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Epidemiological modeling

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lower classes and emergence of non-conformism and dissent once a religious belief has reached

dominance), we assume that the rate of loss of belief increases as the number of humans colonized by a

ere arh inherf reprvolutiole requg theor mu

advocated by Cavalli-Sforza and Feldman (1981) and Boyd and

odels7).ls to

develop between human populations that live in isolation from

Contents lists available at ScienceDirect

.e

Journal of Theor

Journal of Theoretical Biology 267 (2010) 676684into contact after they have diversied in isolation. Such anE-mail address: doebeli@zoology.ubc.ca (M. Doebeli).each other (e.g. on different continents), because differentclimatic and biological circumstances may lead different culturalcontent to thrive in different areas of the globe. Much thought hasbeen given to understanding what happens when cultures come

0022-5193/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jtbi.2010.09.013

Corresponding author at: Department of Zoology, University of BritishColumbia, Vancouver, BC, Canada V6T 1Z4.tion of intellectual content such as ideas and theories.Such a perspective of cultural evolution has been lucidly

address the problem of cultural diversication. It seems perhapsrelatively easy to understand how cultural differentiation canand theories are more successful at such reproduction throughtransmission than others, hence there is typically differentialreproductive success. As a consequence, there is cultural evolu-

cultural evolution, we believe that such epidemiological mmight be well suited to study cultural dynamics (Dietz, 196

Here we apply mathematical epidemiological modeand it can be passed on to the brains of other individuals, therebyreproducing itself. Such reproduction typically occurs withmodication and through mixing and amalgamation with otherideas present in the human population. For a multitude ofpotential reasons, both cognitive and selective in nature (Henrich,2009; Henrich et al., 2008; Richerson and Boyd, 2005), some ideas

1981), it seems that to date this epidemiological perspective hasnot been rigorously adopted as a basis for the mathematicalmodeling of cultural evolution. There is a large body of theoreticaland mathematical literature on the modeling of the epidemiolo-gical dynamics of pathogens in humans and many other animals.While one naturally has to be careful in adopting such models for1. Introduction

Evolution can occur whenever ththat produce other such units whicof the parent units. If the units oreproductive output, there will be etual content can satisfy these simptheory can be viewed as colonizinhuman (or animal). It can developcontent, and we use evolutionary theory to show that this frequency dependence can lead to the

emergence of coexisting clusters of different cultural types. The different clusters correspond to

different cultural traditions, and hence our model describes the emergence of distinct descendant

cultures from a single ancestral culture in the absence of any geographical isolation.

& 2010 Elsevier Ltd. All rights reserved.

e units of reproductionit some characteristicsoduction vary in theirnary change. Intellec-irements. An idea or abrain of an individualtate within that brain,

Richerson (1985), and by now there is a rather large body ofliterature on cultural evolution (see e.g. Richerson and Boyd,2005). In particular, population genetic models have beenadopted early on to study cultural evolution (Boyd and Richerson,1985; Cavalli-Sforza and Feldman, 1981). On the other hand, anepidemiological perspective of cultural content colonizing humanbrains has been advocated in various verbal narratives by anumber of researchers (e.g. Dietz, 1967; Sperber, 1996). However,despite some early attempts (e.g. Cavalli-Sforza and Feldman,particular religious phenotype increases. This generates frequency-dependent selection on culturalA model for the evolutionary diversica

Michael Doebeli a,b,, Iaroslav Ispolatov a,b

a Department of Zoology, University of British Columbia, Vancouver, BC, Canada V6T 1b Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V

a r t i c l e i n f o

Article history:

Received 19 January 2010

Received in revised form

29 July 2010

Accepted 8 September 2010Available online 18 September 2010

Keywords:

Cultural evolution

Religion

a b s t r a c t

We address the problem o

human hosts and using div

evolution of pathogens o

dynamics known from the

humans. Rates of transmis

of loss of ideas (loss of

interactions between hosts

conformism can be negat

journal homepage: wwwon of religions

1Z4

ltural diversication by studying selection on cultural ideas that colonize

ication of religions as a conceptual example. In analogy to studying the

ymbionts colonizing animal hosts, we use models for hostpathogen

ical epidemiology. In these models, religious content colonizes individual

of ideas between humans, i.e., transmission of cultural content, and rates

ef) are determined by the phenotype of the cultural content, and by

rrying different ideas. In particular, based on the notion that cultural non-

frequency-dependent (for example, religion can lead to oppression of

lsevier.com/locate/yjtbi

etical Biology

as well as a very recent split of the Anglican church based on

M. Doebeli, I. Ispolatov / Journal of Theoretical Biology 267 (2010) 676684 677approach would attempt to determine the winners among apreexisting set of different cultures (e.g. Diamond, 2005; Limet al., 2007). This approach is roughly equivalent to studyingspecies selection between already established species andforegoes the question of how diversity arose in the rst placewithin a single culture.

However, cultural differentiation also seems to occur whenpeople adopting diverging cultures live together, which appearsto have been an important mode of cultural diversicationthroughout history. Several instances of diversication in religionmay serve as paradigms for such processes. For example, the splitof the protestant from the catholic church in the 16th centuryoccurred from within an essentially entirely catholic culture, anddespite some subsequent spatial segregation of the divergingreligions (due, among other things, to violent conicts), the tworeligions essentially coexisted since the split. It has been arguedthat this split was caused by a decline in the moral authority ofthe catholic leadership (Tuchman, 1985), i.e., by processesoccurring within the catholic church that led certain people toovercome the peer pressure to conform to the existing doctrine,and generate and/or become susceptible to new religious ideas.Thus, processes within the catholic church may have createdconditions that favoured the emergence of a dissident religiousstrain. In a more recent example, Whitehouse (1995) has observedan ongoing splitting off of minor sectarian movements from amainstream religious organization in Papua New Guinea. In thesociological literature, such a new branch of religion that isformed from within and in coexistence with the old religion isoften referred to as a sect (Stark, 1996), while a followinggenerated by an original set of ideas, sufciently unrelated to anyexisting religion (at all, or typical to a particular geographicalarea) is dened as a cult. In this language, our the modelspresented here would (loosely) refer to the formation of new sectsrather than new cults.

In particular, we propose to model cultural diversicationarising without spatial segregation. We use the emergence of newreligions, or sects, as a schematic backdrop for informing ourmodeling assumptions, but conceptually, our approach can beused to describe general scenarios for the evolution of new sets ofcultural content (e.g. languages or ideologies) from a singleancestral culture. Borrowing from the epidemiological literature,our models incorporate human individuals as hosts for religiouscontent. Just as in traditional epidemiological models, religiouscontent can be lost by a human host, and it can be transmittedbetween human hosts. We use established mathematical techni-ques from evolutionary theory to describe the dynamics of hostpopulations that harbour a variable set of religious ideas that are,for simplicity, described by a continuous real variable. The traitvalue of a religious idea determines the ideas propensity of beinglost by their human hosts, as well as their success in colonizingsusceptible hosts. The basic question we address concerns thedistribution of religious values: when does a unimodal distribu-tion, corresponding to a human population adopting a singlereligion, split into different modes, corresponding to coexistingsubpopulations adopting different religions. Overall, the purposesof our paper are twofold. On the one hand, following mostlyverbal narratives of epidemiological terminology in the theory ofhuman culture, we advocate a more explicit use of quantitativeepidemiological modeling to understand the dynamics of culturalevolution. On the other hand, we illustrate the potentialusefulness of such modeling by presenting models for gradualcultural diversication of an ancestral culture into different andcoexisting descendant lineages. In analogy to organismic evolu-tion, where the theory of adaptive diversication in the absence ofgeographical isolation has received considerable attention in the

past decade (Dieckmann and Doebeli, 1999), such processes ofdifferent positions towards homosexual bishops and marriages.From a complimentary perspective, the loss of belief due to

overcrowding of the ancestral religion can be seen as thedeepening of a conict between non-conformists and progres-sively less exible mainstream ideologies. As the number offollowers of a religion grows, the initially diverse set of ideasinevitably solidies into a number of increasingly dogmaticprinciples that are easy to communicate to new adepts and thushard to alter or even doubt. A vivid example of such dogmatiza-tion is the emergence of mainstream Christianity from a largenumber of early branches with an initially diverse vision andinterpretation of what later became known as Biblical events.cultural diversication occur due to frequency-dependent selec-tion on cultural content. Our models are both general andsimplistic, and we view them as a conceptual proof of principal.Nevertheless, we hope that these models serve the purpose ofillustrating how the general mechanism of frequency-dependentnon-conformism can give rise to cultural diversity without spatialsegregation. Studying potential diversication as a consequenceof non-conformism appears to be interesting, because seeds ofnon-conformism seem to be present in almost any culturalsetting, ranging from languages to politics, economics, andreligion.

2. Model setup

Religions are sets of ideas, statements and prescriptions ofwhose validity and applicability individual humans can becomeconvinced. Thus, individual minds are the hosts of religious ideas,and these ideas can exert considerable inuence on the behaviourof their hosts. In principle, understanding the dynamics of religioncan be achieved by understanding the interaction betweenreligious ideas and their hosts, i.e., by understanding how religionaffects not only the behaviour of individual hosts, but also thesocial structure of host populations, and how human behaviour,cognitive and social structures in turn affect the transmission ofreligious ideas among host individuals. For example, hostpopulations of a given religion are often hierarchically structured,with relatively few hosts enjoying high social status, and manyhosts enjoying fewer benets from adopting the given religion. Asthe number of host individuals adopting a given religion grows,this social structure may give rise to unrest, particularly in thelower social ranks. As a consequence, individuals may be enticedto start or adopt alternative, unspoiled religions, which offerless repression, and in which they can attain improved social andspiritual status, both in terms of intellectual satisfaction andhierarchical position. For example it has been suggested thatsocial unrest led to the split of the protestant church from thecatholic church in the 16th century (Tuchman, 1985). In that time,political developments led to ever increasing nancial needs ofthe catholic church, which burdened its followers throughtaxation and other means, e.g. the sale of indulgences. This inturn led to unrest and spiritual decay and contributed to thesecession of a more democratic and less repressive religion. Inother words, hosts of Catholicism tended to lose faith due toeffects that Catholicism itself generated in their host society.Moreover, belief-losing hosts became susceptible to a similar, butdistinct type of religion that promised to improve the conditionsof these hosts, probably at least in part because the new religionwas not very common, and hence did not have the samedetrimental effects on its hosts as ancestral one. Other potentialexamples of such secessions include the diversication of Judaisminto more traditional Orthodox, Reform and Humanistic branches,Another example is the consolidation of a very multifaceted early

M. Doebeli, I. Ispolatov / Journal of Theoretical Biology 267 (2010) 676684678Marxism into rigorous ideologies of Stalinist USSR and MaoistChina with physical extermination of even the slightest dissent. Atmore advanced stages of the development of the ancestralreligion, it then becomes more promising for non-conformists tocede and start an independent school of thought rather than toargue for changes in the mainstream ideology. Of course, manyother forces impinge on the well-being of hosts of particularreligious ideas. For example, common religions may offerprotection and a well-established machinery of spiritual andeconomic support, and rare ones may suffer persecution, so thatmany dissident movements may not be successful. Thus, there arenumerous aspects of religious life that are described in terms ofpositive frequency dependence, which compete with any trendsthat are due to negative frequency dependence. Nevertheless, itseems reasonable that some fraction of followers of a religionwould tend to value freedom from dogmatism and repressionhigher than certain levels of protection and comfort, thusbecoming susceptible to dissent. Here we propose that mechan-isms such as the ones alluded to above could cause negativefrequency-dependent selection on religious ideas, and, as aconsequence, adaptive cultural diversication.

Under the perspective of host individuals being colonized byreligious ideas, it is natural to attempt a formal analysis of theevolution of religion using epidemiological models. Such modelsare very well studied in the context of disease dynamics (Otto andDay, 2007). In the simplest case, there is only one type of religionpresent, and the corresponding model describes the dynamics oftwo variables, each associated with a subpopulation of the totalhost population: S denotes the density of susceptible individualsin the host population, i.e., individuals that are not yet hosts to thegiven religion, and C denotes the density of colonized hosts, i.e.,host individuals whose minds have adopted the given religion.

Our analysis is based on the following ecological model forthe dynamics of susceptible and colonized hosts:

dS

dt rSS 1

SCKS

tSC lC

dC

dt rCC 1

SCKC

tSClC 1

Here we have assumed that both susceptible and infected hostsgrow logistically. Thus, in the absence of religious colonization,susceptible hosts have an intrinsic growth rate rS and growlogistically to carrying capacity KS, and in the absence ofsusceptibles, hosts colonized by the religion have an intrinsicgrowth rate rC and grow logistically to carrying capacity KC. Forsimplicity, we assume that offspring of religious hosts are alsoreligious (in principle, part or all of these offspring could rst jointhe susceptible class), and that offspring of susceptible hosts alsobelong to the susceptible class. The two death terms forsusceptible and colonized hosts are coupled by assuming thatgrowth depends on the sum of the two populations S and C. Inaddition, susceptible hosts adopt religion, i.e., become colonized,at a per capita rate tC that is proportional to the number ofreligious hosts. However, religious people also lose their belief at aper capita rate l and become susceptible once again, leading to adecrease in C at a rate lC and a corresponding increase in S.

To introduce variability in religion and thus to allow forreligious diversication, we expand this model by making thevery simplistic assumption that religion types are characterizedby a one-dimensional trait xwith distribution C(x). For example, ithas been proposed by Whitehouse (1995) that religions can becharacterized by either having a high frequency of low-arousalrituals, or having a low frequency of high-arousal rituals. Onecould then describe religions simply by the frequency of their

rituals. Mathematically, C(x) dx is the population density of hostscolonized by religious ideas with values in the interval (x,x+dx).To introduce frequency-dependent selection on religious content,we rst introduce a measure of overcrowding by dening, forany given type x, the function

Ax ZyaxyCydy 2

where axy is a unimodal function of the form

axy a0exp 1

2

jxyjsa

ba !3

The exponent ba in axy is a positive real number. For example,if ba 2, axy is a Gaussian function. Technically speaking, A(x)is a convolution of the density distribution C(y) with the kernelaxy. Such a convolution corresponds to a weighted sum overall densities C(y), with the weights axy. Since axy has amaximum at xy and decreases to 0 as the distance jxyjincreases, the densities C(y) of hosts colonized with religioustypes y that are very different from the focal type x have littleweight, and hence matter little for calculating the quantity A(x),where as the density of hosts colonized by types that are moresimilar to the focal x have more weight in calculating theovercrowding A(x) at x. In general, if the distribution C(x) isunimodal with a single maximum at xx0, then overcrowdingA(x), Eq. (3), tends to be large for x close to x0, i.e., for common x,and, conversely, A(x) tends to be small for x very different from x0,i.e., for rare x.

We then assume that the per capita rate of loss of the religiousidea, l, is a function of overcrowding, lAx, where l(z) increasesmonotonically with increasing z. For simplicity, we assumel(z)z. This implies that the rate of loss is high for hosts carryingreligious ideas x for which A(x) is large, whereas the rate of loss issmall for hosts carrying religious ideas x for which A(x) is small.Thus, hosts are more likely to lose common religious content thanrare religious content. As mentioned above, one rationale for thisassumption is that once a religion becomes common, the socialstructure may change such that the benets gained from adoptingthe religion decrease for the majority of hosts, so that, on average,hosts of such religion become more likely to lose belief.

With religious variability, the differential equation for C must bereplaced by a partial differential equation describing the dynamics ofthe distribution C(x). To model this, we assume that offspring ofhosts colonized by religious type x on average also carry type x, butwith a certain probability the religion carried by the offspringundergoes a small mutation (as e.g. when children adopt religiousnotions that are slightly different from those of their parents, seeHenrich, 2009). Thus, the offspring of a parent with religion y has aprobability Ny,sm to lie in the interval (x,x+dx), where Ny,sm is anormal distribution with mean the parental type y and mutationalvariance sm. We are aware of the fact that this is an extremelysimplied view of the cognitive process of learning religions(Henrich, 2009). In fact, it has been argued that cognitive biasesare the main determinants of cultural evolution, but this argumenthas been largely defused (Henrich et al., 2008). Nevertheless, thesimple models presented here do not address the importantquestions of cognitive bias and selective learning, and insteadassume a very simple form of transmission of cultural content. Itshould be pointed out, though, that it is entirely possible to includecognitive biases and more complicated forms of transmission(e.g. learning from the population mean) in epidemiological modelsof the type studied here.

With this in mind, the epidemiological dynamics for reli-giously variable host populations becomes

dS rSS 1SRxCxdx SZ txCxdxZ AxCxdx 4dt KS x x

decreases with increasing distance from the optimum x0, this

M. Doebeli, I. Ispolatov / Journal of Theoretical Biology 267 (2010) 676684 679@C

@t rC

ZyNy,smCydy

rCCx SRxCxdx

KC

txSCxAxCx

5To include mutation at birth, we have separated the birth anddeath term of the logistic equation, with rC

RyNy,smCydy describ-

ing all offspring that are born to parents with all possible religioustypes y and whose type mutated to x. In the death term,

RxCxdx

is the total population size of religious hosts.To describe recruitment of a new host to a religion, we assume

that only a susceptible individual can be converted to a bearer of acertain trait x. This is in agreement with the observation made inStark (1996) that both cults and sects recruit new members mostsuccessfully from the non-religious population rather than byconverting strong believers. We have made the additionalassumption that the transmission rate tx is a function of thereligious trait x. This function is assumed to reect some intrinsicproperties of religious ideas that determine their likelihood oftransmission to susceptible hosts. For example, some ideas mightentice their carriers to proselytize more than others, but suchactivities might come at certain costs. The function tx is assumedto reect the balance of such costs and benets. Specically, weassume that this function is unimodal, so that there is a uniqueoptimal religious type in terms of transmissibility. This intro-duces a stabilizing component of selection on the trait x, restrictingthe proliferation of very extreme ideas characterized by largedeviation of x from the optimum x0. The unimodality of tx isobviously a simplication, and in general the transmission functioncould have multiple local maxima (especially in multi-dimensionalphenotype spaces). However, the unimodality assumption is madehere for the sake of argument and to show that even the simplestassumptions can lead to interesting outcomes.

Specically, we will use the form

tx t0exp 1

2

jxx0jst

bt !6

where the exponent bt is a positive real number. Note that forbt 2, tx is a Gaussian function. The rate at which hostscolonized with type x convince susceptible individuals of theirreligion is txCx, so that the total per capita rate of transmissionfor susceptible hosts is

RxtxCxdx. Note again that this assumed

process of convincing is extremely simplied. In reality, even ifa person is successful in convincing another person of somereligious ideas, it is likely that in this newly convinced person, thereligion takes a slightly different form from that present in theperson from which the transmission occurred. Thus, culturalmutations may occur during transmission from colonized tosusceptible hosts, the magnitude of which in general may dependon the nature of religion and the power of rituals (Henrich, 2009).In addition, a newly susceptible person might adopt a religiononly after listening to various models, so that the adopted religionwould represent some average over already colonized hosts. Inour models, such effects could for example be implemented byadding a diffusion term D@2C=@x2, describing random steps in xspace during the transmission processes. However, our simula-tions show that as long as the diffusion constant D is not verylarge, adding such a diffusion term does not affect the dynamics ofthe model qualitatively, which is why we have not incorporateddiffusion into the description given here.

For colonized hosts with religious type x the per capita rate ofloss is A(x) as described above, so that the total rate of loss isRxAxCxdx. For simplicity, we assume rSrCr and KSKCK inthe sequel. We note that with the above assumptions, thereligious trait x only affects rates of loss and transmission, but itdoes not affect the birth and death rates of colonized hosts, which

are assumed non-evolving (constant) on the timescale relevant tocan be roughly interpreted as diversication in religion occurringif the advantage gained from rarity outweighs the disadvantagedue to having lower transmissibility.

Second, whether diversication, if it occurs, results in unim-odal or multimodal equilibrium distributions depends on theexponents ba and bt, i.e., on the nature of the functions axy andthe diversication of culture. Thus, selection on religion is notmediated by differential viability and/or reproductive success inthe host population, but instead by differential loss and gain ofideas by colonized and susceptible hosts which in its turndepends on the density of these hosts.

3. Results

The dynamical system given by Eqs. (5) and (6) is in generalanalytically intractable but can always be solved numerically. Suchsimulations reveal two basic dynamic regimes. In the rst one, allcolonized host are concentrated in a narrow vicinity of themaximum of the transmission rate tx. In this state, religiousvariation is controlled only by mutation, i.e. random deviations ofthe hosts from the optimal religion type, as illustrated in Fig. 1a. Inthe second regime, frequency-dependent selection on religioustypes leads to the maintenance of religious variation. Maintenanceof variation in turn occurs in two different ways. At equilibrium,the distribution of colonized hosts, C(x), is either a unimodalfunction with a large positive variance (much larger than expectedfrom mutation alone), as shown in Fig. 1b, or the equilibriumdistribution is multimodal, as shown in Fig. 1c. Multimodal patternformation as shown in Fig. 1c corresponds to the emergenceof different religions, and hence to religious diversication.

In fact, even if religious diversity ultimately manifests itself ina unimodal distribution as in Fig. 1b, starting from homogeneouspopulations essentially containing only one type of religion, thisequilibrium distribution is reached through a series of bifurca-tions into distinct religious strains, as can be seen in Fig. 1b. Overtime, different strains give rise to new strains, a process whicheventually lls in the religious space and results in a unimodalequilibrium distribution. This can be seen more clearly using theindividual-based models introduced below (see Fig. S1b).

It is worth noting that the case of Gaussian functions tx andaxy shown in Fig. 1b is special in the sense that for thesefunctions, it is possible to nd an analytical expression for theequilibrium distribution of colonized host. Specically, it is nothard to see that the equilibrium distribution C(x) must satisfy theequation

RaxyCydy txS, and if both tx and ax are

Gaussian, this equation has a Gaussian solution C(x) with widths

s2ts2a

p. This is the width of the equilibrium shown in

Fig. 1b. However, the existence of such an equilibrium is a specialproperty of the Gaussian case, and nding analytical expres-sions for equilibrium distributions in cases where the exponentsbt and ba appearing in the functions tx and axy are not equalto 2 is in general impossible. In particular, it is in general not truethat such equilibrium distributions are unimodal, as the examplein Fig. 1c shows.

Whether diversication occurs, and whether diversicationmanifests itself in unimodal or multimodal distributions, dependson the parameters of the model. First of all, diversication occurswhen sa is small enough compared to st. This is revealed bynumerical simulations, and in Appendix A we will use theframework of adaptive dynamics to provide some analyticaljustication for this threshold. Because sa is a measure for howfast religious types can gain an advantage by being different fromcommon types, and st measures how fast transmissibilitytx. Generally speaking, multimodal distributions, and hence

M. Doebeli, I. Ispolatov / Journal of Theoretical Biology 267 (2010) 676684680-3 0 3x

0

100

200

t

-100

1000

2000

3000

4000

t

0

10000

20000

30000

40000

50000

C(x)diversication into multiple distinct religious strains, requirelarger exponents in these functions. For example, in Fig. 1bshowing unimodal diversication, these exponents were set to 2,i.e., both functions axy and tx were of Gaussian form. Asmentioned, in this case the dynamical system given by Eqs. (5)and (6) has an equilibrium distribution of colonized hosts that isitself Gaussian, and hence unimodal. However, increasing theseexponents to ba 3 and bt 3, as in Fig. 1c, results in multimodalequilibrium distributions. Thus, platykurtic functions axyand tx that fall off less sharply from their maximum tend tofavour multimodal diversication. This is in agreement with resultsfrom models of adaptive diversication in organismic evolution(Doebeli et al., 2007; Leimar et al., 2008; Pigolotti, 2010).

Some of the above arguments can be made more precise usingthe framework of adaptive dynamics (Dieckmann and Law, 1996;Geritz et al., 1998; Metz et al., 1996) to describe the culturalevolutionary dynamics resulting from the epidemiological assump-tions described in the Model setup. Adaptive dynamics has provento be a very useful tool for identifying various scenarios ofevolutionary diversication and speciation in organismal biology(e.g. Dieckmann and Doebeli, 1999; Dieckmann et al., 2004), andAppendix A describes how this theory can applied in the presentcontext. In contrast to the often analytically intractable partialdifferential equation models used above, adaptive dynamics theoryallows one to derive certain analytical results about the conditionsthat lead to cultural diversication. For example, one can easilyprove that if the two exponents ba and bt occurring in the functionsax,y (Eq. (4)) and in the function tx (Eq. (6)) are equal to 2, thendiversication occurs whenever saost (see Appendix A).

-3 -2 -1

Fig. 1. Evolution of the religious distribution C(x) obtained via numerical solution of Ediversication, i.e., a narrow unimodal distribution; (b) diversication in the form

distributions, with each mode representing a separate emerging religion. In panel (d)

parameters used in panel (a) (solid line), panel (b) (dashed line), and panel (c) (dot-das

a0 0:003 for all panels (these parameters were chosen so as to maintain a suitable posa 1, bt ba 2; panel (b): st 1, sa 0:5, bt ba 2; panel (c): st 1, sa 0:5, b0 10x

-3 0 3x

0

100

200

300

400

500

600

700

t4. Discussion

The purpose of this paper is to translate verbal narratives ofcultural epidemiology into quantitative epidemiological modelingof cultural evolution, and to illustrate the potential usefulness ofsuch models for understanding processes of cultural diversica-tion that are due to frequency-dependent selection on culturalcontent, rather than based on geographical isolation or otherdiversifying effects. Naturally, cultural evolution is an extremelycomplicated and multifaceted process, where diversication,shifts and mergers (Hochberg, 2004) between cultures are theproduct of cooperation and competition between a multitude offorces driving and inhibiting diversity. These forces can be veryindirect, such as religion-induced xenophobia that reduces thespread of epidemics and thus reinforced isolation and diversica-tion (Fincher and Thornhill, 2008). Our goal in this paper is not toinvestigate all possible reasons for the diversication of systemsof ideas, schools of thought, and religions, but to suggest alogically appealing and observationally relevant description of aso far underexplored, but potentially important driving forcebehind such diversication.

Thus, as any complex phenomenon, in reality the diversity ofsystems of ideas cannot be explained by a single mechanism orprocess, and here we suggest one particular candidate for such aforce and provide simple quantitative arguments for its feasibilityas a mechanism for cultural diversication. The mechanismwe consider is negative frequency-dependent selection, orovercrowding, reecting an inherent tendency to dissentas ideologies become more dominant, rigid, and repressive.

0 1 2 3x

qs. (5) and (6). For different values of the parameters evolution results in (a) no

of a broad unimodal distribution; (c) diversication in the form of multimodal

the equilibrium distribution C(x) attained in the long-time limit is plotted for the

hed line). Parameter values were KCKS104, rCrS1, sm 0:02, t0 0:006 andpulation size in the individual-based models used for Fig. S1). Panel (a): st 0:5,t ba 3.

M. Doebeli, I. Ispolatov / Journal of Theoretical Biology 267 (2010) 676684 681Tuchman (1985) has argued that such dissent has been a majordriving force for the split of protestantism from the catholicchurch in the early 16th century.

Our hypothesis neither builds on nor contradicts the majorityof existing views of cultural diversication. It simply provides anew theoretical point of view, hopefully lling some gaps in thecomplete description of this complex phenomenon. In presentingour model we kept only the details that are relevant for theparticular mechanism of diversication under consideration, andwe did not consider other features that may exist and playimportant roles in the evolution of culture and religions, yetwould not alter the basic conclusions of our particular conceptualapproach. This allows us to show that in principle, negativefrequency dependence is sufcient to drive religious diversica-tion and allows one to develop mathematical intuition about whatmight enhance or inhibit such diversication.

Consequently, we have applied the theory of evolutionarydiversication to cultural evolution, with the evolution of religionas a schematic backdrop. Using a simple mathematical modeladopted from the epidemiological literature, we have shown thatin principle, a sufcient overcrowding of followers of a main-stream religion can lead to splitting and diversication ofreligions, which manifests itself either as a broadening of theoriginal religion into a wide continuous ensemble of religioustypes, or in splitting into several separate confessions or sects. It isimportant to realize that this type of diversication occurs notbecause of spatial separation between different cultures, butbecause of frequency-dependent selection on religious types thatis mediated by interactions between human carriers of differenttypes. Thus, this type of diversication occurs in situ from a singleancestral religion. The historic record contains many examples ofboth types of diversication occurring in our models: emergenceof partially overlapping sects that differ from each other, forexample, in the details of the interpretation of holy texts (such asthe proliferation of protestant denominations in the UnitedStates), and major splits that lead to the emergence of separatereligions, such as between the Catholic, Protestant, and EasternOrthodox churches. The initial branches often further diversify,for which the fragmentation of the Protestant church, whichpeaked in the 19th century, may be a good example. Some of thebranches may later merge again, such as in the reunication of theRussian Orthodox Church and the Russian Orthodox ChurchOutside of Russia, which were divided by the 1917 revolutionbut reunited very recently. The sequential branching and laterreunication can be observed in the behaviour of our model (e.g.Fig. 1b). A more recent example of the type of culturalevolutionary branching modeled here may be occurring in PapuaNew Guinea, where Whitehouse (1995) described the coexistenceof a mainstream religious cult with periodically emergingsectarian splinter groups. We think that it might be interestingand fruitful to investigate the driving force for the emergence ofsuch modes of religiosity (Whitehouse, 1995) based on theperspectives of frequency-dependent selection and evolutionarybranching in religious content.

Even though we have used religion as a guiding example forour modeling approach, it should be pointed out that our modelsmay serve more generally as a conceptual basis for understandinggeneral processes of cultural diversication that are due tofrequency-dependent non-conformism, e.g. in politics or in theevolution of language, as well as in more mundane areas, such asthe evolution of fashion trends. Thus we purposefully avoided anyreference to supernatural concepts (Atran, 2002; Boyer, 2001),intrinsic to religions, but usually irrelevant in more generalsecular cultural contexts. Evidently, the inclusion of such religion-specic concepts could affect certain quantitative aspects of our

model (such as transmission rates and the probability of beliefloss, see Henrich, 2009 for an example), yet the underlyingnegative frequency dependence paradigm and qualitative conclu-sions would remain similar. Our models illustrate that diversify-ing processes should be expected to operate whenever thelikelihood of secession from a dominant culture increases withgrowth of dominance of the mainstream culture. Intuitively, it isnot hard to imagine that the attractiveness of a culture diminishesas the culture becomes more dominant, dogmatic, and perhapsoppressive, and that the desire to stand out and be differentincreases in more conformist cultures.

In this paper we have deliberately avoided the term meme(Dawkins, 1976) for denoting a unit of cultural or religiouscontent, because in the literature, this term is rather loaded andits use subject to considerable controversy. For example, the useof memes in cultural evolution is sometimes criticized becausememes are supposedly discrete entities that replicate faithfully. Incontrast, in the present context, memes of religious contentwould be continuously varying rather than discrete entities, withdifferent memes given by different real numbers. Also, suchquantitative memes are not necessarily replicated faithfully, sincethe copy y of meme x that is transmitted from one human host toanother might have type yax. In fact, the assumptions fortransmission in our model are very simplistic, and one couldenvision much more complicated transmission schemes, in whichfor example the meme colonizing a new host is an average of asample of memes from already colonized hosts, and in which theacquisition of new memes by susceptible hosts is biased bycognitive processes. In traditional evolutionary terms, the issue ofwhether memes are categorical and discrete or quantitative isanalogous to whether one studies evolutionary models withdiscrete or continuous trait values, and of course, depending onthe problem studied, either approach is feasible. Similarly, theissue of the modes of cultural transmission would, in classicalevolutionary biology, roughly correspond to questions about themodes of reproduction and inheritance, and in principle, evolu-tionary models are not restricted to certain classes of reproduc-tion and inheritance. For example, if cultural reproduction, i.e.,transmission, occurs through an averaging process, this wouldcorrespond to a process of blending inheritance (Cavalli-Sforzaand Feldman, 1981). Such cultural mixing could in principleimpede the emergence of new cultural strains (similar to whenrandom mating impedes speciation in organismic evolution). Inthis situation, cognitive constraints, such as an inability to learnfrom averaging of distinctly different models, could lead tocultural assortment during transmission and thereby promotediversication (similar to when assortative mating promotesspeciation in organismic evolution). It would seem of interest topursue such extensions in future work. Overall, we think that ourmodels for cultural evolution could well be formulated in terms ofa generalized theory of memes, in which memes are quantitative,and in which transmission of memes is not necessarily faithfuland may involve cognitive as well as selective biases. This viewseems to be in agreement with a recent paper by Henrich et al.,2008, who tried to clarify a number of misconceptions about thetheory of cultural evolution.

We think that our paper makes two potentially novelcontributions. First, it uses explicit epidemiological models todescribe the dynamics of cultural evolution, and second it studiesthe process of cultural diversication from a single ancestralreligion in well-mixed, i.e., geographically unstructured populations.Sets of cultural ideas, such as languages, fashions, and ideologies,clearly exhibit reproduction and heredity through their transmis-sion between human hosts. Of course, these ideas ultimately needtheir human bearers for survival and reproduction (for example, abooks content only comes alive once the book is read). Just as

the survival and reproduction of symbionts and pathogens is tied

then investigates the fate of rare mutant types that appear in the

morphic for type x, the distribution C(z) is a delta function with total

M. Doebeli, I. Ispolatov / Journal of Theoretical Biology 267 (2010) 676684682to their effects on their hosts, the evolutionary fate of culturalideas is tied to their impact on human individuals. And just asviewing individual organisms as hosts of evolving symbionts orpathogens offers the appropriate perspective for studying theevolution of those symbionts and pathogens, viewing humanindividuals as hosts of evolving cultural content offers a usefulperspective.

Formal epidemiological models for cultural evolution do notseem to have been used very much in the past, even though thefeasibility of the approach was already pointed out in Cavalli-Sforza and Feldman (1981) and Dietz (1967). An interestingexception is Lafferty et al. (2008), who use explicit epidemiolo-gical models to analyze the spread of (terrorist) ideologies. Justlike ours, their models are very general and conceptual, but theyare also potentially fruitful, because framing the spread ofideology in an epidemiological context allows one to identifythe conditions that are most favourable to such spread, and themeasures that would allow one to control the spread of detrimentalideologies, e.g. by identifying the various factors that impinge onthe basic reproductive number R0 of an infectious ideology.

Epidemiological terminology and narratives have been used incultural contexts by well-known scholars such as RichardDawkins and Daniel Dennett. Indeed, the fact that such prominentgures use epidemiological metaphors for culture seems to callfor a more quantitative investigation of cultural epidemiology. Inour opinion, one advantage of using this modeling framework isthat it shifts the perspective from considering the adaptive valueof culture for human hosts to considering the adaptive value ofcultural traits for the cultural meme itself. This perspective iscommon when studying the epidemiology of microbial pathogensor symbionts, where an evolutionary understanding of microbialtraits is primarily based on their adaptive value for the microbes(which is often mediated through their effects on the humanhosts). A similar perspective might be useful for studying culturalepidemiology. This is not to say that other approaches cannot beput to very good use when studying the adaptive value of culturaltraits for the cultural memes themselves. For example, Boyd andRicherson (1985) have already pointed out, using models adoptedfrom population genetics, that cultural traits can increase infrequency even if they are maladaptive for their human hosts.Nevertheless, we think that there is potential in the formalepidemiological approach for making progress in understandingcultural evolution based on adaptation in cultural memes.

Regarding processes of cultural diversication, a more tradi-tional approach consists of viewing different human populationsas carrying different cultural ideas, and of investigating competi-tion between such human populations. In the language of hostpathogen models, this would correspond to considering differenthost populations carrying different pathogens and asking which ofthe host populations can outcompete the other. Because in thisperspective success is based on characteristics imparted orimposed by the pathogen on a group of hosts, this perspective isakin to group selection. In contrast, studying the evolution ofpathogens or the evolution of cultural content in a single hostpopulation is based on individual selection on the pathogens or onthe cultural content. In our model, this difference to traditionalapproaches is reected in the fact that the religious trait x doesnot affect survival and reproduction in the host. Thus, selection onculture is not mediated by differential viability or reproductivesuccess in the host population, but by the fact that differentreligious ideas have different rates of being transmitted tosusceptible hosts, and different rates of being lost from colonizedhosts (Henrich, 2009). Loss of belief is frequency-dependent,because the rate of loss depends on overcrowding, which is thedriving force of diversication. Boyd and Richerson (1985) have

already pointed out that frequency-dependent selection canweight C(x) centered at x. Therefore, A(x)C(x) (Eq. (2)). Eqs. (5) and(6) then become a system of two ordinary differential equations:

dS

dt rS 1 SCx

K

txSCxa0CxCx 7

dCxdt

rCx 1 SCxK

txSCxa0CxCx 8

It is easy to see that this system has a unique equilibrium

S,C Ka0a0tx

,Ktx

a0tx

9

at which both S40 and C40. Moreover, the Jacobian matrix ofsystem (7), (8) at the equilibrium S,C has two negativeeigenvalues, and the equilibrium is globally stable in the sensethat the system will converge to this equilibrium from any initialcondition with both densities 40.

Let us assume that the host population is colonized by a singleresident type x, and that the resident dynamics given by Eqs. (7)and (8) has settled at its equilibrium S,C. This equilibriumconstitutes the environment for a rare mutant type y that appearsresident population, e.g. because one of the hosts colonized by theresident religious type has slightly changed their belief and is nowhost to a slightly altered mutant type.

To do this, we rst have to consider the dynamics ofmonomorphic resident populations. If the population is mono-maintain cultural polymorphisms. However, these earlier treat-ments did not describe the gradual emergence of a diversied setof coexisting cultural lineages out of a single ancestral lineage, asexemplied by cultural evolutionary branching.

Just as in hostpathogen or hostsymbiont systems, coevolu-tion between humans and culture may be very important, and onecan easily envisage many extensions of the model presented hereto more complicated scenarios, in which the effects of culture onindividual hosts as well as on the demographics of entire hostpopulations are described in more mechanistic detail (e.g. interms of propensity of host reproduction, sacrice, and suscept-ibility to diseases, Fincher and Thornhill, 2008), and in whichgenetic evolution in the host occurs as a response to constraintsimposed by cultural content, which in turn changes culturalopportunities. We believe that the perspective of cultural ideas asthe evolutionary units will be very useful for such studies.Cultural content is best viewed not as xed and pre-existing, butas evolving due to its effect on human individuals, who ultimatelydecide whether to accept or reject such content and howvigorously to spread it upon acceptance.

Acknowledgments

M.D. gratefully acknowledges the support of NSERC (CANADA)and of the Human Frontier Science Program.

Appendix A

To augment the analysis based on partial differential equationdescribed in the main text, here we use the mathematicalframework of adaptive dynamics (Dieckmann and Law, 1996;Geritz et al., 1998; Metz et al., 1996) to describe the evolutionarydynamics of culture based on the epidemiological assumptionsdescribed in the Model setup section of the main text. In theadaptive dynamics framework, one considers monomorphicresident populations consisting of a single religious type, andin the host population. If the mutant is rare, its logistic growth

term is determined by the total resident density SC, itstransmissibility is ty, and its rate of loss of belief isAy a0ayxC. Therefore, the growth of the population ofhosts colonized by the mutant religion type y is

dCydt

rCy 1 SCK

tySCyayxCCy 10

The invasion tness f(x,y) of a rare mutant y in the resident x liesat the basis of adaptive dynamics analyses and is dened as theper capita growth rate of y-types, i.e., by the right-hand side ofEq. (10) divided by C(y):

f x,y r 1 SCK

tySayxC 11

According to general theory (Dieckmann and Law, 1996), theadaptive dynamics of the religious trait x is then given by theselection gradient

Dx @f x,y@y

y x

tuxS 12

function is necessarily 0 at a singular point. Thus, genericallythe invasion tness function either has a maximum or aminimum at x0. It is shown from adaptive dynamics theory(Geritz et al., 1998) that if x0 is a tness minimum, this generatesthe phenomenon of evolutionary branching. Once the resident isat x0, every nearby mutant can invade. Moreover, two nearbymutants on either side of the singular value x0 can coexist,leading to religion populations consisting of two coexistingstrains. Finally, in each of these two strains selection favourstrait values lying further away from the singular point, whichmeans that the two strains will diverge evolutionarily.The phenomenon of convergence to a singular point that is atness minimum and subsequent emergence and divergence ofcoexisting strains is called evolutionary branching, and thesingular point is called an evolutionary branching point. Forexample, if we assume that the two exponents ba and btoccurring in the functions ax,y (Eq. (3)) and in the function tx(Eq. (6)) are equal to 2, then one can show that the singular pointx0 is a tness minimum if

mod

an e

ts in

M. Doebeli, I. Ispolatov / Journal of Theoretical Biology 267 (2010) 676684 683More precisely, the adaptive dynamics of the trait x is

dx

dt mDx 13

where m is a quantity describing the rate at which resident typesgive rise to mutational variants.

The analysis of the evolutionary dynamics given by (13)proceeds in two steps. First one nds stable equilibria of thedynamical system (13), and then one checks the evolutionarystability of these equilibria, as follows. Equilibria of (13) are pointsx* in trait space satisfying D(x*)0. In the present case, i.e., withD(x) given by (12), there is only one such point: the maximum ofthe function tx, x*x0. Dynamic stability of this so-calledsingular point is determined by the derivative of D(x) at thesingular point, i.e., by dD/dx(x0). In the present case, thisderivative is proportional to the second derivative of tx at x0,which is negative. Therefore, the singular point x*x0 is a locallystable attractor for the dynamics (13), and it follows that startingfrom any initial resident value x, the religious trait will convergeto the value x0.

However, despite this convergence stability the singular pointx0 need not be evolutionarily stable. Evolutionary stability isdetermined by the shape of the invasion tness function aroundthe singular point. Note that by denition of the singular point asa solution of D(x*)0, the rst derivative of the invasion tness

-3 0 3x

0

1000

2000

t

-100

1000

2000

3000

t

Fig. S1. Evolution of the religious distribution C(x) obtained from individual-baseddiversication. (b) Continuous sequential branching in the Gaussian case results in

equilibrium distribution in Fig. 1b. (c) Branching stops after two bouts and resuldeterministic model in Fig. 1c, which exhibits ve clusters, the individual-based modesaost 14

In particular, evolutionary branching occurs if sa is small enoughcompared to st.

A single bout of evolutionary branching leads to coexistence ofdiverging strains, and it is in principle possible to analyze the(two-dimensional) adaptive dynamics of these coexisting strainsusing invasion tness functions. In a typical scenario, the twocoexisting strains evolve to a new equilibrium (i.e., a new singularpoint in two-dimensional trait space), and this singular point mayor may not be a branching point. If it is, further bouts ofevolutionary diversication occur, resulting in the coexistence ofmore than two strains. If it is not, evolution comes to a halt in adiversied population consisting of two distinct and coexistingstrains. Adaptive dynamics after diversication into two coexist-ing strains can in principle be studied analytically (in a similarway as above, see e.g. Dieckmann and Law, 1996), but it is alsoillustrative to study the evolutionary dynamics in individual-based models. In such models, the various terms on the right handside of Eqs. (4) and (5) are interpreted as rates at which birth,death, transmission and loss of belief occur, resulting in astochastic model for the evolutionary dynamics. The detailedsetup of these models is described in Appendix B. Fig. S1 showsdifferent scenarios of evolutionary branching occurring in theindividual-based model.

0 10x

-3 0 3x

0

1000

2000

3000

t

els. Parameter values are the same as in the corresponding panels of Fig. 1. (a) No

ssentially unimodal distribution of religious types, corresponding to the unimodal

the coexistence of three distinct religious clusters. Note that in contrast to thel only shows three clusters due to nite population size.

Appendix B

To construct individual-based stochastic models that corre-spond to the deterministic model given by Eqs. (4) and (5), wehave to distinguish the different types of events that can occur at

probability distribution with mean 1/E, where E is the total eventrate. Thus, if the total event rate E is high, the time lapse Dtbetween one event and the next is small, and vice verse if the totalevent rate is low. Starting from some initial population containingS0 susceptible hosts and C0 hosts colonized by religious typesx01, . . . ,x

0N0

at time 0, iteration of the computational steps describedabove generates the stochastic cultural evolutionary dynamics ofa nite population in continuous time.

M. Doebeli, I. Ispolatov / Journal of Theoretical Biology 267 (2010) 676684684birth rate of colonized hosts, BC, is rCC, where S and C are thenumber of susceptible and colonized hosts present in thepopulation at any given time (note that in contrast to Eqs. (4)and (5), where S and C are population densities, and hence realnumbers, in the individual-based models S and C are integers). Forboth susceptible and colonized host individuals, the per capitadeath rate is rS(S+C)/KSrC(S+C)/KC, and total death rates DS andDC for susceptible and colonized individuals are rSS(S+C)/KS andrCC(S+C)/KC, respectively. For a host colonized by religious type x,the per capita rate at which this type is transmitted to susceptiblehosts is txS, where tx is the transmission function (6). The totalrate of transmission, T, is therefore

PitxS, where the sum runs

over all colonized hosts. Finally, the per capita rate of loss ofreligion of host individuals colonized by religious type x is givenby c(A(x))A(X) (Eq. (2)), so that the total rate of loss, L, isPiAx.

The individual-based model is implemented as follows. At anygiven time t, all individual rates as well as the total rates BS, BC, DS,DC, T and L are calculated as described above. Then the type ofevent that occurs next, birth or death of a susceptible or acolonized host, transmission of religious content, or loss ofreligious content, is chosen with probabilities proportional tothe total rates for these events (i.e., with probabilities BS/E, etc.,where E BSBCDSDCTL is the total event rate). For theevent chosen, the individual to perform this event is chosen withprobabilities proportional to the individual rates for the chosenevent. For example, if loss of belief is the chosen event, individual iis chosen to lose belief with probability A(x)/L, where x is thereligious type of individual i. This individual is then removed fromthe population of colonized hosts, and the number of susceptiblehosts is augmented by 1. Similarly, if the chosen event istransmission, individual i among the colonized hosts is chosenfor transmission with probability txS=T , where x is again thereligious type of individual i. Individuals for birth and deathevents are chosen analogously. If an individual dies it is removedfrom the population (and the numbers S or C are updatedaccordingly). If a susceptible individual gives birth the number S isaugmented by 1. If an individual colonized with type x gives birth,a new colonized host is added to the population carrying a type xuthat is chosen from a normal distribution with mean the parentaltype x and a certain (small) width sm. This reects mutationduring transmission of religious content from parent to offspring.

Performing one individual event in the manner describedabove completes one computational step in the individual-basedmodel, which advances the system from time t to time tDt inreal time. To make the translation from discrete computationalsteps to continuous real time, Dt is drawn from an exponentialReferences

Atran, S., 2002. In Gods we Trust. Oxford University Press, Oxford.Boyd, R., Richerson, P.J., 1985. Culture and the Evolutionary Process. University of

Chicago Press, Chicago.Boyer, P., 2001. Religion Explained: The Human Instincts that Fashion Gods, Spirits

and Ancestors. Basic Books, New York.Cavalli-Sforza, L., Feldman, M., 1981. Cultural Evolution. Princeton University

Press, Princeton.Dawkins, R., 1976. The Selsh Gene. Oxford University Press, Oxford, UK.Diamond, J.M., 2005. Guns, Germs, and Steel. W.W. Norton.Dieckmann, U., Doebeli, M., 1999. On the origin of species by sympatric speciation.

Nature 400, 354357.Dieckmann, U., Doebeli, M., Tautz, D., M.J. (Eds.), 2004. Adaptive Speciation.

Cambridge University Press, Cambridge, UK.Dieckmann, U., Law, R., 1996. The dynamical theory of coevolution: a derivation

from stochastic ecological processes. J. Math. Biol. 34, 579612.Dietz, K., 1967. Epidemics and rumours: a survey. Journal of the Royal Statistical

Society 130, 505528.Doebeli, M., Blok, H.J., Leimar, O., Dieckmann, U., 2007. Multimodal pattern

formation in phenotype distributions of sexual populations. Proc. R. Soc.London B 274, 347357.

Fincher, C.L., Thornhill, R., 2008. Assortative sociality, limited dispersal, infectiousdisease and the genesis of the global pattern of religion diversity. Proc. R. Soc. B275, 25872594.

Geritz, S., Kisdi, E., Meszena, G., Metz, J., 1998. Evolutionarily singular strategiesand the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12,3557.

Henrich, J., 2009. The evolution of costly displays, cooperation and religion:credibility enhancing displays and their implications for cultural evolution.Evol. Human Behav. 30, 244260.

Henrich, J., Boyd, R., Richerson, P.J., 2008. Five misunderstandings about culturalevolution. Human Nat. 19, 119137.

Hochberg, M.E., 2004. A theory of modern cultural shifts and meltdowns. Proc. R.Soc. B 271, 313316.

Lafferty, K.D., Smith, K.F., Madin, E.M.P., 2008. Natural security: a Darwinianapproach to a dangerous world. In: Sagarin, T.T.R.D. (Ed.), University ofCalifornia Press, Ltd., London, England, pp. 186206.

Leimar, O., Doebeli, M., Dieckmann, U., 2008. Evolution of phenotypic clustersthrough competition and local adaptation along an environmental gradient.Evolution 62, 807822.

Lim, M., Metzler, R., Bar-Yam, Y., 2007. Global pattern formation and ethnic/cultural violence. Science 317, 15401544.

Metz, J., Geritz, S., Meszena, G., Jacobs, F., van Heerwaarden, J., 1996. In: van Strien,S., Verduyn Lunel, S. (Eds.), Stochastic and Spatial Structures of DynamicalSystems. North Holland, Dordrecht, Netherlands, pp. 183231.

Otto, S.P., Day, T., 2007. A Biologists Guide to Mathematical Modeling in Ecologyand Evolution. Princeton University Press, Princeton.

Pigolotti, S., Lopez, C., Hernandez-Garca, E., Andersen, K.H., 2010. How Gaussiancompetition leads to lumpy or uniform species distributions. Journal ofTheoretical Ecology 3 (2), 8996.

Richerson, P.J., Boyd, R., 2005. Not by Genes Alone: How Culture TransformedHuman Evolution. University of Chicago Press, Chicago.

Sperber, D., 1996. Explaining Culture: A Naturalistic Approach. Blackwell, Oxford,UK, Cambridge, MA.

Stark, R., 1996. The Rise of Christianity: A Sociologist Reconsiders History.Princeton University Press, Princeton.

Tuchman, B., 1985. The March of Folly: From Troy to Vietnam. Ballantine Books.Whitehouse, H., 1995. Inside the Cult. Oxford University Press, Oxford, UK.that the total birth rate of susceptible hosts, BS, is rSS, and the totalthe level of individuals: birth, death, loss of belief, and transmis-sion of belief. Each of these events occur at certain rates. Forexample, all host individuals have a per capita birth rate rSrC, so

A model for the evolutionary diversification of religionsIntroductionModel setupResultsDiscussionAcknowledgmentsReferences