ecologcal modellng xxx (2006) xxx xxx avalable at www.scencedrect.com journal homepage: www.elsever.com/locate/ecolmodel Spato-temporal communty dynamcs nduced by frequency dependent nteractons Margaret J. Eppsten a,, James D. Bever b, Jane Molofsky c a Department of Computer Scence, Unversty of Vermont, 327 Votey Bldg., 33 Colchester Ave., Burlngton, VT 05405, USA b Department of Bology, Indana Unversty, Bloomngton, IN 47405, USA c Department of Botany, Unversty of Vermont, Burlngton, VT 05405, USA artcle nfo abstract Artcle hstory: Receved 31 May 2005 Receved n revsed form 4 January 2006 Accepted 9 February 2006 Keywords: Frequency dependence Spatal models Cellular automata Communty dynamcs Coexstence nvasveness A mathematcal model ncorporatng the effects of possbly asymmetrc frequency dependent nteractons s proposed. Model predctons for an dealzed two-speces annual plant communty wth asymmetrc lnear frequency dependence are explored usng () analytc mean feld equlbrum predctons, () determnstc, dscrete-tme, fnte-populaton, mean feld predctons, and () stochastc, dscrete-tme, cellular automata predctons for a varety of szes of the spatal nteracton and dspersal neghborhoods. We defne speces nteracton factors, rangng from 0 to 1, whch ncorporate both frequency ndependent and frequency dependent terms. The maxmum compettve ablty of a speces s reduced unless speces frequency s optmal based on speces-specfc frequency dependence coeffcents, rangng from 1 to +1. Assumng that maxmum compettve ablty s dentcal for two speces, they can coexst ndefntely when they have equal absolute magntude or both have suffcently negatve frequency dependence. Although smaller scales of spatal nteractons reduce the regon of the parameter space n whch stable coexstence s predcted, the tme to extncton of one speces can be sgnfcantly ncreased or decreased by the localty of nteractons, dependng on whether the losng speces has postve or negatve frequency dependence, respectvely. The senstvty to ntal condtons n the communty at large s dramatcally reduced as the spatal scale of nteractons s decreased. As a consequence, smaller spatal nteracton neghborhoods ncrease the ablty of ntroduced speces to nvade establshed communtes n regons of the parameter space not predcted by mean feld approxmatons. In the loser postve, wnner postve regons, smaller scales of nteracton dramatcally ncreased nvasveness. In the loser postve, wnner negatve regons of the parameter space, nvason success decreases, but tme to extncton of the resdent speces durng successful nvasons ncreases, wth an ncrease n the spatal scale of nteractons. The loser negatve, wnner postve regons were relatvely nsenstve to ntal condtons, so nvason success was relatvely hgh at a varety of spatal scales. Surprsngly, nvasons n parts of ths regon are most often successful wth ntermedate neghborhood szes, although the maxmum tme that the losng speces could persst before beng drven to extncton ncreases wth an ncrease n the spatal scale of nteractons. These results Correspondng author. Tel.: +1 802 656 1918; fax: +1 802 656 0696. E-mal address: Magge.Eppsten@uvm.edu (M.J. Eppsten). 0304-3800/$ see front matter 2006 Elsever B.V. All rghts reserved. do:10.1016/j.ecolmodel.2006.02.039 ECOMOD-4320; No. of Pages 15
2 ecologcal modellng xxx (2006) xxx xxx are explaned by understandng cluster formaton and densty and the relatve local nterspecfc dynamcs n cluster nterors, exterors, and boundares. In summary, frequency dependent nteractons, and the spatal scale on whch these nteractons occur, can have a bg mpact on spato-temporal communty dynamcs, wth mplcatons regardng speces coexstence and nvasveness. The model proposed heren provdes a theoretcal framework for studyng frequency dependent nteractons that may shed lght on spato-temporal dynamcs n real ecologcal communtes. 2006 Elsever B.V. All rghts reserved. 1. Introducton Compettve nteractons abound n the natural world, and these have hstorcally been consdered one of the major organzng prncples n ecologcal communtes. Consequently, there s a rch body of lterature developng a theoretcal framework for the study of compettve nteractons (e.g., Volterra, 1926; Lotka, 1932; Tlman, 1982, 1994; Pacala and Levn, 1997; Neuhauser and Pacala, 1999; Chesson, 2000). These models manfest negatve densty dependence, as populatons compete for fnte resources n the envronment. Speces coexstence n these resource-governed models requres nche dvergence, wheren drect nter-specfc competton s reduced and speces lmt themselves more than they lmt other speces. Hubbell (2001) argues that ecologcally equvalent speces can coexst over long tme ntervals, although ecologcal drft wll eventually cause extnctons unless counterbalanced by mgraton. In Hubbell s neutral model, recrutment nto the communty s proportonal to the relatve abundance of speces at some spatal scale. There are also many examples of postve or negatve frequency (or densty) dependent spatal nteractons that are not caused by competton for a pre-exstng fnte pool of resources (Clarke, 1969; Connell, 1983; May and Anderson, 1983; Condt et al., 1992; Wlson and Agnew, 1992; Ronshem, 1996; Smthson and McNar, 1996; Holmgren et al., 1997; Bever, 1999; Weltzn and McPherson, 1999; Catovsky and Bazzaz, 2000; Harms et al., 2000; Renhart et al., 2003; Callaway et al., 2004). Such nteractons affect growth and reproducton, and therefore ultmately affect relatve compettve abltes, of speces n ecologcal communtes. In expermental communtes of annual and bennal plants, relatve compettve abltes of four naturally co-occurrng plant speces were shown to be dfferentally postvely or negatvely affected by aggregaton of con-specfcs, thereby promotng communty dversty (Stoll and Prat, 2001). Negatve frequency dependence s often cted as a mechansm for the mantenance of dversty n ecologcal communtes (Wlls et al., 1997; Wlls and Condt, 1999; Harms et al., 2000; Chesson, 2000; Wrght, 2002; Bever, 2003). Interactons that occur through ntermedares such as through pollnators (Ågren, 1996; Smthson and McNar, 1996) or mycorrhzae (Ronshem, 1996; Bever et al., 1997; Ronshem and Anderson, 2001; Bever, 2002) and/or through predators (Clarke, 1969) or pathogens (May and Anderson, 1983; Westover and Bever, 2001) often create frequency dependent nteractons. For example, subsurface communtes of symbotc mycorrhzae may floursh near certan speces, renderng the sol more favorable to growth of others of the same speces (postve frequency dependence). Conversely, accumulaton of speces-specfc sol pathogens can ncrease seedlng mortalty n the area (negatve frequency dependence). Because spatal structure n communtes can have dramatc mpacts on plant communty dynamcs (Czárán and Bartha, 1992; Herben et al., 2000), there has been an ncreasng recognton of the need for spatally explct models of ecologcal nteractons (Balzter et al., 1998; Berec, 2002; Wu and Marceau, 2002). Spatally explct formulatons of Lotka Volterra competton models have shown that changng the scale of spatal nteractons n homogeneous envronments alters the regon of the parameter space where coexstence at equlbrum can be acheved (Neuhauser and Pacala, 1999; Murrell et al., 2002). Spatal heterogenety of resources can clearly ncrease communty structure and dversty. However, n many cases envronmental heterogenety may actually be nternally generated (or augmented) through ecologcal feedbacks (Czárán and Bartha, 1992; Herben et al., 2000; Bascompte and Rodríguez, 2000; Feagn et al., 2005). There s a growng body of evdence that frequency dependent effects medated by both botc and abotc changes to the subsurface envronment may play an mportant role n nvasveness by exotc plant speces, and must be understood and consdered for effectve conservaton and restoraton of plant communtes (Wolfe and Klronomos, 2005). Recently, some spatally explct models have begun to elucdate the mportance of frequency or densty dependent effects on spato-temporal dynamcs n plant communtes. In a sngle speces model wth an Allee effect (postve ntraspecfc densty dependence at low frequences), the ablty of small local ntal adult dstrbutons to establsh and persst was shown to be very senstve to the shape of the dspersal kernel (Etenne et al., 2002). Feagn et al. (2005) found that both an external envronmental gradent and facltatve successon (postve nter-specfc frequency dependence) were requred n order to accurately smulate communty organzaton n a sand dune plant communty. Wang et al. (2003) found a non-lnear response between weed control and weed patch sze, due to mplct aggregaton effects of large patches that made weeds nsde patches more resstant to external controls. More generally, Molofsky et al. have examned how communty structure s affected by the spatal scale and strength of symmetrc frequency dependent nteractons, both negatve (Molofsky et al., 2002) and postve (Molofsky et al., 2001; Molofsky and Bever, 2002). For example, they have shown that postve ntra-specfc frequency dependence can promote stable coexstence of speces through the formaton of sngle-speces clusters, f the nteractons are spatally localzed. Ther models assume that the magntude and drecton of frequency dependence s dentcal for all speces n the
ecologcal modellng xxx (2006) xxx xxx 3 communty, whch s unlkely to be the case for any real communty. In ths paper, we extend the general theoretcal framework for examnng the effects of postve or negatve frequency dependent nteractons to communtes n whch the strength and/or sgn of these nteractons may dffer for each speces. We develop predctons of the proposed model for dealzed two-speces communtes of annual plants based on analytcal mean feld stablty analyss, determnstc fnte-populaton mean feld smulatons, and spatally explct stochastc cellular automata smulatons. Through these models, we explore the nterplay between the magntude, drecton, and spatal scale of lnear ntra-specfc frequency dependent nteractons, and how such nteractons wll affect communty dynamcs. Whle future work wll extend our analyss to nclude non-lnear nteractons, nter-specfc frequency dependence, speces-specfc dfferences n maxmum habtat sutablty, and multple speces, heren we explore the spatal and temporal dynamcs of speces extncton, coexstence, and nvasveness n varous regons of the frequency dependence parameter space for two speces wth equal maxmum compettve abltes and speces-specfc ntra-specfc frequency dependence. Fg. 1 The lnear frequency dependence nteracton factor I for speces as a functon of frequency of speces (Eq. (2.3)), shown for fve representatve values of frequency dependence. 2. Model development 2.1. Determnstc mean feld model For smplcty, we consder a two-speces communty of annual plants n whch the only compettve nteracton s for space, and where plants of each speces requre the same amount of space to grow. The degree to whch each speces s suted to the habtat s capped by a frequency ndependent maxmum ˇ 0, but can be modfed downward by a frequency dependent nteracton factor I t [0,...,1], based on feedbacks wth the envronment. Note that we denote speces by subscrpt and tme by superscrpt. The tme varyng habtat sutablty H t for a gven speces s then smply the product: H t = ˇI t (2.1) We assume that the number of seeds of speces avalable for germnaton at tme t + 1 s drectly proportonal to speces densty D t at tme t. The expected relatve communty-wde frequency F t+1 of each speces at tme t + 1 can be determned, based on ther relatve products of habtat sutablty and prevalence H t Dt, by the followng dscrete-n-tme approxmaton: F t+1 H D t = H 1 D t 1 + H 2D t 2 (2.2) When ndvduals of each speces are equal compettors (H1 t = Ht 2 ), Eq. (2.2) predcts that current populaton frequences wll be mantaned, whereas a speces wth a hgher habtat sutablty factor wll ncrease n relatve frequency. The functonal form of the nteracton factors I wll depend on the nature of the partcular feedback nteractons. In realty, there may be several dfferent but smultaneous types of nterac- tons wthn and between varous speces. If these nteractons operate va the same ntermedary, then they would be addtve. However, f multple nteractons are ndependent of each other, for example operatng at dfferent perods n the lfe hstory of the plants, then ther effects could be multplcatve. Here, we consder a sngle type of smple lnear ntra-specfc frequency dependence, where the nteracton factor s defned as follows: { 1 I t + F t =, f > 0 1 + F t, f (2.3) 0 where s a parameter rangng from 1 to +1 and represents the degree to whch the habtat sutablty for speces s negatvely or postvely affected by the frequency of speces n the communty. Eq. (2.3) s graphed n Fg. 1, and llustrates the declne n habtat sutablty at sub-optmal frequences, for dfferent representatve frequency dependence coeffcents. A prevously proposed model for representng symmetrc frequency dependent nteractons n a two speces annual plant communty s as follows (Molofsky et al., 2001; Molofsky and Bever, 2002): H t = 0.5 + (F t 0.5) (2.4) where s also a frequency dependence parameter rangng from 1 to +1. However, n Eq. (2.4) the average habtat frequency (averaged over all possble frequences) s assumed to always be 0.5, and the maxmum habtat sutablty s dependent on, as follows: H t 1 +, (2.5) 2 whereas n Eq. (2.1) the maxmum habtat sutablty s governed by the ndependent varable ˇ. Thus, Eq. (2.4) can only sample a subset of the parameter space represented by Eqs.
4 ecologcal modellng xxx (2006) xxx xxx (2.1) and (2.3). In Molofsky et al. (2001) and Molofsky and Bever (2002), Eq. (2.4) was appled to two-speces communtes wth symmetrc frequency dependence (e.g., 11 = 22 ), so both average and maxmum habtat sutablty were also mplctly speces symmetrc. However, Eq. (2.4) cannot be appled for predctng populaton dynamcs where frequency dependence and maxmum habtat sutablty need to be ndependently vared. Eqs. (2.1) and (2.3) thus generalze and extend the stuatons n whch frequency dependent nteractons can be modeled, beyond the prevously proposed theory expressed n Eq. (2.4). In order to focus on populaton dynamcs caused by asymmetrc frequency dependent effects, we apply Eqs. (2.1) and (2.3), assumng that all speces have dentcal maxmum habtat sutabltes (ˇ = 1) acheved at ther optmal frequences, n all model results presented heren. 2.1.1. Mean feld stablty analyss Because the nteracton functon gven n Eq. (2.3) s not contnuous, we analyze each of the four quadrants of the 11 22 parameter plane separately. We hereafter denote these as the ++, +, +, and quadrants, based on the sgn of 11 and 22, respectvely, as llustrated n Fg. 2a. In the + + quadrant, there exsts an nternal neutral equlbrum for speces, gven by: F = 11 + 22 (2.6) Analyss of local stablty (Edelsten-Keshet, 2005) ndcates that ths nternal equlbrum s unstable, whle the two trval equlbra (F = 0,F = 1) are stable. Ths mples that stable fxaton of ether speces s possble dependng upon ntal condtons (.e., postve frequency dependent renforcement). Specfcally, the neutral equlbrum lne separatng the two stable equlbra (Fg. 2b) has slope C, whch s dependent on ntal condtons. For the mean feld approxmaton, ths slope s smply the rato of the ntal frequency of the two speces, as follows: C mean feld = F0 1 F2 0, (2.7) and coexstence s predcted n the + + quadrant by the analytc mean feld model only when: 11 = F0 1 22 F2 0. (2.8) In the quadrant an nternal equlbrum also exsts for two speces and j, j, gvenby: F jj = (2.9) 11 + 22 Analyss of local stablty ndcates that ths nternal equlbrum s stable whle the two trval equlbra (F = 0,F = 1) are unstable. Ths mples that the two speces wll coexst ndefntely, the expected result wth negatve frequency dependence (Fg. 2c). In consderng the + and + quadrants, the two speces wll have equal ftness when 11 = 22, ndependent of ther ntal frequences, and therefore ther relatve abundance wll neutrally drft under ths condton. Besdes ths specal case, there s no nternal equlbrum. Above the 11 = 22 lne one Fg. 2 (a) Quadrant map for the 11 22 parameter space. Analytc mean feld stablty analyss predctons for: (a) + quadrant, (b) + + quadrant, and (c) quadrant. A quadrant map s shown n (d) for reference. The + quadrant (d) s symmetrc wth the + quadrant shown n (a).
ecologcal modellng xxx (2006) xxx xxx 5 speces wll domnate and below the lne the other wll domnate (as llustrated for the + quadrant n Fg. 2d; the + s symmetrc wth ths and s therefore not explctly shown). 1 + F t I t ( x) j ( x) I, f > 0 j I = 1 + F t j ( x) I, f 0 j (2.11) 2.1.2. Determnstc mean feld model predctons We mplemented a determnstc mean feld model for a fnte populaton of 100,000 annual plants to make predctons across the entre 11 22 parameter plane (n frequency dependent ncrements of 0.1). The determnstc mean feld model comprses a Matlab 7.0 mplementaton of Eqs. (2.1) (2.3) (where densty predctons were based on frequences then rounded to whole ndvduals) embedded wthn a loop that steps through tme untl the predctons converge. In Fg. 3a, we llustrate model results when F1 0 = F0 2. Here, black denotes regons where speces 1 wns (drves speces 2 to extncton), whte denote regons where speces 2 wns, and grayscale ndcates the relatve proportons of the two speces n regons where coexstence s predcted. In regons where one speces domnates over the other, the speces wth the lower absolute frequency dependence s the wnner. In retrospect ths result s obvous, because speces wth lower absolute frequency dependence have fewer constrants on whch cells they can occupy and are therefore able to outcompete speces wth the same maxmum habtat sutablty but greater frequency dependence. Note that above the 11 = 22 lne, the loser has postve frequency dependence, whle below the lne the loser has negatve frequency dependence. 2.2. Stochastc spatally explct model Up untl ths pont, we have presented the populaton growth model as a mean feld model. Eqs. (2.1) (2.3) can be made spatally explct as follows: F t+1 ( x) = ˇI t ( x) I D t ( x) D ˇ1I1 t ( x) I D t 1 ( x) D + ˇ2I2 t ( x) I D t 2 ( x) D 1 1 2 2 (2.10) where x represents a locaton vector n space (e.g., x, y spatal coordnates), and the general notaton f ( x) f means to evaluate the functon f for speces, wthn a spatal neghborhood, centered around locaton x, where the sze, shape, and weght of the neghborhood bass functon s specfc to speces and the functon f for the applcaton n queston. The neghborhoods for non-compettve nteractons ( I ) and dspersal (D ) may be speces-specfc and dstnct for each type of nteracton. Stochastcty may be ntroduced (as n Molofsky et al., 2001; Molofsky and Bever, 2002) by replacng the determnstc predcton of frequency by a stochastc predcton, wheren speces wll occupy locaton x at tme t + 1 wth probablty: P t+1 ( x) = F t+1 ( x) (2.12) where F t+1 ( x) s computed by Eq. (2.10). If the denomnator of Eq. (2.10) s zero, then the probablty n (2.12) s smply set to zero and the cell at locaton x s treated as empty for the next generaton. If = 0 and ˇ =1,, then ths probablty reduces to: P t+1 ( x) = D t ( x) D D t 1 ( x) D + D t 2 ( x) D 1 2 (2.13).e., n Eq. (2.13) the probablty that speces wll occupy the cell at x n the next generaton s smply determned by ts proportonate densty n the neghborhood, sensu the voter model (Holley and Lggett, 1975; Hubbell, 2001). The above equatons are easly extended to handle more than two speces and/or non-zero nter-specfc frequency dependence, but we do not report on these extensons here. Fg. 3 (a) Fnte-populaton determnstc mean feld and (b) 3 3 cell neghborhood model predctons, startng wth equal proportons of each speces n a strped dstrbuton. Grayscale represents the predcted frequences of each speces, where black means 100% speces 1 and whte means 100% speces 2.
6 ecologcal modellng xxx (2006) xxx xxx Stochastc spatally-explct models are mplemented n Matlab v.7.0 as follows. Eqs. (2.11) and (2.12) are smulated on a Cartesan grd of dscrete cells, wheren each of the dscrete cells can be occuped by at most one ndvdual from any of the speces n the communty. All operatons are hghly vectorzed n Matlab to acheve computatonal effcency. In the remander of ths manuscrpt we report on experments for synthetc two speces annual plant communtes as follows. We employed 100 100 cell grds wth no unnhabtable cells and non-perodc boundary condtons (.e., wth no torodal wraparound, so local speces frequences around cells near boundares were smply computed over smaller neghborhoods), because these more accurately represent boundary condtons n natural communtes of fnte extent. Pror expermentaton had shown that larger grds and use of perodc boundary condtons result n quanttatve dfferences n tme to extncton, but do not qualtatvely alter relatve model behavor n the dfferent parts of the parameter space. The only compettve nteracton s for space, wheren the speces that occupes a gven cell (wns the competton) s stochastcally based on Eq. (2.12). Each speces exhbts one frequency dependent nteracton based on the frequency of ts own speces, although the strength and drecton of those nteractons vares ndependently for each speces. The physcal envronment n the communty s consdered to be homogeneous, wth each speces equally well-adapted to t. We smulate determnstc death of these annual plants, so 100% of the cells become avalable at the start of each year. In the experments reported here, we assumed unform and equvalent square neghborhoods of cells of a gven wdth and centered on x, for determnaton of both densty dependent seed dspersal D t ( x) D and frequency dependent nteractons I t ( x) I of an ndvdual n a gven cell located at poston x.we explored coeffcents of frequency dependence ( 11, 22 ) that sampled the parameter space for each combnaton of n the range [ 1, +1], n ncrements of 0.1. It should be noted that, for most of the parameter combnatons tested, all cells are typcally occuped each year (wth the excepton of certan cases wth very strong negatve nteractons, resultng n some cells havng zero probablty of beng occuped by ether speces). Thus, snce we permt at most one ndvdual per unt area n our models, most of the smulated nteractons reported here could alternatvely be vewed as ether frequency dependent or densty dependent. 3. Coexstence studes For coexstence studes, the communty was ntalzed wth equal frequences of the two speces;.e., F1 0 = F0 2. We explored both a strped ntal dstrbuton, where the left half of the envronment was ntally fully populated by speces 1 and the rght half was ntally populated by speces 2, and a random dstrbuton, n whch equal frequences of the two speces were randomly located across the doman. Although equlbrum coexstence predctons were the same from ether strped or random ntal dstrbutons, the ntal dstrbuton dd affect the non-equlbrum dynamcs. We ran coexstence smulatons for all 21 2 = 441 possble combnatons of the 21 coeffcents { 1, 0.9,..., 0.9, +1} usng the spatallyexplct stochastc model wth both 3 3 cell and 100 100 cell neghborhood szes. For consstency, ths set of experments (reported on n Fgs. 3 and 4) all started from the strped ntal dstrbuton. We refer to models run wth 100 100 cell neghborhoods as stochastc mean feld models, snce nteractons are communty-wde. Addtonal selected experments (reported on n Fg. 5), startng from the random dstrbuton, were run and are reported on for other neghborhood szes n order to more clearly elucdate pattern formaton and trends relatng to the spatal scale of nteractons. All smulatons were run untl ether one speces went extnct or for a maxmum of 2000 generatons (years). Pror expermentaton wth up to 10,000 generatons had shown that results dd not sgnfcantly dffer from the 2000 generaton predctons. Predctons of the determnstc mean feld model for the coexstence studes are shown n Fg. 3a. The stochastc mean Fg. 4 Non-equlbrum dynamcs startng from a strped ntal dstrbuton. (a) Tme to extncton events n the loser negatve regons as a functon of neghborhood sze; x-tcks correspond to the dashed so-dfference contours ( 11 22 ) shown n (b). (c) Tme to extncton events n the loser postve regons as a functon of neghborhood sze; x-tcks correspond to the sold so-dfference contours shown n (b). In (a) and (c) the dashed lnes represent the average of the 3 3 cell neghborhood predctons and the sold lnes represent the average of the stochastc mean feld predctons.
ecologcal modellng xxx (2006) xxx xxx 7 Fg. 5 (a) The representatve locaton 11 = 0.6, 22 = 0.5 n the + + quadrant, (b) the tme scale of extncton at the representatve locaton as a functon of neghborhood sze, (c) a representatve pattern of tght clusters of speces 1 (black) shown at generaton 20 of one 3 3 cell neghborhood smulaton; the arrows represent competng forces at the cluster boundares that help the clusters to persst, and (d) a representatve pattern of loose clusters of speces 1 (black) shown at generaton 20 of one 9 9 cell neghborhood smulaton, shortly before speces 1 went extnct; the arrows llustrate the rapd shrnkage of these clusters. All data n ths fgure s startng from a random ntal dstrbuton. feld predctons (not shown) appear nearly dentcal to Fg. 3a, wth the excepton that coexstence s no longer predcted along the 11 = 22 lne n the + + quadrant because, n the stochastc model when frequency dependence s postve and symmetrc, drft ultmately always enables one speces or the other to gan an advantage and drve the other to extncton. In addton, the number of generatons untl extncton events anywhere n the parameter plane s generally shorter n the stochastc than n the determnstc model. When spatal nteractons are lmted to 3 3 cell neghborhoods, the equlbrum predctons appear smlar to the stochastc mean feld predctons, wth the excepton that the regon of coexstence n the quadrant shrnks (Fg. 3b). Ths s consstent wth the fndngs of Neuhauser and Pacala (1999) for spatal Lotka Volterra compettve nteractons where ntra-specfc competton s stronger than nterspecfc competton. Such compettve nteractons exhbt negatve negatve densty dependence and so behave smlarly to negatve negatve frequency dependent nteractons. Unlke n the stochastc mean feld model, the 3 3 cell neghborhood stochastc model does predct coexstence (for at least 2000 generatons) along the 11 = 22 lne n the + + quadrant. Ths s because the equal and postve frequency dependence of both speces results n the formaton of sem-stable clusters that promote coexstence. Ths confrms the pror fndngs of Molofsky and Bever (2002) usng the smlar model emboded n Eq. (2.4). The apparent smlarty of the equlbrum frequency predctons shown n Fg. 3a and b, for the determnstc mean feld and stochastc 3 3 cell neghborhood spatally explct models, respectvely, s somewhat msleadng. Ths s because both communty structure and the tme scale for the extncton events vary dramatcally as a functon of the spatal scale of nteractons, and vary n dfferent ways for dfferent parts of the parameter space, as follows. In the loser negatve regons (below the 11 = 22 lne n the + and + quadrants), the average tme to extncton decreases from an average of 548 generatons n the stochastc mean feld model to an average of 208 generatons n the model wth 3 3 cell neghborhoods. In ether case, tme to extncton n these regons s nversely proportonal to the absolute dfference n the frequency dependence of the two speces ( 11 22 ), as shown n Fg. 4a, where tme to extncton s plotted as a functon of ths dfference. Note that the data
8 ecologcal modellng xxx (2006) xxx xxx ponts n Fg. 4a correspond to observed numbers of generatons pror to extncton along the dashed so-dfference contours n the loser negatve regons shown n Fg. 4b. Although the trend s nosy, the tme to extncton s generally faster for the 3 3 cell neghborhoods (astersks and dashed lne, Fg. 4a) than n the stochastc mean feld model (open crcles and sold lne, Fg. 4a). Ths s because small local neghborhoods have hgher apparent frequences of the lower frequency speces, so when the losng speces has negatve frequency dependence t s at an even greater relatve dsadvantage when the neghborhood sze s small, and consequently the speces that s less frequency dependent domnates more quckly. In contrast, n the loser postve regons (above the 11 = 22 lne) ths trend s reversed. Here, the number of generatons to extncton events ncreases from an average of 19 generatons n the stochastc mean feld model to an average of 326 generatons (startng from a strped dstrbuton) n the stochastc model wth the 3 3 cell neghborhoods. When plotted as a functon of dfference n absolute frequency dependence (correspondng to the sold so-dfference contour lnes shown n Fg. 4b), the tme to extncton drops exponentally as a functon of the absolute dfference n frequency dependence, and s dramatcally slower wth small neghborhoods (astersks and dashed lne, Fg. 4c) than n the stochastc mean feld model (open crcles and sold lne, Fg. 4c). We further elucdate the effect of neghborhood sze on tme to extncton n the loser postve regon by closer examnaton at one representatve locaton wth a small dfference n frequency dependence (Fg. 5a). In Fg. 5b, we plot tme to extncton as a functon of neghborhood sze, for 10 stochastc smulatons at each of a varety of neghborhood szes startng from a random equal dstrbuton of the two speces. Tme to extncton s seen to drop exponentally as a functon of neghborhood sze (Fg. 5b). Ths s because smaller nteracton neghborhoods facltate the formaton and retenton of protectve clusters of postve ntra-specfc frequency dependent speces, offsettng small frequency dependence dsadvantages and enablng them to coexst for hundreds or even thousands of generatons wth the less frequency dependent speces that s predcted to domnate. For example, consder the tght clusters of speces 1 (black), shown at generaton 20 of one representatve 3 3 neghborhood smulaton n Fg. 5c. In ths example, both speces have postve frequency dependence, so n the nteror of the clusters of speces 1 the envronment s more favorable to speces 1, and ndvduals of speces 2 are out-competed there. However, at the cluster boundares where frequences are roughly 50/50 wthn the small local neghborhoods, condtons are relatvely more favorable for speces 2 because ths border frequency s above the crtcal value (Eq. (2.6)) necessary for majorty advantage. Thus, the clusters of speces 1 are slowly eroded at the boundares and speces 2 ultmately wns, although ths may take hundreds of generatons, especally f ntal clumpng s present (as n the strped ntal dstrbuton). For larger neghborhood szes, the percent of each cluster that acts as erodable boundary s also larger, so clusters degrade more rapdly. For example, Fg. 5d shows that by generaton 20 of one representatve 9 9 neghborhood smulaton, the very loose clusters of speces 1 (black) are already nearly oblterated. At neghborhoods greater than about 12 12 cells, the clusters are so loose as to offer no protectve local envronment, so extnctons proceed as rapdly as n the stochastc mean feld model (Fg. 5b). 4. Invason studes For nvasveness studes, the communty was ntally fully populated by speces 1. Then, at the start of the smulaton, varous numbers of ndvduals from speces 2 were randomly placed n the doman, such that ntal rato of speces 1 to speces 2 ranged from approxmately 2:1 to 10,000:1. In the last case, a sngle ndvdual of speces 2 was placed near the center of a communty fully populated by speces 1. We ran nvasveness studes for the same 21 2 = 441 possble combnatons of the 21 coeffcents { 1, 0.9,..., 0.9, +1} usng the determnstc mean feld model, the stochastc mean feld model, and the stochastc 3 3 cell neghborhood model. The converse case (where speces 1 nvades speces 2) was not explctly consdered snce the results are symmetrc. When maxmum habtat sutablty s equal for all speces (as they are n experments here), nvason wll only be possble n those regons of the parameter space n whch the nvadng speces has a frequency dependence advantage;.e., has the lower absolute value of frequency dependence. The analytc mean feld predctons ndcated that, n the + + quadrant, the slope of the lne separatng the regons n whch each speces would domnate should be lnearly dependent on the rato of ntal frequences of the two speces (Eq. (2.7)). Ths result was confrmed by the predctons of both the determnstc and stochastc mean feld models (although the results of the latter were obvously noser due to random perturbatons). However, when the neghborhood sze was reduced to 3 3 cells, the dependence of the slope C was approxmately logarthmc n the rato of the ntal communty-wde frequences (Fg. 6). In other words, wth small neghborhood szes the outcomes are much less senstve to the ntal condtons n the communty at large. Ths has sgnfcant mplcatons for the ablty of speces 2 to nvade speces 1 n the + + quadrant. For example, speces 2 was not able to overcome even a mld 10:1 ntal frequency dfference anywhere we examned n the + + quadrant of our 10,000 ndvdual mean feld smulatons. However, when the nteracton neghborhood was reduced to 3 3 cells, speces 2 was repeatedly able to drve speces 1 to extncton above about the 11 =2 22 lne when startng from a 100:1 ntal frequency dsadvantage. Even when startng wth only a sngle ndvdual of speces 2 (a 10,000:1 ntal frequency dsadvantage), speces 2 was occasonally able to nvade n the + + quadrant above the 11 =10 22 lne. The reason that smaller neghborhoods promote nvasveness n the parts of the + + quadrant where speces 2 (the nvader) has a frequency dependent advantage s that the relatve frequency of speces 2 appears hgher n small local neghborhoods than n the communty at large. For example, n our study one ndvdual of speces 2 ntroduced nto a communty of 10,000 ndvduals of speces 1 has a local frequency of 1/9 wthn the 3 3 cell neghborhood centered on the locaton of the propagule, as opposed to ts communty-wde frequency of 1/10,000. Thus, even a 10:1 frequency dependent advantage wll favor
ecologcal modellng xxx (2006) xxx xxx 9 Fg. 6 Senstvty to ntal condtons n the + + quadrant as a functon of neghborhood sze. In the mean feld model, the slope of the neutral equlbrum lne s drectly proportonal to the ntal proportons of the two speces. In a 3 3 cell neghborhood model the relatonshp s approxmately logarthmc n the ntal proportons of the communty of 10,000. Above ths lne speces 2 wll nvade and drve speces 1 to extncton, below the lne speces 1 wll resst nvason. the nvader n the ts local 3 3 cell neghborhood, and once a foothold s establshed the cluster of speces 2 wll rapdly expand and ultmately drve the more frequency dependent (resdent) speces extnct. As n the mean feld model, the dependence on ntal condtons n the + + quadrant s lnear, wth slope C (recall Eq. (2.7)). However, when nteractons are lmted to local neghborhoods, the establshment of the ntroduced speces, and therefore the slope C, s governed by the frequency of the neghborhood wth the most ntroduced propagules. Wthout loss of generalty, f we assume that speces 1 s the resdent speces and speces 2 s the ntroduced speces, ths can be formalzed as the mnmum relatve neghborhood frequency of the resdent at the tme of the ntroducton (t = 0), over all local nteracton neghborhoods I n the doman, as follows: ( ) F 0 1 ( x) I 1 C I = mn 1 I 2 x F2 0( x) I 2 (4.1) Note that Eq. (4.1) converges to Eq. (2.7) as the sze of the nteracton neghborhoods I 1 and I 2 ncrease and approach the doman sze. Although the analytc nfnte-populaton mean feld analyss ndcated that only the + + quadrant should be dependent on the ntal condtons (Fg. 2), our experments showed that outcomes of even the determnstc mean feld fnte-populaton models could be nfluenced by ntal condtons across much of the 11 22 parameter plane. For example, n the + quadrant, the ntercept of the lne separatng the loser postve (resdent goes extnct) from loser negatve (nvader goes extnct) regons was observed to vary n drect proporton to the ntal frequences of the two speces n the determnstc fnte-populaton mean feld model (Fg. 7), even though ths quadrant s not predcted to be senstve to ntal condtons by the analytc mean feld model. Above ths lne, speces 2 was able to nvade and drve speces 1 to extncton (loser postve), but at or below ths lne the nvadng speces 2 was not able to ncrease ts frequency above the ntal proportons, due to the dscrete frequency ncrements n fnte populatons. In an nfnte-populaton mean feld model (no roundng to whole ndvduals), a large dsadvantage n speces proportons that was offset by even a small frequency dependence advantage was slowly reduced untl a crtcal pont of nstablty was reached, at whch tme the less frequency dependent speces rapdly drove the other to extncton (loser negatve). However, n the fnte-populaton mean feld model, predcted ncreases of less than 0.5 ndvduals were truncated to 0, wth the result that the ntal proportons never changed and a stable equlbrum was mantaned. In the stochastc mean feld model, drft always caused one or the other speces to wn and, although the lne separatng the loser postve and loser negatve regons was rregular due to random events, the average percent of the + quadrant n whch the loser was postve (.e., n whch speces 2 could drve speces 1 to extncton) was the same as n the determnstc mean feld model. Snce real plant communtes are fnte, these devatons from the mean feld predctons may lend meanngful nsght on the potental of postvely frequency dependent resdents to resst nvason by negatvely frequency dependent speces that have a frequency dependence advantage, where such resstance s not predcted by the analytc mean feld model. For the same reasons as n the + + quadrant, smaller neghborhood szes n the + quadrant rendered the model much less senstve to ntal condtons, wth the ntercept of the lne separatng the loser postve and loser negatve regons roughly proportonal to the logarthm of the rato of ntal proportons for the 3 3 cell neghborhoods n ths 10,000 member communty (Fg. 7). Even wth a 10,000:1 ntal frequency dsadvantage, speces 2 was able to occasonally nvade speces 1 n the + quadrant, gven a suffcently large frequency dependence advantage. We explored the nvason dynamcs n ths regon by closer examnaton of one representatve pont n the
10 ecologcal modellng xxx (2006) xxx xxx Fg. 7 Senstvty to ntal condtons n the + quadrant as a functon of neghborhood sze. In the mean feld model, the ntercept of the neutral equlbrum lne s drectly proportonal to the ntal proportons of the two speces. In a 3 3 cell neghborhood model the relatonshp s approxmately logarthmc n the ntal proportons n the 10,000 member communty at large. Above ths lne speces 2 wll nvade and drve speces 1 to extncton, below the lne speces 1 wll resst nvason. + quadrant, located at 11 = 0.8, 22 = 0.1 (Fg. 8a). We ran 100 smulatons n whch we ntroduced one ndvdual of speces 2 nto the center of an establshed communty of speces 1, at each of a number of neghborhood szes, and found that the percent of nvason success decreased lnearly wth neghborhood sze (R 2 = 0.78), wth a maxmum nvason success rate of 14% for the 3 3 cell neghborhoods (Fg. 8b). Although nvason success was hgher for small neghborhoods, the tme t took for the nvaders to completely overtake the resdent speces was generally hgher for small neghborhood Fg. 8 (a) The representatve locaton 11 = 0.8, 22 = 0.1 n the + quadrant, (b) nvason success as a functon of neghborhood sze, (c) generatons to extncton events as a functon of neghborhood sze; above the dashed lne the resdent went extnct, below the dashed lne the nvader went extnct, and (d) one representatve nvadng tght cluster of speces 2 (black) 40 generatons after the ntroducton of a sngle ndvdual of speces 2 n a 3 3 cell neghborhood model.
ecologcal modellng xxx (2006) xxx xxx 11 szes (Fg. 8c). The reason for these behavors s llustrated n Fg. 8d, whch shows a representatve tght expandng cluster of nvaders, at generaton 40 of a successful nvason event, wth neghborhood sze of 3 3 cells. As wth the coexstence results, smaller spatal nteractons facltate the formaton of tghter clusters. Soon after an nvader s ntroduced, stochastc events can cause the nvaders to de out before a cluster can form (Fg. 8c, below the dashed lne). For the small 3 3 cell neghborhoods, nvasve clusters become establshed wthn about 20 generatons f the nvadng speces can survve that long. For larger neghborhoods ths can take up to about 80 generatons. However, once a cluster of the nvadng speces s formed, the nvadng speces wll eventually overtake the resdent speces. Ths result s somewhat counter-ntutve, because the nvadng speces n ths regon has negatve frequency dependence whle the resdent had postve frequency dependence. However, consder the dynamcs at the boundary of the growng cluster of nvadng speces 2 (black, Fg. 8d). The area just outsde the cluster boundary s very favorable to speces 2, because ts negatve frequency dependence favors occupyng cells n neghborhoods where t s rare. Speces 1, whch has postve frequency dependence, does best n areas t prevously domnated but s vulnerable at the cluster boundary. The area nsde the cluster of speces 2 s unfavorable to both speces, but speces 2 can tolerate t better than speces 1 snce speces 2 has lower absolute frequency dependence. Consequently, once such clusters get started they grow slowly but relentlessly. Because the range of dspersal was lmted to the nteracton neghborhood n these smulatons, the number of generatons to total extncton of the resdent speces was hghest wth the smallest neghborhoods (Fg. 8c, above the dashed lne). In the loser negatve regon of the + quadrant, the nvason dynamcs were qute dfferent, as llustrated n Fg. 9 for the case where 11 = 0.5, 22 = 0.4 (Fg. 9a). Invason success was much hgher n the loser negatve regon of the + quadrant than n the loser postve regon of the + quadrant (compare Fg. 9b to Fg. 8b). Ths occurs because, where the establshed speces 1 has negatve frequency dependence, t s easer for ndvduals of the nvadng speces 2 to avod early extncton (Fg. 9c, below the dashed lne), because the negatve frequency dependence of the resdent speces has a tendency to make spaces avalable to the nvader. Surprsngly, t turns out that n ths case ntermedate szed neghborhoods were the most favorable to nvadng speces, wth a maxmum observed nvason success rate of 36% occurrng wth the 11 11 and 13 13 cell neghborhoods (Fg. 9b). We beleve ths non-lnear relatonshp may occur because of the followng. When neghborhood sze s very small, denser clusters are formed by vrtue of lmted dspersal and the tendency of the postvely frequency dependent nvader to grow near others of ts own knd. Insde dense clusters the frequency of the resdent s relatvely low; ths s favorable to the resdent, whch has negatve frequency dependence. Conversely, the terrtory outsde the cluster s relatvely unfavorable to the nvader. Ths keeps the nvadng clusters from growng rapdly and ncreases the chance that stochastc events wll kll off the nvader before the cluster grows enough to establsh a Fg. 9 (a) The representatve locaton 11 = 0.5, 22 = 0.4 n the + quadrant, (b) nvason success as a functon of neghborhood sze, (c) generatons to extncton events as a functon of neghborhood sze; above the dashed lne the resdent went extnct, below the dashed lne the nvader went extnct, and (d) one representatve nvadng loose cluster of speces 2 (black) 40 generatons after the ntroducton of a sngle ndvdual of speces 2 n an 11 11 cell neghborhood model.
12 ecologcal modellng xxx (2006) xxx xxx foothold on ts way to domnance. On the other hand, f the spatal scale s too large, then the postvely frequency dependent nvaders become wdely scattered early on, makng them more susceptble to early stochastc de out. Thus, for ths parameter combnaton n the loser negatve regon of the + quadrant there appears to be an optmal neghborhood sze for nvaders that balances these two effects, as shown by the rapdly expandng loose cluster at generaton 40 of one representatve 11 11 cell neghborhood smulaton, on ts way to successful nvason (Fg. 9d). When a successful nvason does occur n ths regon, however, t proceeds more rapdly when the nteracton neghborhood sze s smaller (Fg. 9c, above the dashed lne),.e., although tghter clusters of nvaders are more dffcult to establsh n ths regon, once they do become establshed they are advantageous to the nvadng speces, whch s postvely frequency dependent. In comparson to the tme to extncton n the loser postve regon of the + quadrant (Fg. 8c), complete extnctons take much longer n the loser negatve regon of + quadrant (Fg. 9c). Ths s because the negatve frequency dependence of the resdent enables t to reman vable even when dspersed over large dstances, and coexstence can occur for hundreds or even thousands of generatons. In the quadrant the coexstence results of all models were almost completely nsenstve to ntal condtons, as expected, wth the excepton that when there was only a sngle nvadng ndvdual t was () subject to extncton by random drft n the stochastc models, and () was not able to ncrease n frequency n the upper regons of the quadrant n the determnstc fnte-populaton model, for the same reasons dscussed wth regards to the + quadrant. 5. Dscusson Supported by a well-developed theoretcal framework, much of plant ecology has focused on the mportance of competton n structurng communtes. However, there s growng evdence that other botc nteractons can also have mportant effects on plant populatons, n part by generatng frequency dependence n plant populaton growth rates. Ths work advances the feld by developng and employng a model to explore the effects of asymmetrc frequency dependent nteractons at varous spatal scales on the spatal and temporal dynamcs of communtes. As wth prevous nvestgatons of local scale competton (Pacala and Levn, 1997; Neuhauser and Pacala, 1999; Bolker and Pacala, 1999; Deckmann et al., 2000) and symmetrc frequency dependence (Molofsky and Bever, 2002; Molofsky et al., 2002), we have found both qualtatve and quanttatve shfts n communty structure resultng from the spatal scale of ecologcal nteractons. In the absence of drft, speces that are ntally equally frequent can coexst ndefntely when ether () they have equal absolute frequency dependence or () they both have suffcently negatve frequency dependence. Smaller scales of spatal nteractons reduce the regon of the parameter space n whch ndefnte coexstence s predcted. Ths s consstent wth observatons made on spatal Lotka Volterra models (Neuhauser and Pacala, 1999), and s due to the fact that calculatng frequences over small neghborhoods has the effect of ncreasng the apparent frequency of the less frequent speces n those neghborhoods, relatve to ther frequency n the communty at large. We defne frequency dependent nteracton factors rangng from 0 to 1, such that the maxmum ftness of a speces s reduced unless frequency s optmal for that speces. Thus, outsde of the regons where ndefnte coexstence can occur, startng from equal frequences the speces wth lowest absolute frequency dependence wll ultmately drve the other to extncton. Ths result stems from the fact that less frequency dependence mples fewer constrants on whch cells an ndvdual s lkely to occupy, assumng dentcal maxmum habtat sutablty. However, the dynamcs of the spatal structure of the communty and the resultng rate of extncton are dramatcally affected by the spatal scale of the nteractons. When both speces have postve frequency dependence, small spatal nteracton neghborhoods promote cluster formaton, whch n turn promotes coexstence. Ths s consstent wth prevous model outcomes for symmetrc nteractons (Molofsky et al., 2001; Molofsky and Bever, 2002) but extends these results to asymmetrc frequency dependence. Throughout the loser postve regons of the parameter space (.e., where the losng speces has postve frequency dependence that s stronger than the postve or negatve frequency dependence of the wnnng speces), the tme to extncton ncreases exponentally as the scale of spatal nteractons decreases, because of the ablty of the losng speces to form protectve clusters that stave off extncton. When two speces have smlar, but unequal, postve frequency dependence, they can coexst for hundreds of generatons, even though mean feld stablty predctons ndcate that one speces wll domnate. Conversely, n the loser negatve regons of the parameter space (.e., where the losng speces has negatve frequency dependence that s stronger than the postve frequency dependence of the wnnng speces), the tme to extncton decreases wth decreasng spatal scale of nteractons, because the clusters formed by the wnnng speces can grow aggressvely. The most surprsng and ntrgung results, however, came from the nvasveness studes, where one or more representatves of an nvadng speces were ntroduced nto an establshed communty of the other speces. In these studes, the sze of spatal nteracton neghborhoods and the use of fnte-populaton models dramatcally affect predctons regardng how frequency dependence can potentally affect the nvasveness of speces. The analytc mean feld predctons ndcate that when both speces have postve frequency dependence, any ntal dsadvantage n frequency must be offset by an equally large frequency dependence advantage, f the ntroduced speces s to successfully nvade. Ths predcton was confrmed by the mean feld cellular automata smulatons, wth the mplcaton that t s vrtually mpossble for one speces wth postve frequency dependence to nvade an establshed communty of another postvely frequency dependent speces. However, the senstvty to ntal condtons n the communty at large was dramatcally reduced wth decreasng spatal scale of nteractons, snce the speces need only overcome a frequency dsadvantage wthn
ecologcal modellng xxx (2006) xxx xxx 13 any local neghborhood n order to become establshed. Thus, smaller spatal nteracton neghborhoods promote nvasveness when both nvader and resdent have postve frequency dependence, whch s not predcted by mean feld approxmatons. When speces have frequency dependence of opposte sgns, the analytc mean feld predcts that outcomes should be ndependent of ntal condtons. Ths mples that even a sngle ndvdual of a speces wth lower absolute frequency dependence should be able to nvade and drve to extncton resdent speces wth hgher absolute frequency dependence of the opposte sgn. However, our fnte-populaton mean feld and cellular automata models dd not concur wth ths and outcomes proved senstve to ntal frequences of the two speces over these regons of the parameter space, although, as n the postve postve case, ths senstvty to ntal condtons was dramatcally affected by decreasng the spatal scale of nteractons. In the loser postve, wnner negatve regons of the parameter space, nvason success was nversely proportonal to neghborhood sze, wth the smallest scales promotng the greatest nvasveness. Contrary to ntuton, nvaders wth negatve frequency dependence were able to form and explot nvasve clusters. However, the tme that the losng speces could persst before beng drven to extncton tended to decrease wth an ncrease n the spatal scale of nteractons. The loser negatve, wnner postve regons were, n general, less senstve to ntal condtons. Surprsngly, nvasons n at least part of ths regon were more often successful wth ntermedate neghborhood szes, although the maxmum tme that the losng speces could persst before beng drven to extncton ncreased wth an ncrease n the spatal scale of nteractons. The nvason results have mportant mplcatons for understandng the establshment of new speces nto resdent communtes. Specfcally, our results provde nsghts nto why a small populaton of alen speces can nvade and domnate a resdent plant communty. The nterplay between the sgns of the feedback and the sze of the nteracton neghborhood dramatcally affect both the ablty of an ntroduced speces to become establshed and the speed wth whch an establshed ntroduced speces ncreases n frequency. Our results show that, whle mean feld predctons suggest that postve frequency dependence n an ntroduced speces would nhbt ther establshment n a communty of postvely frequency dependent natves, when frequency dependence occurs locally, as would be the case for sol feedback (Bever, 2003), ntroduced speces may become establshed under much broader condtons. Once establshed, an nvader wth postve frequency dependence can overtake the resdent speces more quckly than when the nvader has negatve frequency dependence. We also demonstrate theoretcally that negatve frequency dependence n a resdent communty fosters early establshment of an ntroduced speces, but whether the ntroduced speces becomes nvasve depends on the relatve sgn and strength of frequency dependence n the ntroduced speces. Recent lterature has shown that some nvasve speces may have undergone a shft n ther nteractons wth ther sol communty from negatve feedback (mplyng negatve frequency dependence) n ther natve habtat to postve feedback (mplyng postve frequency dependence) n the new habtat (Renhart et al., 2003; Callaway et al., 2004). Our model predctons are consstent wth these observatons, n that they mply that establshed resdent communtes of coexstng speces are more lkely to have negatve frequency dependence, and such communtes are easly nvaded by speces wth mldly postve frequency dependence. However, more mportantly our work dentfes that the relevant varables are the relatve sgns, magntudes and spatal scales of the nteractons of resdent and ntroduced speces n the target communty, rather than shfts n frequency dependence of ntroduced speces between ther natve and new habtats. The predctons n ths paper are based on several smplfyng assumptons such as equal maxmum habtat sutabltes for all speces, rectangular and equvalent nteracton and dspersal neghborhoods, and lnear frequency dependence. Future work wll consder the effects of frequency dependence on nvasveness under more general condtons. In summary, frequency dependent nteractons can have a bg mpact on speces coexstence and nvasveness, and must be consdered to fully understand communty dynamcs. The spatal scale on whch these nteractons occur can dramatcally affect communty structure and populaton dynamcs. Understandng cluster formaton and densty and the relatve local nter-specfc dynamcs n the nterors, exterors, and boundares of self-organzng clusters of con-specfcs can provde nsghts nto the mechansms governng emergence of communty-wde spato-temporal dynamcs. Our results also hghlght the mportance of consderng non-equlbrum dynamcs, snce the tme horzon of envronmental changes may well be shorter than the tme horzon to acheve equlbrum condtons. The model proposed heren provdes a theoretcal framework for studyng frequency dependent nteractons that may shed lght on spatal and temporal dynamcs n real ecologcal communtes. Acknowledgements We are grateful to Frédérc Guchard, whose dscussons provded nsght nto our results, and to the edtor and two anonymous revewers for ther helpful suggestons. Ths work was supported n part by a plot award funded by DOE-FG02-00ER45828 awarded by the US Department of Energy through ts EPSCoR Program. references Ågren, J., 1996. Populaton sze, pollnator lmtaton and seed set n the self-ncompatble herb Lythrum salcara. Ecology 77, 1779 1790. Balzter, H., Braun, P.W., Köhler, W., 1998. Cellular automata models for vegetaton dynamcs. Ecol. Model. 107, 113 125. Bascompte, J., Rodríguez, M.A., 2000. Self-dsturbance as a source of spatotemporal heterogenety: the case of the tallgrass prare. J. Theor. Bol 204, 153 164. Berec, L., 2002. Technques of spatally explct ndvdual-based models: constructon, smulaton, and mean-feld analyss. Ecol. Model. 150, 55 81.