Branching and Self-Organization in Marine Modular Colonial Organisms: An Ecological Approach.

Transcription

1 Banhing and elf-oganizaion in Maine Modula Colonial Oganisms: An Eologial Appoah. Juan A. ánhez (), Howad R. Lase (), J. Daio ánhez () and Mihael J. Woldenbeg (3). () Depamen of Biologial ienes, 9 Cooe Hall, Univesiy a Buffalo (The ae Univesiy of ew Yo), Buffalo, Y 46, UA. js5@buffalo.edu; hlase@buffalo.edu phone: ; fax: () Depaameno de Maemaias, Univesidad aional de Colombia, Bogoá, Colombia. sanhez@buffalo.om phone ; fax (3) Depamen of Geogaphy, Univesiy a Buffalo, (The ae Univesiy of ew Yo), Y 46, UA. (76) x9 geomie@asu.buffalo.edu Key wods: banhing, modula oganisms, self-oganizaion, gogonian oal, olony developmen, banh inefeene. Running head: Banhing in modula oganisms.

2 Absa Regadless of he elevane and univesaliy of banhing paen in modula olonial oganisms, hee is no lea explanaion abou is developmen no wha maes hese oganisms o peseve shape duing gowh. Modula oganisms suh as gogonian oals banh subapially and depiing hieahial mohe-daughe elaionships among banhes. Mohe banh size fequeny disibuion followed a saling powe-law dependene suggesing selfoganized iialiy. hape is peseved in hese ee-lie newos by mainaining a onsan aio beween oal banhes and mohe banhes (). I is assumed ha, an inege in ems of numbe of banhes, is dynamially mainained by he poduion of mohe banhes (fom an old daughe banh) when he numbe of oal banhes is off he neighbohood of. Using ha simple ule, we modeled boh he ininsi poess of banhing along wih he global-eologial effes by adaping a disee logisi equaion. I exhibied a vey pediable ajeoy ending wihou fixed poins o sable equilibium, onveging o a fixed numbe of banhes. The banhing ajeoy was sigmoid wih a apid exponenial phase ha ended in a sho asympoi peiod, whih has been obseved by a numbe of empiial sudies on hydoids and oooals. Diffeen olony ahieues may have vey diffeen qualiaive behavios depending on he elaionship beween banhing () and. The inlusion of anohe paamee aouning fo banh inefeene and allomey, had endosemen fom size and esoue apue onsains. The omplex and dynami naue of banhing is sill modulaed iialiy by he ineaion beween and bu global eologial onsains pevail houghou olony developmen. Inoduion The sudy o banhing and ee-lie newos onens a gea numbe of disiplines as well as boh physial and biologial sysems. Banhing newos ae pesen fom he inies

3 vessels in he human body o inse wing o he oninenal size of he Amazon o Mississippi basins. Many maine modula olonial oganisms, suh as sponges, oals and hydoids among ohes, ae also banhing suues hemselves (i.e. Walle and eingaebe, 985). The gea impoane of he sudy of banhing fo hese oganisms is ha hei main ineaion wih he envionmen is hough banhing. Life-hisoy and finess in sessile modula oganisms ae elaed o he olony shape and size (i.e. Hughes, 983; Hughes e al., 99). Despie he elevane and univesaliy of he poblem of banhing, hee is no lea explanaion abou how is banhing dynamially ahieved duing developmen no wha maes hese oganisms o peseve fom o o sop gowing (see eviews in Buss, and Lase and ánhez, ). ome of hem ae indeed lassified as having indeeminae gowh (i.e. plasi aenuaing gowh: ebens, 987). This sudy, based on empiial obsevaions, aims o explain he limis of banhing in modula olonial oganisms aoding o a few paamees ha inea dynamially and how his sysem self-oganizes o peseve shape houghou developmen. Alhough being among he mos pimiive and basal meazoans, modula olonial oganisms suh as nidaians onain omplex developmenal mahiney suh as homeoi genes (i.e. Finney and Maindale, 997). Genes, suh as Cnox- fo insane, seem o be ieaively expessed houghou developmen of hydoid olonies, poviding a simple basis fo an ininsi mehanism of gowh bu no lea disinion fo poduing a new banh o exending an old one has been found (Cawigh e al., 999; Cawigh and Buss, ; Cawigh, pes. om.). Colonial gowh seems o be paially onolled and shaped by envionmenal simuli (Buss, ). Banhing in hese oganisms, onsequenly, is no jus a developmenal mae bu an eologial poess oo. Compue models demonsae ha wih a few ules and envionmenal inpu is possible o mimi olony gowh of modula olonial oganisms wih emaable ealism 3

4 (see eview in Kaandop and Kluebe, ). I is heefoe easonable o hin ha banhing in a modula olonial oganism be a ombinaion of boh inenal and exenal faos ha shape dynamially he sysem lie in any ohe eologial poess (e.g. logisi populaion gowh, hos-paasie, pedao-pey ineaions, e.). A hallenging goal fo a model of banhing in modula oganisms is o inlude he ineplay beween an ininsi gowh poess and global eologial effes. The sudy of banhing newos, in onas, has usually involved he sudy of odeing, whih depis hieahial elaionships among banhes as diffeen odes. Banhing in nonolonial sysems and ives has had onsideable heoeial aenion. Odeing sysems ae lassified ino wo goups: enifugal and enipeal. Cenifugal, is when odes inease in he same dieion as a gowing ee. Keill (63-79) developed elaboaed enifugal saling laws based on measuemens of ass of he aeial sysem assuming dihoomous banhing of he aeies (Woldenbeg, 997). Alhough enifugal appoahes wee vey ealisi on he hieahy of banhes and nodes, i was eplaed almos ompleely by enipeal shemes. Hoon (945) poposes a enipeal odeing sheme assigning odes fom he peiphey owads he un, whee unbanhed ibuaies ae assigned ode one and exessive side banhes have o be ignoed o ea lie a onsan. When wo banhes of ode one mee, hey eae a banh of ode wo and so on. ahle s (95) modifiaion of Hoon s mehod is he mos widely used odeing sysem in he sudy of ive and biologial ee-lie newos. Boh sysems ae well suied o eaing empiial geomei seies o numbe of banhes, mean banh lengh, mean basin aea, e. fo asymmeial ees. A simila enipeal sysem allows moe odes han ahle o Hoon sysems and he geomei seies ae no as genealized (Hosfield, 98). In spie of he dominane of he ahle ode sysems fo ives and 4

5 biologial sysems (i.e. Woldenbeg e al., 993), hee is some disonen wih he fa ha he Hoon-ahle odeing sysem does no hange when lowe ode banhes join he highe ode banh. These geneaional sysems ae well suied fo modeling he gowh of symmeially bifuaing newos. Unfounaely, few, if any of hese sysems ou in naue (bu see eview in Kaandop and Kluebe, ). Insead, naually ouing ees have asymmeial banhing. Anohe majo poblem of Hoon-ahle odeing has been exessive side banhing found in naue (e.g. pinnae banhing newos: Fig. ). Tounaga (978) poposed an elegan way o assign ode o asymmeial ees wih many side banhes. Using his modifiaion of he Hoon-ahle odeing sysems, saisial symmey has been eenly found beween geologial and geophysial banhing paens (Tuoe and ewman, 996) as well as beween living and non-living sysems (Pelleie and Tuoe, ; Kaandop and Kluebe, ). ome modula olonial oganisms suh as gogonian oal ees have been analyzed using he adiional Hoon-ahle odeing (Bazeau and Lase, 988; Mihell e al., 993) wheeas some ohe ould no due o exessive side banhing (anhez e al., submied). Using he Tounaga odeing sheme, a ommon feaue of ives and leaves newos was he nealy pefe lineaiy in he numbe of banhes () pe ode (i) when using semi-log sale (Pelleie and Tuoe, ), i, whee The mean banh lengh (L) in ems of ode in ives and leaves is also a linea funion using semi-log sale, L βi, whee β.34. Howeve, ohe modula oganisms suh as gogonian oals do no exaly follow he same paen. Fo he simple pseudo-dihoomous speies (e.g. andelabums Fig. B) hee was no appaen hange among o wihin odes in he semi-logaihmi plo, whih suggess ha all odes have appoximaely he same banhing popeies (e.g. gowh aes). On he ohe hand, 5

6 omplex pinnae speies (e.g. feahes Fig. ) exhibi a gea deal of wihin- and among-ode vaiaion ha ould be due o he diffeenial gowh aes obseved beween main and side banhes (i.e. Lase e al., in pep.). eveheless, L(i) fo oals was vey diffeen fom whih was found in ives and leaves (L βi ). Theefoe, he behavio of numbe of banhes and mean banh lengh pe ode ae no univesally simila in all banhing paens of modula oganisms suh as gogonian oals (see anhez e al., submied). This ould be due o he diffeen naue of he maine animal newos o simply beause hey do no eally bifuae duing banhing, whih seems o be he ase a leas fo gogonian oals. In his pape, we povide new heoeial gound on he odeing and dynamial behavio of banhing and olony developmen. We aimed o explain he poblem of banhing wih selfoganizaion and eologial gounds. Paiula goals of he sudy wee () o examine he banhing and odeing poess in modula oganisms using gogonian oals as model sysem, () o idenify ii paamees fo he banhing dynamis and (3) o popose a model explaining suh dynamis. Paiulaly fo suh model we dedued analyially an expession ombining a diffeene logisi equaion (i.e. May, 976; Case, ) wih he poess of banhing and odeing obseved empiially. Finally, he onvegene of he disee suession of banhing was examined o see if he model povides an explanaion fo deeminae gowh and when/how a olony sops banhing. This model does no aim o explain boh mehanisi and exenal onols on banhing bu o help undesanding he numeial behavio of banhing as a disee poess. We waned o show ha alhough he inenal onols fae he hanging envionmen of he olony foming a dynami eologial ineaion olony fom is peseve unil he end of developmen. 6

7 Colonial gowh in maine modula inveebaes Muliple obsevaions on olony gowh/developmen fom seveal maine modula oganisms have a ommon feaue: gowh deeases wih size/age. Fo insane, omplee obsevaions on he olony gowh ineis of he banhing hydoid Campanulaia flexuosa have shown ha hee is a deline in gowh as size ineases (ebbing, 98). I is woh noing ha hose hydoid olonies afe sopping gowh eep desobing hydans wihou beoming senesen. Even moe ineesing is he numeial behavio of he gowh ae of C. flexuosa hough ime, whih exhibied yli non-linea behavio always deeasing afe a high pea (i.e. ebbing, 98, Figs. -). Colony gowh in gogonian oals, ou model sysem, has also been doumened o sop/deease in a deeminae fashion when he olonies ae eahing eain size. Fo insane, gowh of he Paifi gogonian oal Muiea alifonia deeases as funion of heigh (Gigg, 974). ea whip gogonians Lepogogia spp. fom he Gulf of Mexio also exhibi eduion of gowh ae wih heigh (Mihell e al., 993). Colonies of he Medieanean gogonian oal Paamuiea lavaa exhibi also size-speifi negaive gowh (Coma e al., 998). imilaly, Lase e al. (in pep.) showed in an exensive suvey of he Caibbean gogonian Pseudopeogogia elisabehae how olonies deease and finally sop gowing nea eain size. Gowh fom he Medieanean sea-fan Euniella lavolinii adjuss o sigmoid gowh models (Velimiov, 875). Ohe oooal suh as he deep-sea sof oooal Anhomasus iei pesens a sigmoid Gompez gowh ajeoy ending asympoially a he bigges/oldes olonies (Codes e al., ). Consequenly, i is easonable o hin ha if olonies ae deeasing gowh when eahing eain size, hee ould be suh a fixed numbe indiaing he maximum apaiy of banhes befoe omplee inefeene. These obsevaions 7

8 also suggesed a sigmoid- o logisi-lie gowh ajeoy, whih mus be me by he pediions of a ealisi model of banhing. Banhing in modula oganisms: a self-oganized poess Modula olonial oganisms suh as gogonian oals banh sub-apially and depiing hieahial mohe-daughe elaionships among banhes, whih ould be obseved on young olonies of Pseudopeogogia bipinnaa (Fig. A-B; see also Lase and ánhez, in pess). This paen also ous among speies wih sub-apial gowh (poduion of banhes below he apex), whih ae he ase fo mos banhing nidaians as well as mos plans (see eview in Pusiniewiz, 998) and fungi (i.e. Waes e al., ). Mohe banhes esemble a pine banh wih many pinnules and ae self-simila (e.g. fis mohe main sem, seonday mohe banh, eiay mohe banh, e.). Daughe banhes, he pinnules, have deeminae gowh in sho peiods of ime (Lase e al., in pep.). The paen defining he mohe-daughe odeing sheme is fa fom having fixed banhing aios. I has indeed a paaboli-lie behavio in ems of banhes pe ode and a logisi- o sigmoid-lie appeaane when aumulaing he numbe of banhes pe ode. Howeve, any of he olonies follow he same developmenal ajeoy and i is easonable o hin ha hee is no suh deeminisi ule as he ase of enipeal of enifugal odeing and bifuaion shemes. Hundeds of olonies of he gogonian oal Pseudopeogogia elisabehae wee examined duing wo yeas showing ha any daughe banh an un in a mohe banh bu is exa loaion unde naual ondiions is uneain (unpublished). imilaly, evey olony banh disibuion (e.g. loaions of mohe banhes, numbe of daughe banhes, e.) has is unique ideniy lie a fingepin bu hey peseve simila speies-speifi fom. These obsevaions suggesed us o hin a diffeen 8

9 dieion o undesand banhing. This sysem seems o gow in a moe auonomous and individualisi way. Banhing on modula oganisms seems o be iially onolled by a paamee indiaing he aio beween oal banhes () and mohe banhes () o (Fig. ). Colonies fom wo diffeen gogonian speies (Gogoniidae: Pseudopeogogia bipinnaa; Plexauidae: Plexaua flexuosa) mainain a linea elaionship beween oal banhes and mohe bahes, whih slope is (Fig. ). hape is peseved in hese ee-lie newos by mainaining a onsan aio beween oal banhes and mohe banhes (). I is assumed ha, an inege in ems of numbe of banhes, is dynamially mainained by he poduion of mohe banhes (fom an old daughe banh) when he numbe of oal banhes is off he neighbohood of. I means ha evey ime mohe banhes ae poduing daughe banhes hey ae moving he sysem off, whih evenually eahes again afe new gowh fons (e.g. mohe banhes) ae podued and so on. I is assumed ha is elaed o a banhing heshold o an ininsi mehanism of banhing, whih is speies- o olony fom-speifi. To explain he banhing poess dynamially le us desibe banhing as a disee euen poess. Dynamial models may no always be ansmied in ems of diffeenial equaions fo hose whih vaiables hange diseely (Case, ). udying slow-gowing modula oganisms suh as Pseudopeogogia spp., hee is no way o alulae an insananeous gowh ae. This, indeed, should be alulaed fom disee peiods of ime (i.e. Goh and Chou, 995; Coma e al., 998; Lase and ánhez, in pess). Finie disee equaions also show he ieaive naue of olony developmen whee he gowing vaiables ae a funion of he pevious ondiions o X f (X ) (i.e. Kaandop, 994). Theefoe, le us onside a populaion of banhes (daughe and mohe banhes) inegaed as a olony fom a 9

10 pimodial mohe banh. Unde iniial ondiions, he numbe of banhes pe mohe banh ineases daughe banhes afe evey ieaion fom o (banhing ae). As he numbe of daughe banhes ineases fom ieaions of o, i eahes evenually banhes (o is lose o), ha povoes he poduion of mohe banhes fom an old daughe banh. I indeed podues gandhilden banhes espe o and so on (Fig. A-B). Theefoe he fuhe poduion of mohe banhes o gowh avalanhes has his euene fom: o, (). Expession () desibes he poess of banhing poduing self-simila mohe banhes in a self-oganized poess. Fo insane, a hypoheial speies wih and podued 9 mohe banhes afe ieaions and hei size fequeny disibuion (assuming an inemen of daughe banhes pe mohe banh afe evey ieaion) follows a powe law (Fig. 3 C). If banhing is leaded by a iial sae he size fequeny disibuion of banhing fons size (daughe banhes mohe banh -, D) mus be dependan of a saling powe law of he fom D(n) ~ n -τ, whee n is he fequeny of D(n) daughe banhes pe mohe banh and -τ is a faal exponen. Ineesingly, if we ompae he mohe banh size fequeny disibuion of diffeen olonies fom Pseudopeogogia bipinnaa and Plexaua flexuosa, i is obseved a vey simila paen as he obseved ieaing expession () and only noing qualiaive diffeenes beween speies due o diffeen faal saling powes (-.6 and -.4: Fig. 3 A and B). Consequenly, hee is an exponenial deease in banh size (o numbe of daughe banhes mohe banh - ) as hei fequeny in he banhing sysem ineases.

11 These heoeial and empiial obsevaions sugges he gea impoane of paamees suh as fo banhing peseving shape. An idea o explain his finding is he behavio of some dynami sysems ha evolve sponaneously fom appaenly saionay sages wihou uning of paamees (self-oganized iialiy, see eview in Ba, 996). uh sysem evoluion aound a paamee is due o he iial effe of suh paamee as vaiables inease suh as he ase of he slope in he sand pile/avalanhe model (i.e. Calson and windle, 995). Banhing and olony developmen an be edued as he poduion of new and ovelapping gowh fons of mohe banhes. Banhing an be onsideed as a self-oganized iialiy phenomenon osillaing as is being appoahed/eeaed. ine banhing oninues as well as he effe of in he poess, hese avalanhes of mohe banhes will be moe fequen and he olony will eep banhing a many mohe-daughe hieahies simulaneously lie in a self-oganized iialiy (e.g. Fig. C). Besides o be an aual phenomenon, his poess explains saisfaoily empiial obsevaions on banhing fo gogonian oals and he paamee eveals a mophospae when ompaed wih module size fo he diffeen olony ahieues of 4 Caibbean oooal speies (anhez e al., in pep.). Howeve, his ininsi banhing mehanism does no seem o explain when his poess should end. Expession () pedis an exponenial poduion of banhes (Fig. D), whih is no obseved empiially. The apid gowh by means of new gowh fons of mohe banhes bings along boh allomei and banh inefeene onsains ha ould influene o esoue apue and gowh aes. Consequenly, i is easonable o hin ha hee is an independen effe fom he ininsi banhing poess ha pevens olonies o gow exponenially. The following model aims o sele boh he ininsi poess of banhing wih he global-eologial effes.

12 An eologial appoah This model shapes he ininsi mehanism ha podues and lins mohe and daughe banhes in a olony explained above () by inluding global eologial effes. The model pedis a eduion of banhing as he numbe of mohe banhes eahes a maximum numbe o hage apaiy (). As we have seen peviously, he wo vaiables (mohe and daughe banhes) ae phenomena well disinguished duing he gowh of banhing olonies. Alhough hee is a enifugal odeing involved in he poess of olony developmen, his model pedis he poduion of banhes only wih he infomaion of mohe-daughe banh aio, whih is independen of odeing. Howeve, he exponenial inemen fo he numbe of mohe banhes obseved in () does no oun fo owdedness effes o he densiy-dependan/allomei onsains fo he apidly ineasing olony size (e.g. Fig. 3 D). Couning fo liely banh densiy-dependan onsains, a maximum numbe of mohe banhes due o banh inefeene, o, ould shape dynamially banhing in a logisi-lie fom. Hee, i is assumed ha he naual bigges olonies of a populaion exhibiing low (o asympoi) banhing ae in he neighbohood of suh numbe. ow he aim is o poay banhing in a logisi-lie dynami poess bu sill dependan of mohe banh poduion (as empiially obseved). Unde suh iumsanes we an modify a populaion disee logisi equaion lie his (): (). Whee is he oal numbe of banhes in a olony (boh daughe and mohe) and is he numbe of mohe banhes. Even hough expession () inludes he ininsi banhing ae i does no show how he banhing poess ous. Theefoe, he model is a sysem ha needs he euene expessions () and () o fully depi he dynami ineaion of banhing.

13 3 Resuls Aoding o he esuls fom inegaing expession () as shown in he Appendix A, we have expession (3) fo any value of giving only he iniial ondiions. (3). This expession is also needed in ode o find he analyi soluion fo all, ou vaiable of inees, in ems of, he leading vaiable ha behaves aoding o a mohe-daughe elaionship. ow we an eplae (3) o edue () o iniial ondiions (4): (4). Following he analyi soluion fo aoding o iniial ondiions (expessions 5-, Appendix B). Then we have poduion of banhes in ems of paamees and iniial ondiions (): (). Thee is also a speial ase when, o and we an quie edue he expession (): ( ) ( ) 3 (). Ou empiial obsevaions showed us ha all ee-lie modula oganisms suh as gogonian oals sa always wih one single soue banh ( ), whih is he only banh oo a iniial

14 4 ondiions ( ). Then we an have he model only in ems of paamees afe he fis ieaion ():, o, (). This shows ha only afe he seond ieaion he oganism sa having daughe banhes o banhing as he eason of he disee banhing ae wih a mino effe of he hage apaiy, beause hee is only one mohe banh and low inefeene is expeed. Afe he seond ieaion he banhing ineases in a moe omplex way unde he full sope of paamees, when i is pesumably eahing he neighbohood of (3). 4 (3). Fixed poin, onvegene, and qualiaive behavios. If expession () is he soluion fo a sysem desibing he quaniaive behavio of banhes duing olony developmen, i is expeed a pediable ouome pe speies beause i is a poess o build a disee suue insead of a oninuous sabilizing populaion. Theefoe, he poess should have a vey sable

15 ajeoy befoe o eah a fixed poin ohewise should onvege o a ommon numeial neighbohood fo eah olony (o speies) shaing he same banhing paamees. Iniially, if we wan o now a fixed poin fo we should ge: lim *, as:, whee. Theefoe, he lim exiss only if n lim lim < < < < <. Howeve, and ae by definiion posiive paamees (numbe of banhes). Theefoe, > and >, hen will neve be lowe han and does no have a fixed poin. If hee is no fixed poin fo a banhing olony, hen, wha is he fae of his dynami sysem? We an sudy he onvegene of he ieaive sysem fom a diffeen appoah. Fo insane, sine hee is popoionaliy beween he vaiables (mohe and oal banhes) o (4): (4). This expession ould be also used o examine he onvegene fo all hough ime. Fo insane, if we subsiue (4) in (), we obain (5): (5). 5

16 Le us suppose iniially ha he suession is onvegen and hen hee is he lim n L. Thus, aing he limi a boh sides of (5): lim lim lim. } Ν Howeve, wih { i is a sub-suession of{ } Ν, hen, lim lim L, and, lim L. Hene, we an alulae he value of L in he expession (6): L L L (6). ine, hen L and we an simplify (6) obaining: L L L This is he ase whee he suession { } Ν possibiliies (see appendix C fo onvegene ieion): is onvegen o L and we have wo I) If > in some ineval (e.g. when he uve is ove he saigh line ) hen he suession { } Ν ineases as long as is values emain in ha ineval. II) If < in an ineval (e.g. when he uve in unde he saigh line ) hen he suession { } Ν deeases as longs as is values emain on ha ineval. 6

17 The onvegene an also be ooboaed by using he quaniaive soluion of expession () and hypoheial values of paamees. Oveall hee is iniially a seady inemen in he numbe of new banhes podued beween disee peiods of ime ha uned abuply afe eahing maximum poduion pe peiod of ime (Fig. 4 A). The uning poin was usually above he line bu i osses he ajeoy a he uning poin when he value o was muh smalle han (Fig. 4 A). The umulaive numbe of banhes ime seies, indiao of olony size, had a sigmoid ajeoy wih a sho peiod of low gowh followed by exponenial inemen and ending afe a sho asympoi peiod (Fig. 4 B). Hypoheial speies also showed how he ineplay beween paamees and esuled in modified qualiaive behavios of he banhing ajeoy (Fig. 4). Thee ae wo main qualiaive behavios. One is when << oesponding o feahe-lie pinnae olonies suh as Pseudopeogogia bipinnaa, whih banhes podue many daughe banhes befoe han one of hese sa poduing gandhilden banhes (Fig. 5 A). The ohe ouome was hose speies wih vey simila values fo and, whee almos evey banh will fo sue have daughe banhes (Fig. 5 B). These ae pseudo-dihoomous o andelabum-lie olonies, suh as Plexaua flexuosa, ha pesen many ovelapping geneaions of mohe banhes. The model has a simplified vesion due o (expession : Fig. 3 B). Even hough he fae of banhing seemed vey deeminisi, Rein diagams of hese wo ouomes show how he pinnae olonies migh eep gowing in a yle of peiod aound he uning poin (Fig. 5 A). The absene of a fixed poin a he ineep wih he line sugges ha his is a vey unliely even bu qualiaively possible. oneheless, i is lea ha he ase of pseudo-dihoomous olonies is moe pediable han pinnae ones beause he values of unde he uning poin ae sill in he neighbohood of he maximum banh poduion in he lae (Fig. 5 A). This 7

18 ind of behavio migh be also obseved if exenal peubaion affes he iial paamee (e.g. gazing, disubane, e.). Disussion Banhing in modula olonial oganisms an be undesood as self-oganized poess ha podues fas pulses of banhing. This poess is shaped by he hanging envionmen of he gowing olony (e.g. banh inefeene and global olony size). Lie in many eologial sysems, banhing may have size- and densiy-dependan onsains explaining eduion of gowh. Colonial gowh, hough shaped logisially, onveges vey pediably o hei expeed maximum numbe of banhes, when banhing sops ompleely by is own banh inefeene. I was heoeially demonsaed ha hee is no fixed poin duing he poess of banhing and ha he euene suession onveges o, whih is nohing bu he maximum numbe of oal banhes. Consequenly, he banhing of modula olonial oganisms unde he sheme exposed above follow a vey pediable ouome. This is a damai diffeene wih a logisi populaion model whee hee may be sable poins and/o peiodi yles and he populaion peishes (i.e. May, 976). The banhing ajeoy was sigmoid wih a apid exponenial phase ha ended in a sho asympoi peiod. The new banhes podued duing eah disee peiod of ime was nealy symmei above and below he line indiaing a half-life uning poin whee gowh sas o deease. oneheless, he diffeen olony ahieues may have vey diffeen qualiaive behavios depending on he elaionship beween gowh and poduion of new gowh fons as new mohe banhes. The model pediions adjused o wha wee obseved by seveal sudies on gowh of maine modula inveebaes, whih wee logisi- o sigmoid-lie gowh ajeoies. 8

19 Banhing and self-oganized iialiy. The model pesened hee showed a poess whee a simple modula sysem an dive iself wihou he need of fine uning of any paamee. This self-oganized poess allowed o banh hough ime peseving olony shape. Alhough his ind of behavio has been idenified mosly on physial sysems (e.g. sandpiles, avalanhes, foes fie, e.), hee is an ineasing numbe of appliaions fo biologial sysems inluding banhing poess. elf-oganized iialiy (OC) has been found in he avalanhes of alveoli aivaion duing lung inflammaion, whih is a self-simila banhing suue (see eview in Csaho, ). Banhed polyme gowh has been explained by a OC sae of a egulaing ule fo he aggegaion of monomes (Andade e al., 997). Even he punuaed equilibium evoluion model, leading o evoluionay banhing, an be explained as a OC phenomenon (see eview in Ba and Pazusi, 995). I was supised o find ha a model as OC explains moe adequaely he dynami poess of banhing in modula olonial oganisms han adiional appoahes suh as odeing. Hee was exposed by he fis ime how he fom and developmen of banhing modula oganisms an be paially explained by OC. I was lea how he saling of mohe banhes size (boh empiial and heoeial) followed powe law fequeny dependene. I was idenified a iial paamee () ha eeps he sysem in a sponaneous dynamis wheeas peseving fom. hould mohe banhes podue moe daughe banhes, o, daughe banhes podue moe mohe banhes? This is he iial sae ha eeps he olony aively gowing. Bu, wha biologial mehanism ould povide suh self-oganized sae? Buss () model of hydoid olony gowh by inussusepions is onepually a OC, whee olonies eep adding modules esponding o eain heshold of inenal fluid ension. A simila appoah has been poposed using edox onol fo he same ind of hydoids (Blasone, 999), whih also 9

20 sugges he pesene of hesholds iggeing olonial expansion. Alhough we do no now he ineplay beween esoue anspo and developmenal genes expession in gogonian oals, he iial sae in a olony ould indiae eain diffeenial beween esoue availabiliy and suplus ha ould igge new gowh fons. I is, as a esul, esable o find a physiologial o poeonomial oespondene o he pulses of gowh obseved in gogonian oals. Cowdedness and banh inefeene. ize inemen bings along a seies of onsains ha affe he olony design and module inefeene (see eview in Lase and ánhez, in pess). pae is he pimay limiing onsain fo lonal sessile axa (Jason, 977). Fo insane, he design of banhing olonies of modula oganisms suh as heilosome byozoans have been shaped boh o peven dagging and beaage, and, o minimize owdedness and module inefeene (Cheeham and Thomsen, 98; Cheeham, 986). Duing he gowh of byozoans, banhes iniially divegene and pogessively onvege beginning o inefee wih hei funions, whih seems o limi he maximum olony size (Cheeham and Haye, 983). ebbing (98) suggesed ha Campanulaia flexuosa sops gowing when he spaes fo asexual poduion of zooids ae ompleely filled o in lose poximiy o ohe zooids. In gogonian oals, fo insane, expeimenal evidene shows ha owding among banhes impedes he apue of esoues a he shaded banhes (self-shading: Kim and Lase, 997). Thee ae indeed allomei onsains fo esoue apue duing modula gowh beause inenal modules begin o be esoue-depleed by he expansion of new exeio modules (Kim and Lase, 998). ome oooal olonies (sof oal aophyon) sa gowing and alifying fom he base up o he banhes (Tenoi E., Cenal Queensland Univ., pesonal ommuniaion). Boh sof and gogonian oals have die onneions fom evey module down o he olony base (Baye, 96). In he ase of gogonian oals i an be

21 plausible ha he gowh and exension of hei inenal axial hannels (solenia) be lined o he poduion of new banhes (unpublished). If esoues ae being depleed as he olony olonizes is peiphey, hee should be suh maximum exension poin (e.g. ) when no suplus is hen povided o he base and gowh and banhing would sop. The inlusion of a new paamee onolling banhing in modula oganisms suh as, o maximum banh apaiy, had suppos fom size and inefeene onsains, whih ae eviden phenomena duing olony gowh. The omplex naue of banhing is sill modulaed iialiy by he ineaion beween and bu global eologial onsains pevail houghou olony developmen. eveheless, die empiial obsevaions o fulfill a omplee eologial heoy of banhing ae needed. ome biologial ohe aspes of modula olonial oganisms suh as epoduion and egeneaion wee no ove in his model and ould have and impoan effe on banhing. aighfowad paamees have been idenified and a numbe of model oganisms seem appopiae fo suh ess. Fuue obsevaions on olony gowh ould gealy inease he nowledge in his field by inluding banhing paamees insead of heigh/widh o ohe indie measuemens of modulaiy. Anowledgmens J. A. ánhez anowledges Fulbigh-Laspau-COLCIECIA fo a dooal sholaship and gea suppo duing The Complex ysems umme hool a he Cenal Euopean Univesiy, Budapes, Hungay, he ana Fe Insiue (ew Mexio, UA), G. Yan (Univesiy a Buffalo, UY), and A. Cheeham (mihsonian Insiuion) gave o J.A.. new insighs and good disussion in he sudying of banhing. The aional Cene fo Eologial Analysis and ynhesis (CEA), Univesiy of Califonia, ana Babaa, woshop Modeling of gowh and fom in sessile maine oganisms 999 (J. Kaandop and J.E. Kuble) povided

22 gea feedba and disussions. The Bahamas Field aion (999-), Geae Reseah Cene-College of he Bahamas, an alvado, Bahamas, povided field failiies fo obseving gogonian oals. Commens and disussions fom C. Mihell, D. J. Taylo,.D. Cains, M.A. Coffoh and G. Yan (Univesiy a Buffalo, UY) gealy helped duing ealy sages of he sudy. APEDIX A Mohe banhes n fo all giving only iniial ondiions. (). If we eep ieaing equaion () by disee peiods ba in ime o ge he iniial ondiions we go:. And by muliplying hem membe o membe,

23 We have a simplified fom of () fo all given he iniial ondiions ieaed o imes (3): o, (3). APPEDIX B Analyial soluion fo aoding o iniial ondiions and. We have his sysem of disee diffeene equaions: (). (). Replaing () ino () (4). Fo bee manipulaion of (4) we an define (5) as: (5). Then ou euene equaion (4) ges he fom (6): o: ( ) (6). 3

24 4 This is a elesopi fomula given he fis values o, saing fom and hen, e., and i has he following soluion: ) (, ) (, ) (, 4 3 ) (,. Adding and equaling membe o membe i is obained (7): ( ) ( ) [ ] (7). The summaion of is a geomei pogession and i may have a value B : Β... (8). Then if we muliply B by we have:... Β o now, Β Β o: ( ) Β. If

25 5, hen Β... Liewise:.... Theefoe: ( ) (9). Using he oiginal paamees and aoding o (5), hen we have poduion of banhes in ems of paamees and iniial ondiions (): ( ) o, (). APPEDIX C Convegene ieion. In summay he onvegene ieion an be poayed as follows: Being L, obained above, he fixed poin of he funion ) ( x x x f is he ineval I, hen

26 (i) If f (L) >, hen he suession { } Ν does no onvege o L, exeping he ase whee he suession has a onsan value L, i means ha he suession would have been edued o {,,,, L, L, L, }. (ii) If < f (L) <, hen he suession { } Ν onveges o he limi L, in a monoonous way in he neighbohood of he fixed poin L. Addiionally: ) We say ha { } Ν onveges o L in he neighbohood of L if { } Ν when belongs o he neighbohood of L. ends o L ) If, f ( x) x x, hen f '( ) lim x f ( x) lim x x f ( ) lim x ( x ) ( x ) ( x ) ( ) x Theefoe f '( ). ine and ae posiive numbes may no be highe han. Thus, < and beause < f () < and heefoe < <, hen we should have ha < when hee is a neighbohood fo of { x R δ < x < δ } ( δ, δ ) /, suh as < f ( x) < x when x (, δ ) > f ( x) > x when x ( δ, ) onsequenly, if ( δ, δ ) hen, n 6

27 < n < n when n (, δ ) > n > n when n ( δ, ) Theefoe, he suession { } Ν ineases (o deeases) and is delineaed by if (, δ ) (o ( δ, ) espeively), and due o suession heoy, he suession { } Ν } Ν suession { in ( δ, δ ) neighbohood of. onveges in he neighbohood of. Howeve, is he only limi of he onsequenly { } Ν onveges o monoonously in he Lieaue Cied Andade J., J.., L.. Luena, A.M. Alena and J.E. Feias elf-oganizaion in gowh of banhed polymes. Physia A 38: Ba, P How naue wos: he siene of self-oganized iialiy. Copenius, ew Yo. Ba, P. and M. Pazusi Complexiy, oningeny, and iialiy. Poeedings of he aional Aademy of ienes UA 9: Baye, F.M. 96. The shallow wae Oooallia of he Wes Indian egion. udies of he Fauna of Cuaçao : Blasone,.W Redox onol in developmen and evoluion: evidene fom olonial hydoids. Jounal of Expeimenal Biology : Bazeau D. and Lase H Ine- and Inaspeifi vaiaion in gogonian olony mophology: quanifying banhing paens in aboesen animals. Coal Reefs 7:

28 Buss, L.W.. Gowh by inussusepion in Hydainiid hydoids. In: Evoluionay Paens: Gowh, Fom, and Tempo in he Fossil Reod., J.B.C. Jason,. Ligand, and F.K. MKinney (eds). Chiago: Univesiy of Chiago Pess. Calson, J.M. and G.H. windle elf-oganized iialiy: sandpiles, singulaiies, and saling. Poeedings of he aional Aademy of ienes UA 9: Cawigh, P. and L.W. Buss Colony inegaion and he expession of he Hox gene, Cnox-, in Hydainia symbiologiapus (Cnidaia: Hydozoa). Jounal of Expeimenal Zoology 85: Cawigh, P., J. Bowshe and L.W. Buss Expession of a Hox gene, Cnox-, and he division of labo in a olonial hydoid. Poeedings of he aional Aademy of ienes UA 96: Case, T.J.. An illusaded guide of heoeial eology. Oxfod Univesiy Pess, ew Yo. Codes, E.E., ybaen J.W. G. and VanDyhuizen.. Repoduion and gowh of Anhomasus iei (Oooallia : Alyonaea) fom Moneey Bay, Califonia, UA. Maine Biology 38 (3): Cheeham, A.H Banhing, biomehanis and byozoan evoluion. Poeeding of he Royal oiey of London B 8: 5-7. Cheeham, A.H. and L.-A. C. Haye Geomei onsequenes of banhing gowh in adeonifom Byozoa. Paleobiology 9(3): 4-6. Cheeham, A.H. and E. Thomsen. 98. Funional mophology of aboesen animals: sengh and design of heilosome byozoan seleons. Paleobiology 7(3): Coma, R., Ribes M., Zabala M. and J.-M. Gili Gowh in a modula olonial maine inveebae. Esuaine Coasal and helf iene. 47(4):

29 Finney, J.R. and M.Q. Maindale Homeoboxes in sea anemones (Cnidaia: Anhozoa): a PCR-based suvey of emaosella veensis and Meidium senile. Biologial Bullein 93: Gigg, R.W Gowh ings: annual peiodiiy in wo gogonian oals. Eology 55: Goh,.K.C., and L.M. Chou Gowh of five speies of gogonians (sub-lass Oooallia) in he sedimened waes of ingapoe. Maine Eology 6(4) Hosfield, K. 98. The siene of banhing sysems. In: ienifi foundaions of espiaoy mediine. J.G. adding and G. Cumming (eds). London: William Heinemann Medial Boos. pp Hoon, R.E Eosional developmen of seams and hei basins: hydophysial appoah o quaniaive mophology. Geologial oiey of Ameia Bullein 56: Hughes, R Evoluionay eology of olonial eef-oganisms, wih paiula efeene o oals. Biologial Jounal of he Linnean oiey : Hughes, T.P, D. Aye and J.H. Connell. 99. The evoluionay eology of oals. Tends in Eology Evoluion 7(9): Jason, J.B.C Compeiion on maine had subsaa: The adapive signifiane of soliay and olonial saegies. Ameian aualis : Kaandop, J.A Faal modelling. Gowh and fom in biology. pinge-velag, Belin. Kaandop, J.A. and J. Kluebe.. The Algoihmi Beauy of eaweeds, ponges and Coals. pinge-velag, Amsedam. 9

30 Kim K. and H.R. Lase Flow-mediaed esoue ompeiion in he suspension feeding gogonian Plexaua homomalla (espe). Jounal of Expeimenal Maine Biology and Eology 5():49-64, Kim K. and H.R. Lase Allomey of esoue apue in olonial nidaians and onsains on modula gowh. Funional Eology. (4): Lase, H.R., M. Bolle & J.A. ánhez. (submied). Deeminae gowh of a modula maine oganism. May, R imple mahemaial models wih vey ompliaed dynamis. aue 6: Lase, H.R. and ánhez (). Allomey and Asogeny of modula oganisms. In: R.. Hughes (ed.), Pogess in Asexual Popagaion and Repoduive aegies. John Wiley (Oxfod). (in pess) Mihell,.D., Dadeau M.R. and W.W. hoede Colony mophology, age suue, and elaive gowh of gogonian oals, Lepogogia hebes (Veill) and Lepogogia vigulaa (Lama), fom he nohen Gulf of Méxio. Coal Reefs (): Pelleie J.D. & Tuoe D.L. () hapes of ive newos and leaves: ae hey saisially simila?, Philosophial Tansaions of he Royal oiey of London B 355: Pusiniewiz, P Modeling of spaial suue and developmen of plans: a eview. ienia Hoiuluae 74: ebens, K. P The eology of indeeminae gowh in animals. Annual Reviews in Eology and ysemais 8: enning, A.R.D. 98. The ineis of gowh onol in a olonial hydoid. Jounal of he Maine Biologial Assoiaion U.K. 6:

31 ahle, A Hyposomei (aea-laiue) analysis of eosional opogaphy. Geologial oiey of Ameia Bullein 63: 7-4. Tounaga E Consideaion on he omposiion of dainage newos and hei evoluion. Geogaphy Repos Toyo Meopolian Univesiy 3: -7; Tuoe D. L. and ewman W. I ymmeies in geology and geophysis. Poeedings of he aional Aademy of ienes UA 93: Tuoe D. L., J.D. Pelleie and W.I. ewman ewos wih side banhing in biology. Jounal of Theoeial Biology 93: Velimiov, B Gowh and age deeminaion in he sea fan Euniella lavolinii. Oeologia 9: Walle D.M. and D.A. eningaebe Banhing and modula gowh: Theoeial models and empiial paens. pp In: Jason J.B.C., L.W. Buss and R.E. Coo (eds.). Eology and evoluion of lonal oganisms. Yale Univ. Pess, ew Haven. Waes, M.K., C. Humphies, I. De Vies and A.J. Giffihs.. A homeosai se poin fo banhing in euospoa assa. Myologial Reseah 4(5): Woldenbeg, M.J James Keill (78) and he mophomey of he mioosm. Pp In: D.R. odda (ed), Poess and fom in geomophology. Rouledge, London and ew Yo. Woldenbeg, M. J., M. P. O'eill, L. J. Quaenbush, and R. J. Penney Models fo gowh, deline and egowh of he dendies of a Puinje ells indued fom magniude and lin-lengh analysis. Jounal of Theoeial Biology 6(4):

32 Figue Legends Figue. Phoogaphs of living olonies of he gogonian oal Pseudopeogogia bipinnaa (an alvado, Bahamas). A-B. Examples of wo young olonies a Deembe 999 () and July, ( ), he aows in depi he daughe banhes ha uned ino a new mohe banh in. C. Adul olony showing muliple gowing mohe banhes (gid x m). Figue. Plos of he numbe of oal banhes vs. mohe banhes olony - wih mean and 95% pediive inevals indiaed. A. Fom olonies of Pseudopeogogia bipinnaa ( 9;.69; P<.5). B. Fom 9 olonies of Plexaua flexuosa ( 5;.88; P<.5) (an alvado, Bahamas). Boh inse phoos: an alvado, Bahamas. Figue 3. A-C. ize fequeny disibuion of numbe of daughe banhes banh - in log-log sale. A. Fom 76 mohe banhes of phoogaphed olonies of Pseudopeogogia bipinnaa (.93; P<.5). B. Fom mohe banhes of 6 phoogaphed olonies of Plexaua flexuosa (.9; P<.5) (an alvado, Bahamas). C. Fom a hypoheial olony ieaing expession () eleven imes, wih (assuming an exension of a evey mohe banh pe ieaion) and, 9 banhes mohe banhes (.96, P<.5). D. Cumulaive ime seies of mohe banhes, daa fom C. Figue 4. Resuls fom he ieaion of he model fo a hypoheial speies wih 3 and. A. Diffeen numbe of new banhes fo and wih diffeen values of (4, 8,,, and ). B. Cumulaive oal numbe of banhes fo diffeen values of (4, 8,,, and ) along 6-monh peiods. Figue 5. Rein diagams showing wo diffeen quaniies behavios fo he numbe of new banhes (ne gowh pe ieaion) fo he map of all and ( 3). A. When 4 and,. B. When and. 3

33 A B C

34 Toal banhes A Toal banhes 4 3 B Mohe banhes Mohe banhes

35 A B Banh Fequeny Banh Fequeny C Daughe Banhes Banh - D Daughe Banhes Banh - Banh Fequeny Daughe Banhes Banh

36 A B (umulaive)

37 5 4 A >> B ( ) ( ) 3