Synthsis btwn dmic and cultural diffusion in th Nolithic transition in Europ Joaquim Fort 1 Complx Systms Laboratory, Dpartmnt of hysics, Univrsity of Girona, ES-1771 Girona, Catalonia, Spain Editd by Marcus W. Fldman, Stanford Univrsity, Stanford, CA, and accptd by th Editorial Board Octobr 2, 212 (rcivd for rviw January 12, 212) Thr is a long-standing controvrsy btwn two modls of th Nolithic transition. Th dmic modl assums that th Nolithic rang xpansion was mainly du to th sprad of populations, and th cultural modl considrs that it was ssntially du to th sprad of idas. Hr w intgrat th dmic and cultural modls in a unifid framwork. W show that cultural diffusion xplains 4% of th sprad rat of th Nolithic transition in Europ, as implid by archaological data. Thus, cultural diffusion cannot b nglctd, but dmic diffusion was th most important mchanism in this major historical procss at th continntal scal. This quantitativ approach can b usful also in rgional analysis, th dscription of Nolithic transitions in othr continnts, and modls of many human sprad phnomna. fronts cultural transmission raction-disprsal raction-diffusion Th Nolithic transition, a major pisod in human history, is dfind as th shift from a huntr-gathrr conomy (alolithic) into anothr on basd on agricultural activitis (Nolithic) (1). In th Nar East, this transition took plac 12, y ago, and from thr it sprad across Europ until 5, y ago (2 4). Archaologists hav providd many data that mak it possibl to masur th of th sprad of th Nolithic transition, but thy disagr on which of th following possibilitis is corrct: (i) it was mainly a dmic procss (rang sprad of farmrs) (5); (ii) it was mainly a cultural on (transmission of th plants, animals and knowldg of farmrs to huntr-gathrrs (6)); or (iii) it was mainly dmic in som rgions and mainly cultural in othrs (7). It is important to not that many authors hav clarly argud for th importanc of both dmic and cultural diffusion. For xampl, Ammrman and Cavalli-Sforza (), whn introducing thir dmic diffusion modl in 1973, wrot that dmic and cultural diffusion ar not mutually xclusiv, and discussd th intractions btwn th Nolithic and alolithic populations that would hav ld to cultural diffusion and gntic clins. Ths authors also mad som crucial statmnts: Th ral qustion may wll b to valuat th rlativ importanc of dmic and cultural diffusion in diffrnt rgions of Europ bcaus in som aras both ar likly to hav contributd to th sprad of farming, but what is ncssary bfor such an attmpt can b mad is th introduction of much mor spcific modls (rf. 4, pp. 6, 135, and 62, rspctivly). This is prcisly th problm to which th prsnt papr aims to contribut: w will hr prsnt a modl, and apply it to dtrmin th importanc of dmic and cultural diffusion on th sprad rat at th continntal scal. W will also outlin how our modl could b applid to solv th sam problm at rgional scals in futur work. Svral aspcts of transitions in human prhistory hav bn analyzd during th past dcad using incrasingly rfind mathmatical modls (9 16). On th othr hand, gntic studis hav ld to an incrasing consnsus that dmic disprsal was important in th Nolithic transition in Europ (17, 1). Howvr, our purpos in this papr is not to analyz th origin and sprad of gns. Instad, w ask a diffrnt qustion: What do th archaological data tll us on th rlativ importanc of dmic and cultural diffusion on th sprad rat of th Nolithic front? An advantag of focusing our attntion on th sprad rat (not on th gns) is that it maks dirct quantitativ comparisons to archaological data possibl. Howvr, up to now, mathmatical modls of population sprad (9 16) hav not bn applid to th controvrsy btwn th dmic and cultural modls of th Nolithic transition. Can a mathmatical modl, basd on anthropologically ralistic principls, shd som light on th rlativ importanc of th dmic and cultural contributions to th sprad rat of th Nolithic transition? Hr w will show that th answr to this qustion is affirmativ, by intgrating dmic modls with cultural transmission thory in a unifid framwork. Rsults To focus on th ffcts of cultural transmission, considr first homognous systms (i.., such that th population dnsitis do not dpnd on position). Lt N stand for th total numbr of Nolithic farmrs and for th total numbr of alolithic huntr-gathrrs. Th volution quations ar (Mthods) >< N = N + f >: = f N ; [1] whr th prims dnot aftr th ffct of cultural transmission. Th positiv and ngativ signs corrspond to th fact that th transmission of th cultural trait (agricultur) incrass th numbr of farmrs and dcrass that of huntr-gathrrs. Th paramtr f in Eq. 1 is th intnsity of cultural transmission, and th intrprtation of γ is as follows. If γ < 1, thn γ is a masur of th prfrnc by huntr-gathrrs to copy th bhavior of farmrs rathr than that of othr huntr-gathrrs (convrsly, if γ > 1 huntr-gathrrs prfr to copy othr huntr-gathrrs rathr than farmrs; Mthods). In contrast to Lotka-Voltrra (4, 12, 19) or othr (16) quations for intracting populations, th volution Eq. 1 has bn drivd from cultural transmission thory (2) (Mthods). In Frquncy-Dpndnt Cultural Transmission, w includ an xplanation on why mor complicatd, frquncydpndnt modls ar not ncssary for our purposs. To analyz th spatial dynamics of th Nolithic sprad, w nd to xtnd this framwork to nonhomognous systms. W will not tak into account gographical factors (mountains, sa travl, tc.), bcaus thy hav bn rcntly shown to hav a small ffct at th continntal scal in purly dmic modls (3). Lt Nðx; y; tþ and ðx; y; tþ stand for th local population dnsitis (pr unit ara) of Nolithic farmrs and alolithic huntr-gathrrs, Author contributions: J.F. dsignd rsarch, prformd rsarch, analyzd data, and wrot th papr. Th author dclars no conflict of intrst. This articl is a AS Dirct Submission. M.W.F. is a gust ditor invitd by th Editorial Board. Frly availabl onlin through th AS opn accss option. 1 E-mail: joaquim.fort@udg.du. This articl contains supporting information onlin at www.pnas.org/lookup/suppl/doi:1. 173/pnas.1266219/-/DCSupplmntal. ALIED HYSICAL www.pnas.org/cgi/doi/1.173/pnas.1266219 AS Novmbr 13, 212 vol. 19 no. 46 1669 1673
rspctivly, at position ðx; yþ and tim t. As is shown in Mthods, th simplst nonhomognous gnralization of Eq. 1 is th following st of coupld raction-diffusion quations: N >< = D N 2 N + FðNÞ + f N t T N + γ >: t = D 2 + FðÞ f ; [2] N T N + γ whr D N and D ar th diffusion cofficints of th N and populations, rspctivly, FðNÞ = a N N 1 NKN and FðÞ = a 1 K ar logistic functions dscribing nt rproduction (with a i th initial growth rat and K i th carrying capacity of population i = N, ), and T is th gnration tim (dfind as th man tim intrval btwn th migration of an individual and on of hr/his childrn (21) and roughly th sam for both populations (22)]. Logistic growth functions ar wll-known to agr with many population data for humans (rf. 9 and rfrncs thrin). Fishr s quation is obtaind from Eq. 2 if f = : Fishr s quation was usd by Ammrman and Cavalli-Sforza in thir dmic diffusion modl (4, ). Howvr, in rcnt yars it has bn shown that raction-diffusion quations can lad to substantial rrors for human populations du to two spcial faturs of human mobility: (i) th ffct of ralistic human disprsal krnls is important and lads to th brakdown of th diffusion approximation (11), and (ii) humans nd to spnd som tim with thir parnts bfor bing abl to disprs and surviv on thir own (cohabitation ffct) (23). Taking both ffcts proprly into account, Eq. 2 is rplacd by th mor ralistic st of intgrodiffrnc, discrt-tim quations (Mthods) < : Nðx; y; t + TÞ = ðx; y; t + TÞ = whr N ~ ðx; y; tþ R T ½Nðx; y; tþš + f ~ ðx; y; tþ R T ½ðx; y; tþš f N ~ x + Δ x ; y + Δ y ; t ϕ N dδx dδ y ~ x + Δ x ; y + Δ y ; t ϕ dδx dδ y [3] R T ½Nðx; y; tþšr T ½ðx; y; tþš R T ½Nðx; y; tþš + γr T ½ðx; y; tþš R T ½Nðx; y; tþšr T ½ðx; y; tþš R T ½Nðx; y; tþš + γr T ½ðx; y; tþš ; [4] R T ½Nðx; y; tþš = a N T K N Nðx;y;tÞ, and R K N + ð a N T 1ÞNðx;y;tÞ T½ðx; y; tþš = a T K ðx;y;tþ ar th nw population dnsitis du to logistic K + ð a T 1Þðx;y;tÞ nt rproduction during th tim intrval T, and ϕ i ðδ x ; Δ y Þ is th disprsal krnl of population i = N,. As shown in Mthods, whn population N xpands its rang into a spac occupid by population, and thir population dnsitis volv according to Eqs. 3 and 4, th front is h 1 a N T + ln + C M s = min j=1 p i ji λrj λ > ; [5] Tλ whr, intrstingly, th cultural transmission paramtrs f and γ do not appar sparatly but combind in thir ratio, C = f γ.inour opinion, this combination is a nic rsult bcaus of its simplicity and bcaus it has a clar intrprtation: C is th avrag numbr of huntr-gathrrs convrtd by ach farmr pr gnration at th ; lading dg of th wav of advanc, i.., for N << (Mthods). Finally, p j is th probability of th N individuals R to disprs at distanc r j (j = 1,2,...,M), and I ðλr j Þ = 1 2π 2π dθ xp½ λr j cosθš is th modifid Bssl function of th first kind and ordr zro. According to thnographic rports, farming is rarly copid at larg distancs by huntr-gathrrs (24, 25). Accordingly, w do not includ nonlocal cultural transmission hr; howvr, taking it into account would not chang our conclusions as long as w us paramtr valus stimatd from mpirical data (Nonlocal Cultural Transmission). To apply Eq. 5 w us th following paramtr rangs, as obtaind from thnographic and archaological obsrvations (aramtr Valus and Obsrvd Nolithic Front Spd Rang), :23 y 1 a N :33 y 1,29 T 35 y, 1: C 1:9, and th following probabilitis and distancs for th disprsal krnl: fp j g = {.42;.23;.16;.;.7;.2;.1;.1}, fr j g = {2.3; 7.3; 15; 25; 35; 45; 55; 1} km. Othr ralistic krnls for prindustrial farmrs yild similar rsults (aramtr Valus and Obsrvd Nolithic Front Spd Rang). Using ths rangs into Eq. 5 w obtain Fig. 1, which plots th maximum and minimum Nolithic front s (full and dashd curvs, rspctivly). Also in Fig. 1, th obsrvd rang of th Nolithic transition in Europ is shown by th hatchd horizontal rctangl and has bn obtaind from archaological data (aramtr Valus and Obsrvd Nolithic Front Spd Rang). Th obsrvd rang of th convrsion intnsity C of agricultur from farmrs to huntr-gathrrs corrsponds to th hatchd vrtical rctangl and has bn obtaind from anthropological data (aramtr Valus and Obsrvd Nolithic Front Spd Rang). For 1 C 2:5 (black ara in Fig. 1), th rang (i.., that btwn th dashd and full curvs) is sn to b consistnt with th obsrvd rang (hatchd horizontal rctangl). Finally, in Fig. 1, w obsrv that th has a finit limit s p as C, which is simply th maximum possibl for individuals (km/yr) 4. 3.5 3. 2.5 2. 1.5 1. maximum minimum simulations obsrvd rang obsrvd C rang consistncy btwn.5 and obsrvd s..1.1 1 1 1 1 1 Fig. 1. rdictd Nolithic front s. Th maximum (full curv) has bn computd using Eq. 5, th maximum obsrvd valu for th growth rat of prindustrial farmrs (a N =.33 y 1 ) and thir minimum gnration tim (T = 29 y). Th minimum (dashd curv) has bn computd using Eq. 5, th minimum obsrvd valu for th growth rat of prindustrial farmrs (a N = :23 y 1 ) and thir maximum gnration tim (T = 35 y). Th hatchd horizontal rctangl corrsponds to th obsrvd of th Nolithic front in Europ (.9<s<1.3 km/y). Th hatchd vrtical rctangl corrsponds to th obsrvd rang of th convrsion intnsity (1.<C<1.9). Dtails on th obsrvations lading to ths obsrvd rangs of a N, T, s, andc, as wll as to th disprsal krnls of prindustrial farmrs, ar givn in aramtr Valus and Obsrvd Nolithic Front Spd Rang. Th symbols ar th s obtaind from numrical simulations of Eqs. 3 and 4 (Numrical Simulations). C s * 167 www.pnas.org/cgi/doi/1.173/pnas.1266219 Fort
moving up to a maximum distanc Δ max pr gnration, namly s p = Δmax T. For th krnl introducd abov, Δ max = 1 km, and w obtain s p = 2:57 km/y for T = 35 y and s p = 3:44 km/y for T = 29 y, which agr up to th third dcimal digit with th valus of s p obtaind from Eq. 5 in Fig. 1. This is a usful chck of Eq. 5, and also shows that our Eqs. 3 and 4 ar physically mor rasonabl than th raction-diffusion modl 2, which prdicts an unboundd as C [for compltnss, in Raction-Diffusion Modl, w show that th raction-diffusion modl 2 lads to similar rsults than Eqs. 3 and 4, so our conclusions do not dpnd on th us of a disprsal krnl (11), instad of convntional diffusion, or on th cohabitation ffct (23)]. Fig. 2 shows th ffct of cultural transmission (i.., th prcnt diffrnc btwn th in Fig. 1 for th valu of C considrd and th of th purly dmic modl, C =, rlativ to th formr). Fig. 2 shows that, for th rang implid by th obsrvations in Fig. 1 (1 C 2:5 ), th cultural ffct is 4 ± %. Convrsly, th dmic ffct is 6%, which implis that, at th continntal Europan scal, th contribution of dmic disprsal (sprad of populations) was substantially largr ( 5% largr) than that of cultural transmission (sprad of idas). Thrfor, dmic diffusion was th most important ffct driving th Nolithic rang xpansion in Europ, but th ffct of cultural diffusion was also important and cannot b nglctd. Discussion W strss that this papr dos not attmpt to answr th qustion of whthr th gns of modrn-day Europans ar primarily of Middl Eastrn farmr origin. Instad, w hav focusd on whthr th archaological data imply that th main mchanism rsponsibl for th sprad rat was dmic or cultural diffusion. To tackl this qustion, hr w hav prsntd a modl of cultur sprad that combins cultural transmission thory with th ffcts of dmic disprsal and population growth. Th modl is basd on physical transport quations and anthropologically ralistic assumptions and paramtr valus, and it lads to an quation for th sprad rat of th wav of advanc, Eq. 5, that dpnds on th numbr C of huntr-gathrrs convrtd by farmr and gnration at th lading dg of th front, which sms rasonabl. Th modl also shows that, at th continntal scal, dmic diffusion was th most important cultural ffct (%) 7 6 5 4 3 2 1 minimum maximum C rang implid by th consistncy (black) ara in Fig. 1.1.1 1 1 1 1 1 Fig. 2. Cultural ffct on th Nolithic front, dfind as th prcnt diffrnc btwn th by th dmic-cultural modl and that by th purly dmic modl (C = ), rlativ to th formr. This figur shows that, for th rang of C consistnt with th obsrvd in Fig. 1 (hatchd rctangl), th ffct of cultural transmission on th sprad rat of th Nolithic transition in Europ was 4 ± %. C procss rsponsibl for th sprad rat of th Nolithic transition in Europ. This framwork unifis dmic front propagation (23) and cultural transmission thory (2), and also shows how Nolithic transitions ar likly to function, drivn by a combination of dmic and cultural diffusion and amnabl to physical modling. Of cours, th 4% contribution of cultural diffusion (as stimatd abov) is a continntal avrag and will vary spatially. Thrfor, th modl should b also applid to rgional analyss. For xampl, th Linarbandkramic (LBK) Nolithic cultur in cntral Europ sprad rat has bn rcntly (26) stimatd as. km/y, which is consistnt with th curvs in Fig. 1 for C (.7.9 km/y), implying a vry small prcntag for th cultural ffct (Fig. 2 for C ). This rsult is ncouraging, bcaus th LBK rang xpansion is widly rgardd as dmic by archaologists (27). Mor dtaild analyss of th LBK data will b ncssary to stimat th statistical rrors in th LBK obsrvd and its cultural ffct. Similar work should b prformd for othr inland Nolithic culturs. On th othr hand, som local Nolithic s wr substantially fastr (26) but, bcaus sa travl was probably important, simulations in ral gographis will b ncssary to prform dtaild comparisons of our modl with such data. Ovrall, this work opns a way to discriminat th rols of dmic and cultural diffusion at rgional scals within Europ, as wll as for Nolithic transitions in othr rgions of th world and for othr historical transitions and cultural diffusion phnomna. Th approach in this papr will b xtndd lswhr to includ intrbrding. Furthr spcific potntial applications includ th Austronsian Nolithic xpansion (2), many xampls of languag substitution, crop disprsals (29), tc. Mthods Homognous Systms. It is wll known from thnographic studis that huntr-gathrrs () somtims bcom farmrs (N), but th rvrs transition is vry rar (4). Thus, th cultural procss N will b includd in our modl, whras N will not. For th numbr of individuals in th nw gnration w writ N = N + I = I ; [6] whr I (th intraction trm) is th numbr of huntr-gathrrs bcoming farmrs pr gnration. Spcial cas. A drivation (2) for th intraction trm I undr cultural transmission introducs n as th numbr of tachrs (othr than parnts) that a individual contacts during his/hr liftim. This drivation also assums that, of ths n tachrs, a proportion u = + is of typ N, so th numbr of tachrs of typ N is nu = n + (in th nxt paragraph, w drop this assumption and gnraliz this modl). If g is th probability that a individual bcoms N du to contact with a singl N individual, th probability that h will bcom N aftr n contacts is1 ð1 gþ nu (2). If g is small, this simplifis to fu with f = ng and u = + (2). Thus, th numbr of individuals bcoming farmrs (N) pr gnration is I = fu = f : [7] N + In systms without nt rproduction (as in rf. 2, but not in our cas), th population siz N + is constant and th first quation of Eq. 6 bcoms th wll-known quation u = u + fuð1 uþ (2). W also not that for N >>, th scond quation of Eq. 6 bcoms = ð1 fþ, which implis that f 1. W hav assumd that th numbr n of tachrs that a individual contacts during hr/his liftim is indpndnt of N and.altrnativly, w could assum that n is proportional to N + (lading to th Lotka Voltrra intraction namly, I = k ). Howvr, a constant valu for n sms mor ralistic bcaus th numbr of tachrs, frinds, tc. pr individual is mpirically obsrvd to b roughly th sam for many diffrnt populations (3). Not also that according to Eq. 7, I bcoms,.g., indpndnt of N if N >>, which sms rasonabl bcaus ach individual cannot intract with an arbitrarily larg numbr of N individuals. ALIED HYSICAL Fort AS Novmbr 13, 212 vol. 19 no. 46 1671
Mor complicatd modls can b considrd, basd on th assumption that th probability g (and thrfor f = ng) dpnds on u = +. Howvr, th main conclusion of this papr would not chang if this ffct wr includd (Frquncy-Dpndnt Cultural Transmission), so frquncydpndnt modls ar not ncssary hr. Gnralizd modl. For our purposs, Eq. 7 has a srious limitation. If N <<, th first quation of Eq. 6 bcoms N = ð1 + fþ N, which combind with f 1 implis that ach N individual can at most convrt a singl individual in thir liftim. This rsult dos not sm rasonabl, in gnral, bcaus thr ar many historical vnts whr a small numbr of immigrants can introduc a nw tchnology rapidly across a socity. Thrfor, w gnraliz th spcial cas in th prvious paragraph as follows. That spcial cas assums that a individual is qually likly to contact with N or individuals (so that th numbr of N tachrs h/sh contacts is n + ). Lt us now assum that, for larning purposs, a individual contacts with only a fraction α of his/hr N nighbors and a fraction β of his/hr nighbors. Thn, th numbr of N tachrs that h/sh contacts is n α α + = n β +, whr γ γ = β α (th spcial cas in th prvious paragraph corrsponds to α = β or γ = 1). Rpating th drivation in th prvious paragraph lads to th following gnralization of Eq. 7: I = f ; [] and this yilds Eq. 1. Th cas γ < 1 corrsponds to a highr tndncy for huntr-gathrrs to slct farmrs rathr than huntr-gathrrs as tachrs (α > β). For N <<, Eq. bcoms I = C N, whr C = f γ. Thn th first quation of Eq. 6 yilds N = ð1 + CÞ N, which shows that C is th numbr of huntr-gathrrs convrtd by ach farmr pr gnration at th lading dg of th wav of advanc ( N << ). Many huntr-gathrrs will b convrtd by ach farmr if C >> 1. Not that I bcoms indpndnt of if N <<, I bcoms indpndnt of N if N >>, and both rsults sm rasonabl (as xplaind blow Eq. 7). Nonhomognous Systms. In this cas, th population dnsitis Nðx; y; tþ and ðx; y; tþ dpnd on spac ðx; yþ and tim t. W discuss th volution quation for N only, bcaus that for is analogous. Th simplst modl is calld th noncohabitation modl. As w shall now s, it includs Fishr s quation as a spcial cas. Th noncohabitation modl (9) is basd on th following assumption for th chang in Nðx; y; tþ during a gnration tim T (dfind as th man tim intrval btwn th migration of an individual and on of hr/his childrn), Nðx; y; t + TÞ Nðx; y; tþ = R R N x + Δ x; y + Δ y ϕn Δx; Δ y dδxdδ y Nðx; y; tþ Nðx; y; tþðx; y; tþ + R T ½Nðx; y; tþš Nðx; y; tþ + f Nðx; y; tþ + γðx; y; tþ : [9] In Eq. 9, th two first trms in th right-hand sid ar du to population movmnt (disprsal), th third and fourth ons ar du to nt rproduction (births minus daths), and w hav addd th last trm to includ cultural transmission, in agrmnt with Eq. 1 (9). Modls with a distribution of gnration tims T yild similar rsults (21). Th disprsal krnl ϕ N ðδ x ; Δ y Þ in Eq. 9 is th probability of migration from an ara cntrd at ðx + Δ x ; y + Δ y Þ at tim t to an ara cntrd at ðx; yþ at tim t + T. Th third trm in th right-hand sid of Eq. 9 corrsponds to logistic dynamics, >< ant K N Nðx; y; tþ R T ½Nðx; y; tþš = K N + ð an T 1ÞNðx; y; tþ >: a T ; [1] K ðx; y; tþ R T ½ðx; y; tþš = K + ð a T 1Þðx; y; tþ which agrs with many population data for humans (rf. 9 and citations thrin). Assuming isotropic krnls and prforming a Taylor xpansion up to scond ordr in Δ x and Δ y and up to first ordr in T yilds Fishr s quation (9) with an additional trm (du to cultural transmission), N t ¼ D N 2 N þ FðNÞþ f N T N þ γ whr D N R R ðδ2 x þ Δ2 y Þϕ NðΔ x ; Δ y ÞdΔ x dδ y is th diffusion cofficint of th Nolithic population and FðNÞ = a N N 1 N its logistic growth function KN [scond-ordr trms in T ar somtims also includd (9), but thy ar not ncssary for our purposs hr]. This stp complts th drivation of th [11] raction-diffusion Eq. 2. Howvr, whn applid to human populations, raction-diffusion quations can lad to substantial rrors for two rasons: (i) It has bn shown that th diffusiv approximation (i.., th scond-ordr spatial Taylor xpansion abov) braks down for ralistic human disprsal krnls (11). (ii) According to Eq. 9, nwborn individuals can appar at (x,y) (trms R T ½Nðx; y; tþš Nðx; y; tþ) whil thir parnts migrat away from (x,y) (first two trms in th right-hand sid). In othr words, som parnts lav thir nwborn childrn alon. Bcaus nwborn humans cannot surviv alon, it is mor ralistic to rplac Eq. 9 by th so-calld cohabitation quation (figur 1 in rf. 11, and rfs. 1 and 23) namly, whr N ðx; y; t + TÞ = N ~ x + Δ x ; y + Δ y ; t ϕ N dδx dδ y ; [12] R T ½Nðx; y; tþšr T ½ðx; y; tþš ~Nðx; y; tþ R T ½Nðx; y; tþš + f R T ½Nðx; y; tþš + γr T ½ðx; y; tþš : [13] This procdur lads to Eqs. 3 and 4, thrby gnralizing th cohabitation modl (23) to includ cultural transmission. W hav assumd that rproduction is followd by cultural transmission (Eq. 13) and thn by disprsal (Eq. 12). Howvr, it is asy to s that th front (drivd blow) is th sam rgardlss of ordr of vnts. For mor dtaild drivations and discussions on th ordr of vnts and th basic cohabitation Eqs. 12 and 13 (without th last trm in Eq. 13), s spcially figur 1 of rf. 11, figur 17 of rf. 1, and rf. 23. Front Spd. To modl th Nolithic transition, w considr farmrs initially only in a givn rgion (an ara locatd in th Nar East, according to archaological vidnc). Bcaus humans disprs and rproduc, farmrs can gradually sprad into othr rgions (.g., into Europ, which was initially occupid by huntr-gathrrs but not by farmrs, again according to archaological vidnc). In th lading dg of th advancing agricultural population front, w may linariz th population dnsitis as Nðx; y; tþ = εðx; y; tþ + Oð2Þ [14] ðx; y; tþ = K δðx; y; tþ + Oð2Þ; whr ε ðx; y; tþ K N, δðx; y; tþ K, and O(2) ar scond-ordr trms. Thn Eq. 1 bcoms >< R T ½Nðx; y; tþš = ant εðx; y; tþ + Oð2Þ δðx; y; tþ [15] >: R T ½ðx; y; tþš = K a T + Oð2Þ: Using Eq. 15 into 13 and prforming a two-variabl Taylor xpansion yilds ~Nðx; y; tþ = an T ð1 + CÞNðx; y; tþ + O 2 ; [16] whr C = f γ ; and finally Eq. 12 bcoms simply Nðx; y; t + TÞ ant ð1 + CÞ N x + Δ x ; y + Δ y ; t ϕ N dδx dδ y : [17] Whn obsrvd disprsal data ar usd, th krnl pr unit lngth φ N ðδþ is dfind as th probability to disprs into a ring of radius Δ and width dδ, dividd by dδ. If individuals of th population N hav probabilitis p j to disprs at distancs r j (j = 1, 2,..., M), φ N ðδþ = XM p j δ ð1þ r j ; [1] j=1 whr δ ð1þ ðr j Þ is th on-dimnsional Dirac δ cntrd at r j (i.., a function that vanishs vrywhr xcpt at Δ = r j ). Bcaus th total probability must b 1, 1 = φ N ðδþdδ; [19] and φ N ðδþ is clarly a probability pr unit lngth. In contrast, th krnl ϕ N ðδ x ; Δ y Þ in Eq. 17 is a probability pr unit ara (bcaus it is multiplid 1672 www.pnas.org/cgi/doi/1.173/pnas.1266219 Fort
by dδ x dδ y, which has units of ara). Th normalization condition for ϕ N ðδ x ; Δ y Þ is thrfor 1 = ϕ N dδx dδ y = 2π ϕ N ðδþδdδ; [2] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi whr w hav usd polar coordinats Δ = Δ 2 x + Δ2 y, θ = tan 1 ðδ x =Δ y Þ, and assumd th krnl is isotropic, ϕ N ðδ x ; Δ y Þ = ϕ N ðδþ. Comparing Eqs. 19 and 2, w s that th disprsal probability pr unit lngth [i.., into a ring of ara 2πΔ dδ) φ N ðδþ is rlatd to that pr unit ara ϕ N ðδþ, as in Fort and ujol (1)], and Eq. 1 yilds φ N ðδþ = 2πΔ ϕ N ðδþ; [21] ϕ N ðδþ = XM j=1 p j δ ð1þ r j 2πΔ : [22] For homognous paramtr valus, th will not dpnd on dirction and can thus b mor asily computd along th x-axis (y = ). Considr a coordinat fram z = x st moving with th wav of advanc (s is th wav ). Th population dnsity of farmrs will b qual to its saturation dnsity K N in rgions whr th Nolithic transition is ovr, and it will dcay to zro in rgions whr fw farmrs hav arrivd (N << K N ). Thus, w assum as usual th ansatz Nðx; y; tþ = N xp½ λ zš for z (with λ > )(1). Thn Eq. 17 bcoms λst = ant ð1 + CÞ Z 2π dθ λδcosθ ϕ N ðδþδdδ; [23] whr w hav applid that Δ x = Δcosθ. Finally, using Eq. 22 and assuming that th minimum is that of th front, w rach Eq. 5 for th front s. Numrical simulations of Eqs. 3 and 4 confirm th validity of Eq. 5 (symbols vs. curvs in Fig. 1). Dtails on th numrical simulations can b found in Numrical Simulations, whr it is also shown that th width of th Nolithic front (as obtaind from th simulations) agrs rasonably wll with th availabl archaological obsrvations. ACKNOWLEDGMENTS. Jan-irr Bocqut-Appl, Marcus Fldman, Thomas N. Hadland, and Nus Isrn ar acknowldgd for providing usful information. This work was supportd in part by Ministry of Scinc Grants Simul- ast-csd-21-34 and FIS-29-135 and th Gnralitat d Catalunya Grup Consolidat 29-SGR-374. 1. Ammrman AJ, Biagi, ds (23) Th Widning Harvst. 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