be sustained in low-productivity environments in oneof two ways: pastoralism or agriculture (Fig. 3 C and J). However,pastoralism requires a high degree of residential mobility, whichcauses a fundamental tension between the two lifeways in thatenvironmental context. Transitioning between the two would re-quire considerable impetus (e.g., total devastation of flocks orharvests). This idea feeds back into the issue of opportunity versusnecessity in subsistence transition.As conditions changeparticularly in major controlling vari-

ablesdifferent subsistence transitions may become more or lesspossible than others. For example, intuitively, a horticultural

society already familiar with domestic plants might opportunis-tically transition to an intensive agricultural strategy, but only ifenvironmental productivity is high enough to allow it. It is alsoimportant to consider, however, that not every macroscale clus-ter was present in the phase space when the data were parti-tioned across major variables (Fig. 3). Although the samerelative patterning of attractors and repellors persisted wherepresent, the overall strength of the different attractors changedbetween partitions. We infer from this consistency that it is easierto make certain subsistence transitions under some socio-envi-ronmental conditions than others. For example, a transition fromhorticulture to intensive agriculture may be more possible in thetropics than it would be in a temperate climate region (holdingpopulation constant) (Fig. 3 D and E), and a transition fromhuntinggathering to shifting cultivation may be easier in lowNPP environments (Fig. 3C). Extrapolating these insights overarchaeological time scales, the interplay of climate or environ-mental change, technological changes, population change, andchanges to the abundance of resources can all affect the resil-ience of particular subsistence attractors and, thus, subsistencetransitions. The key is that controlling variables, like residentialmobility (67, 68) and resource seasonality (69), may have a muchgreater affect on an attractors resilience than others and so evensubtle changes in them could lead to punctuated change.We argue that dynamic optimal foraging models that include

the process of niche construction will be useful in evaluatingwhether and how subsistence attractors emerge in socio-natural

Fig. 3. The results of multiple NMMDS and K-medoids cluster analyses, where the 186 societies were divided into unique subsets by partitions in importantstructuring variables (indicated below). In all of the biplots, clusters are represented by point color, point symbology represents SCCS v858, and two levels ofhierarchical convex hulls are shown. Select SCCS societies are labeled in each plot. (AC) Partitioned by cutoffs in net primary production (NPP) (A, NPP > 4; B, 4 >NPP 1.5; C, NPP < 1.5). (DF) Partitioned by cutoffs in absolute latitude (D, latitude 23.5; E, 23.5 < latitude 50; F, 50 < latitude). (GI) Partitionedby cutoffs in residential mobility (G, impermanent and permanent settlements; H, rotating and semisedentary settlements; I, migratory and semi-nomadic settlements). (JL) Partitioned by cutoffs in total population (J, population > 10,000; K, 10,000 > population 1,000; L, 1,000 > population).

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systems. The centrality of residential mobility in our resultssuggests that the evolutionary ecology of time allocation may beimportant to understand the emergence of subsistence attractorsfrom human-resource feedbacks. For example, changing resil-ience of a hunting and gathering attractor may feedback into theratio of search time to handling costs, creating critical thresholdswhere it suddenly becomes more profitable to remain sedentarythan to search and travel. Another possibility is that the evolu-tion of complex social structures is a response to fundamentaltrade-offs in time allocation. Historic Tuareg pastoralistsinhabited the heart of the Sahara, but they had a complex castesystem of nobles, serfs, slaves, and freedmen that solved the timeconflicts necessary to allow them to do agriculture near oaseswhile also engaging in mobile pastoralism and long-distancetrade of salt and other market resources (70). Dynamic systemsmodels of the ideal despotic distribution are particularly in-teresting for exploring such possibilities (71).

The Transition to Food Production. There is evidence that terminalPleistocene huntergatherer groups were intensively managinglandscapes in many parts of the world (72, 73), including in areasthat would become centers of early agriculture (2, 74). It isnatural to envision the origins of food production as an extensionof this management, which, in many ways, it was. Our analysisillustrates that the transition was not simple, however, nor needit have been gradual. It is unlikely that food production wouldalways have emerged from the same initial conditions, but it isclear that significant socio-natural changes must have accrued topredicate the transition. Landscape management practices byLate Pleistocene hunters, such as moving species to new terri-tories (73) and management of woodlands by burning (72, 73),combined with socio-technical changes, such as decreased resi-dential mobility and increased storage (52), all may have weak-ened the resilience of huntinggathering attractors by changingconstraints on human subsistence and human-resource feedbacks.It is also possible that increasing population densities or climaticchanges may also have narrowed the resilience of hunting andgathering attractors in some areas, increasing the chance of acritical transition (SI Text S2). All of these changes would havealso increased the temporal stability of subsistence practices thatincorporated management of central predomesticated plants.These processes in and of themselves were likely not enough tohave induced a full transition. Exemplifying this complexity, wenote that the intensive wild resource users in the SCCS (e.g., thefisher societies), although likely at the margins of the zone of in-fluence for the hypothesized hunting macroscale attractor, none-theless clustered with hunters in most of our analyses. It is,therefore, likely that, whereas prior system reorganization wasrequired, the actual transition to food production itself occurredonce critical thresholds in the optimal allocation of time for

subsistence tasks or other important constraining variables weresurpassed, and the everyday requirements for a huntergathererlifeway could no longer be met. The pathways of individual groupsacross this threshold were likely unique and related to stochasticevents as well as system-component interaction and the feedbackeffects of accumulated change. But once they transitioned, itwould have been difficult to recover a hunting and gathering wayof life.Viewing human subsistence systems as complex adaptive

phenomena provides a unique opportunity to define both in-ternal and external mechanisms for subsistence change, withoutgiving primacy to one over the other or requiring any one factor.It also provides a basis for understanding why long periods ofpre-domestication cultivation (6) occur for some crops in somecenters of early agriculture, but not for others. Dynamical sys-tems theory combines with evolutionary perspectives to offer aset of governing mechanisms that help to show how majortransitions would occur under particular conditions. Importantly,these mechanisms do not preclude the inclusion of historicalcontingency in explanatory models of subsistence transition. In-deed, under this framework, we can seek to better understandhow each case of novel transition was predicated in system-levelchanges to major controlling variables, their incompatibilities,currencies, and the thresholds that existed before the change.Homo sapiens have been remarkably creative in the de-

velopment and adoption of different subsistence practices in arange of global prehistoric and historic contexts. For many de-cades, anthropologists have studied adaptive subsistence variation,and much headway has been made in describing, categorizing, andmodeling the breadth of diverse economic practices in the humanpast. Researchers have developed a general framing of humansubsistence through time, have shown that humans can undertakevery different subsistence strategies in similar physical environ-ments, and have developed a reasonably good understanding ofthe timing and general circumstances of major subsistence tran-sitions, such as the shift to plant and animal domestication aroundthe world. Drawing on the growing awareness that the transitionfrom foraging to farming was gradual in some places and punc-tuated in others, we have put forth a nonlinear theory of sub-sistence transition, including the transition from foraging tofarming. We argue that modeling the emergence of alternativeattractors and repellors helps define subsistence system variationand may help us understand why the transition from foraging tofarming was at times gradual or punctuated.

ACKNOWLEDGMENTS. Advice from C. Michael Barton, J. Marty Anderies, andLoukas Barton helped us to orient our thoughts. Comments from twoanonymous reviewers helped improve the paper. The University of PittsburghCenter for Comparative Archaeology provided support for some of the research.

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