Competition, extinction, and the sexuality of species

Transcription

1 A. Zool. Fec 38: ISSN X Helsk 2001 Fsh Zoologcal ad Botacal Publshg Board 2001 Dedcated to the memory of W. D. Hamlto Competto, extcto, ad the sexualty of speces Waye M. Getz Departmet of Evrometal Scece, Polcy, ad Maagemet, Uversty of Calfora, Berkeley, CA , USA Receved 25 Jauary 2001, accepted 5 Aprl 2001 Getz, W. M. 2001: Competto, extcto, ad the sexualty of speces. A. Zool. Fec 38: Cosderg the well-kow two-fold cost of males assocated wth sexual reproducto, the mateace of sex despte atural selecto remas a egma for populato bologsts. The prevalece of sex amog eukaryotes s most commoly explaed by hypotheses assocated wth ether the purgg of deleterous mutatos, the geerato of favorable gee combatos, the fxato of beefcal mutatos, or, less frequetly, ecologcal theores dealg wth the coexstece of competg populatos. Almost all these hypotheses gore the fact that stochastc evromets, asexual populatos exhbt hgher rates of extcto tha sexual populatos because the latter geerally explot a wder spectrum of resources tha ther asexual couterparts. Here we develop a model to demostrate, populatos where mutatos from sexual to asexual reproducto are possble, that three reproductve phases sexual, mxed, ad asexual aturally arse amog competg sexual ad asexual les. The partcular phase observed s related to the level of stochastcty the evromet expereced by the populato complex questo (e.g. a partally competg group of cogeerc speces) ad s a mafestato of the teso that exsts betwee the reproductve superorty of asexual populatos ad ther hgher rates of extcto. I term ths explaato the demographc balace hypothess ad suggest the edeostgmatd mtes provde a sutable taxo for testg ths hypothess. Itroducto Gee exchage amog orgasms may be as acet as lfe tself. The evoluto ad mateace of sexual reproducto eukaryotes, however, are egmas for both evolutoary ad populato bologsts. Bologsts have foud t dffcult to expla how the demographc twofold cost of males s mataed uder atural selecto. Specfcally, how s sexual reproducto able to persst whe the trsc reproductve rate of a sexually reproducg populato s

2 316 Getz ANN. ZOOL. FENNICI Vol. 38 half that of a cloally reproducg couterpart (e.g. see Wllams 1975, Mayard Smth 1978, Bell 1982)? At ssue s ot how but why sex arose eukaryotes (Kodrashov 1986, Charlesworth 1993) ad ot how but why reversals to asex occur ad persst (Vrjehoek 1993). Also at ssue s the coudrum that whle sexual reproducto s ubqutous amals, a umber of hgher taxa appear to have bee exclusvely asexual for may mllos of years (Bell 1982, Mayard Smth 1986). I partcular, geetc studes dcate that the bdellod rotfers (class Bdellodea) have abstaed from sex for 30 to 40 mllo years (Welch & Meselso 2000). Datg back to Wesma (1889), may hypotheses have bee voked to expla the mportace of sexual reproducto. (For a crtque of Wesma s orgal deas see Burt (2000)). West et al. (1999), followg Kodrashov (1993), broadly classfy these hypothess or explaatos to two dstct categores: evrometal ad mutatoal. Both of these categores, though, are aalyzed ether a geetc or atural selecto, but ot a demographc, cotext. Mutatoal hypotheses argue that deleterous mutatos are more easly purged uder sexual (amphmxs) tha asexual (apomxes) reproducto. Ths results what s loosely referred to as mutatoal meltdow asexual leages (Lych et al. 1993). Recet data cast doubt, however, o the capacty of sexual reproducto to purge the geome of deleterous mutatos (Keghtly & Eyre-Walker 2000). Evrometal hypotheses post that ether beefcal mutatos, or at least favorable combatos of gees, accumulate more rapdly amphmctcally tha apomctcally reproducg populatos (as revewed Kodrashov 1993, ad West et al. 1999). Legh Va Vale s (1973) Red Quee s hypothess mples that the average ftess of dvduals a populato s mataed at some characterstc fracto of optmal ftess for the speces because the populato s adaptve ftess ladscape s cotuously chagg. Ths, tur, mples that selecto for ew beefcal mutatos ad favorable combatos of gees s cotually ogog. The above metoed characterstc fracto s gog to be much closer to 1 uder sexual tha asexual reproducto because, as emphaszed by West et al. (1999), Evrometal models suggest that sex accelerates adaptato to a chagg evromet by creatg ew gee combatos. Hamlto (1980), whose memory we hoor ths volume, argued the ve of the Red Quee that parastes ad ther hosts are volved a evolutoary arms race whch sex s favored through the geetc varato t creates coupled wth the ftess of rare host geotypes the presece of parastes (see also Lvely 1985, Ebert & Hamlto 1996, Hurst & Peck, 1996; but see Ladle et al. 1993). Ths ad all the other evrometal ad mutatoal hypotheses revewed by Kodrashov (1993) ad West et al. (1999) do ot expla why taxa, such as the bdellod rotfers, have bee asexual for may mllos of years. If sex s advatageous for other speces of rotfers, why s ot advatageous for the asexual bdellods? The aswer to ths questo could well le a curretly eglected thrd category of explaatos, amely purely ecologcal or demographc. From a ecologcal pot of vew, Bell (1982: p. 131) has argued that a heterogeeous evromet a sgle cloe s ulkely etrely to supplat a dverse sexual populato. Ths s ot a argumet that s ultmately caste the cotext of atural selecto but terms of a cotest betwee a cloe whch has the greater reproductve effcecy but a arrower ecologcal competece ad a effcet but broadly competet sexual populato. Bell, spred by a phrase the cocludg paragraph of Darw s Org of Speces, refers to ths explaato for the mateace of asexual populatos as the tagled bak. Case ad Taper (1986), ad more recetly Docaster et al. (2000), used mathematcal models to explore the plausblty of ecologcal explaatos. Case ad Taper, for example, aalyzed the dyamcal propertes of three cosecutve models at creasg levels of resoluto wth respect to the geotypc structure of the populatos they represeted: amely, a lumped model (o geotypc structure), a sgle locus geotype model (pheotype = geotype ths model), ad a quattatve trat model. Ther aalyss revealed that suffcet ecologcal che parttog amog pheotypes allows sexual

3 ANN. ZOOL. FENNICI Vol. 38 Competto, extcto, ad the sexualty of speces 317 speces to coexst ad sometmes eve supplat asexual speces. O the other had, Docaster et al. (2000) used the very smple lumped two speces Lotka-Volterra competto model wth separate brth ad death rate terms to demostrate the exstece of a threshold growth rate for the sexual populato, above whch the vaso [of the asexual populato] s halted by traspecfc competto. As West ad Peters (2000) pot out, Docaster et al. (2000) essetally cofrm Bell s tagled bak hypothess. The same s true of Case ad Taper s aalyss, although they do fd stuatos (regos of model parameter space) whch a sexual populato completely dsplaces a asexual populato ad vce versa. Hece, Case ad Taper do provde a possble explaato for the fact that some taxa are exclusvely sexual, whle others may be mxed or exclusvely asexual. Note that aalyses usg populato models lackg geetc or pheotypc structure, such as the frst of Taper ad Case s three models ad the Lotka-Volterra model used by Docaster et al., essetally reduce the aalyss to a group selectost argumet. A model that s gog to be used to address the ecologcal aspects of the sex versus asex coudrum eeds to explctly corporate the fact that dfferet pheotypes do ot explot the exact same ecologcal che. Wllams (1975) rased the mportat questo (p. 155): Does the presece or absece of sexual reproducto dfferet taxa fluece botc evoluto by alterg rates of extcto, ad f so, how? Bell (1982) aswers ths questo by potg out that f sexual populatos are less sestve to loss of specfc habtats tha asexual populatos, because a sexual populato s lkely to explot a broader array of habtats tha a asexual populato, the loss of a sgle habtat s more lkely to lead to extcto of a asexual tha a sexual populato. The mechasm that I clude the model preseted ths paper s more geeral tha Bell s. I allow habtats (represeted by specfc resource levels) to vary stochastcally, but ot ecessarly dsappear. Ths evrometal stochastcty, together wth the atural demographc stochastcty of populatos (.e. the atural sample varato assocated wth brth ad death processes), results asexual populatos gog extct more ofte tha sexual populatos. Thus, t does ot requre the actual destructo of habtats to cause asexual populatos to go extct more ofte tha sexual oes. From a demographc pot of vew, a teso exsts betwee the reproductve superorty of asexual populatos ad the fact that they are more lkely tha sexual populatos to go extct. Wth the ad of a model, I demostrate below that ths teso s suffcet to expla why we should expect asexual populatos to domate relatvely weakly stochastc evromets, sexual populatos relatvely hghly stochastc evromets, ad the coexstece of the two relatvely moderate stochastc evromets. I use the word relatve because, some taxos, the probablty of extcto s greatly reduced through partcular adaptatos, such as the ablty of bdellod rotfer to survve dehydrato ad freezg (Örsta 1998). Iterestgly, these three reproductve phases asexual (low stochastcty), mxed (moderate stochastcty), ad sexual (hgh stochastcty) replcate a patter observed the dstrbuto of sexual ad asexual speces 27 geera of mtes that are collectvely kow as the edeostgmata group (Norto et al. 1993, see Table 1). I refer Table 1. Dstrbuto of 79 sexual ad 80 asexual speces wth 27 geera costtutg the Edeostgmata group of mtes (Acar). Purely asexual speces Mxed speces (sexual/asexual) Purely sexual speces Number of geera a Mea umber of speces per geus / a Three geera each cotag a sgle asexual speces are omtted because they are ecessarly pure.

4 318 Getz ANN. ZOOL. FENNICI Vol. 38 to the explaato provded by ths teso betwee reproducto ad extcto as the demographc balace hypothess. Returg to what West et al. (1999) refer to as a pluralst explaato for the mateace of sex, demographc factors ca always be added to obta a more complete explaato. The crtcal questo, however, s the relatve mportace of varous processes ay gve stuato ad whether, oe partcular process domates most stuatos. I wll ot aswer ths questo but pot out that, ulke most explaatos relatg to sexual reproducto, the demographc balace hypothess s suffcet to expla why, some taxa, ether sexual or asexual reproducto domates ad, other taxa, coexstece of the two strateges s apparet. Model I ths secto, a dscrete tme populato model s preseted that cludes suffcet structure to explore ssues relatg to vaso, coexstece, ad excluso of sexual ad asexual populatos competg wth dfferet prefereces ad effceces for the same set of resources. Ths model s urelated to models used ay prevous studes ad hece s more fully developed the Appedx. A symmetrc verso of the model cotag far fewer parameters tha the more geeral model s preseted here the text uder a umber of smplfyg assumptos. These assumptos, rather tha reducg the geeral applcablty of the aalyss, allow us to focus o how the reproductve phase of the group of competg populatos depeds o the tradeoff betwee vaso (mutato) ad extcto rates, where each of these rates s represeted by a partcular parameter the model. (Note, the parameter that wll be vared for the extcto rate s the oe assocated wth the level of demographc stochastcty). Determstc symmetrc pheotype model Cosder a sexually reproducg populato that cossts of detfable pheotypes, each at desty x (t), = 1,,, at tme t. Suppose that each pheotype has the same hertablty compoet h ad the th pheotype has a asexual aalogue that whe extat has desty y (t), = 1,,, at tme t. Wth each sexual pheotype ad ts asexual aalogue, all dvduals are cosdered ecologcally detcal (.e. they are ecologcal homologues) the way they compete for resources. For smplcty, cosder a partto of a heterogeeous spectrum of resources to compoets, such that the th compoet, wth desty at tme t deoted by z (t) = 1,,, s preferred over all other compoets by the th ecologcal pheotype, whether sexual or asexual. At ay tme t, from a ecologcal pot of vew, the populato desty of the combed sexual ad asexual populatos s gve by the vector x + y, where x = (x 1,,x ) ad y = (y 1,,y ), ad the resource desty s represeted by the vector z = (z 1,,z ). We assume that the rate at whch a dvdual of pheotype s able to gest the resources t explots (the resources must be coverted to the same commo currecy e.g. calores) s gve by some fucto φ (x + y,z), = 1,,. Aga, for smplcty, cosder the case where all pheotypes have the same desty-depedet brth rate b > 0 ad a resource-gestodepedet survval rate s(φ ) expressed as sa ˆ s( φ)=, (1) 1 + ( K / φ γ ) Here the parameter K > 0 scales the abscssa ad the parameter γ 1 determes how rapdly survvorshp decles the eghborhood of K (Getz 1996). The parameter a scales the ordate axs ad s expressed terms of K ad a maxmum resource gesto rate parameter δ > 0 a maer that esures s( δ )= sˆ. Specfcally, a = 1 + (K/δ) γ whch esures that the survval rate s(φ ) creases mootocally o [0,δ] from ts mmum value 0 to ts maxmum value ŝ. More geerally, all the parameters the fucto s(φ ) could be dexed by, but the symmetrc case treated here they are assumed to be the same for all pheotypes.

5 ANN. ZOOL. FENNICI Vol. 38 Competto, extcto, ad the sexualty of speces 319 Fg. 1. A cartoo represetato of the proportos of tme each ecologcal pheotype (whether sexual or asexual) speds explotg each of the dfferet resources compoets for the cases p = 1/ (the proportos are all the same) ad p = 0.5 (each pheotype speds half ts tme explotg ts preferred resource ad dvdes the remag tme equally amog the remag 1 resources). For each of the asexual pheotypes the chage populato desty from oe ut of tme (geerato) to the ext s modeled by the equato (cf. Eq. A5 the Appedx) 2sba ˆ y ( t + 1) = y () t 1 + ( K φ( +, ) ) γ, x y z = 1,,, (2) where the costat 2 accouts for the fact that asexual females produce twce as may females per geerato as ther sexual ecologcal aalogues ad hece, all else beg equal, ther populato compouds twce as fast each geerato. The equatos for the destes x (t) are complcated by the fact that the sexual pheotypes terbreed. Assumg that all pheotypes are subject to the same hertablty value h, ad assumg that matg s radom the equato s gve by (cf. Eq. A5 the Appedx) sba ˆ x( t + 1) = 1 + K φ( x + y, z) h hx() t + 1 xt() t ( ) γ, = 1,,.(3) All that remas s to specfy the precse form of the resource extracto fucto φ (x + y,z). I the cotext of a lumped populato at desty x explotg a homogeeous populato at desty z, Beddgto (1975) ad DeAgels et al. (1975) proposed the extracto fucto δz φ( xz, ) = d + cx+ z, To keep the umber of model parameters dow to a mmum geeralzg ths fucto to the stuato cosdered here, assume that each dvdual depedet of pheotype speds a proporto 1/ p 1 of ts tme explotg ts preferred resource ad a proporto 0 1 p 1/ 1 explotg each of the remag 1 less preferred resources. Note, f p = 1/, the ths latter equalty mples that each pheotype speds the proporto 1/ explotg each of the remag 1 resource compoets (ths s the complete che overlap case). Further, the the latter equalty mples the populato does ot explot ay of the remag 1 resource compoets so that ter-pheotypc competto s 0 (ths s the o che overlap case) (see Fg. 1). Addtoally, assume that dvduals explot all resources wth equal effcecy d, except for ther preferred resource, whch they explot wth effcecy d/λ, (λ > 1 esures

6 320 Getz ANN. ZOOL. FENNICI Vol. 38 Equlbrum populato sze 100 dvduals are more effcet at explotg ther preferred resource). Wth the above assumptos, ad usg the otato x T = = 1 p x, zt = z ad π = ( 1 p 1), = 1 we ca exted the above scalar from of the extracto fucto φ to obta (see Appedx for more detals) φ( xz, )= πz δ ( 1+ π) d / λ + ( x + ( π 1) x )+ z + ( π 1) z j = 1 j asexual Proporto of tme (p) spet 1 T T z j ( 1+ π) d + ( xt + ( π 1) xj)+ zt zj (4) π Note that as p = 1 t follows that ( 1 + π ) δz π so that φ( xz, ) d / λ + x + z. Basele parameter values sexual Equatos 2 4 costtute a determstc model Fg. 2. The determstc equlbrum sze (uts scaled by settg K = 1) of a sexual populato competg wth a sgle asexual cloe for the basele parameter set s plotted as fucto of the proporto of tme p (1/ p 1 wth = 10) each pheotype speds o ts preferred resource. of a sexual populato wth pheotypes competg wth cloal populatos where each cloal populato s the ecologcal aalogue of oe of the sexual pheotypes. Geerally, we ca evaluate competto betwee the sexual populato ad a coalto of m (1 m ) asexual pheotypes by settg the tal codtos to x > 0 for at least some, ad y > 0, = 1,,m ad y = 0, = m + 1,, (ote, because the values of the parameters are depedet of the pheotype, t makes o dfferece whch set of m tal cloe destes s set to 0). Despte all the smplfyg assumptos, the model stll has eght populato parameters ad resource compoet puts. Notg that the parameter K s a scalg costat that ca be set to 1 wthout loss of geeralty, the followg parameters were selected to be the basele set for the aalyss: = 10, h = 0.5,, K = 1, γ = 2, δ = 10, c = 10, d = 100, ad z = 3, = 1,,10. The remag parameters esure that the model produces a stable coexstece equlbrum betwee the sexual ad oe of the asexual populatos over the whole rage of the che partto parameter p [1/ = 0.1,1] (Fg. 2). Note that the model ca produce chaos, partcularly as the parameter γ creases value (Schoombe & Getz 1998). All umercal smulatos of the model were ru o a Apple G3 PowerPC usg Berkeley Madoa, V , ( The code was checked by comparg umercal ad aalytcal results for some of the smpler stuatos. For the more complcated stuatos, equlbrum solutos were obtaed by rug the model for several hudred geeratos utl the soluto was o loger chagg for the desred umber of sgfcat dgts. I the stochastc smulatos, descrbed below, the radom umber ad ormal dstrbuto geerator fuctos Berkeley Madoa were used. Stochastc extesos Issues of coexstece ad compettve excluso of the sexual populato ad ts partheogeetc ecologcal homologues are greatly affected by the troducto of stochastcty to the model. Two kds of stochastcty were cluded the aalyss. Frst, evrometal stochastcty, was

7 ANN. ZOOL. FENNICI Vol. 38 Competto, extcto, ad the sexualty of speces 321 cluded by treatg the resource compoets z, = 1,,10 as depedet radom varables. I stochastc rus, each of the compoets was draw at each terato of the model from a dstrbuto geerated by averagg the values of three uformly dstrbuted varables o [0,6] (.e. to clarfy, wth the same geerato each of the resource compoets vares stochastcally from the other resource compoets, ad from oe geerato to the ext each resource compoet tself vares stochastcally). Ths procedure results a symmetrc bell-shaped dstrbuto o [0,6] that s more kurtotc (about 0.36) tha a ormal dstrbuto, ad has a mea of 3 ad varace of 1. Secod, demographc stochastcty was cluded by multplyg the trsc growth rate by a ormally dstrbuted stochastc varable W wth mea 1 ad varace σ 2 /x T for the sexual populato ad σ 2 /y for the th asexual populato, but trucated so that all values of W less the 0.01 ad greater tha 3 were respectvely set to 0.01 ad 3. The mplcatos of ths choce are that: () 3ŝb s a upper boud uder the most favorable of codtos, () the populato could ot become egatve, ad () the level of stochastcty s cotrolled by a varace parameter σ 2 wth actual varace versely proportoal to populato sze (as t should be from samplg theory). To obta a more tutve terpretato of the total level of stochastcty perceved by a sexual or asexual populato exstg solato a partcular stochastc evromet, the coeffcet of varato (CV) for the sexual ad asexual populatos was obtaed from smulatos for each populato solato from the others over a perod of 2000 geeratos (Table 3). For extcto studes, populatos were regarded as extct ad set to zero whe the populato sze (x T for the sexual populato ad y for the th asexual populato) dropped below the value gve by the pseudo-extcto threshold parameter θ. I revaso studes, populatos that were 0 were set each tme perod to the value θ INT(V + ε) where INT deotes the teger part of the argumet, ad V s a uformly dstrbuted radom varable o [0,1]. Ths approach mples that the radom values so obtaed are less tha θ wth probablty 1 ε, whch case the populato sze was reset to 0. Coversely, the radom values so obtaed are greater tha or equal to θ wth probablty ε, whch case the system equatos were used to calculate the ext value of the vadg populato. Smulato results Coexstece equlbra Numercal solutos of the determstc Eqs. 2 4 dcate that whe che overlap s relatvely strog (vz. p < 0.37 see Fg. 2) a sexual populato comprsg of te pheotypes s vaded ad excluded by ay of the te cloal ecologcal homologues. Coexstece occurs, however, whe che overlap s relatvely weak, ad both populatos have detcal equlbrum levels whe p 0.49 (Fg. 2). As the degree of che overlap s further reduced (.e. p > 0.49), the sexual populato becomes more domat, but ever excludes the asexual populato. Whe che overlap betwee ecologcal pheotypes s completely elmated (.e. at p = 1) the sexual ad asexual equlbrum values are 70.8 ad 11.2 respectvely (Fg. 2). Though a sexual populato ca effectvely compete agast a sgle cloal le whe che overlap s zero, rrespectve of the degree of che overlap (.e. value of p), the sexual populato s always drve to extcto by a coalto of three or more cloal les (Table 2). Stochastcty ad extctos Mote Carlo smulatos were coducted for the case p = 0.49 that correspods, the absece of stochastcty, to the stuato whch competg sexual ad asexual populatos coexst at the same equlbrum level of approxmately 33 (Fg. 2). Settg the extcto threshold parameter value to θ = 0.01 (.e. approxmately 0.03% of the equlbrum level of 33), both populatos cotue to coexst deftely for relatvely small values of the demographc stochastcty parameter σ 2 eve though evrometal sto-

8 322 Getz ANN. ZOOL. FENNICI Vol. 38 chastcty stll produces moderate coeffcets of varato the sexual ad asexual populatos (Table 3). Vrtually o extctos occur over 1000 geeratos for σ 2 = 2.0, but for σ 2 = 3.5 the probablty of survval of a cloal le for ths perod of tme drops to 63% ad for σ 2 = 5.0 to oly 3% (Fg. 3), whle the probablty of survval of the sexual populato remas at the100% level all three cases. From the determstc equlbrum (Table 2) ad stochastc extcto (Fg. 3) results preseted above, t s evdet that at low levels of stochastcty coaltos of several cloal les drve the sexual speces to extcto, whle at hgh levels of stochastcty coexstece of a sexual populato wth oe or more cloes s dsrupted by the hgh rates of extcto of cloal les. Mutatos ad reproductve phases The ext step the aalyss s to allow sexual dvduals to mutate to dvduals that reproduce partheogeetcally (geetc aspects of ths type of trasformato are dscussed Bell (1982)). Such mutatos are essetally equvalet to aalyzg the vaso of a sexual populato by oe more asexual cloes that are ecologcal homologues of the varous sexual pheotypes. We would ot expect reverse mutatos that s, mutatos from partheogeetc to sexual reproducto to occur because, ths case, cotemporaeous mutatos of partheogeetc dvduals to at least oe male ad oe female are requred. To level the playg feld our aalyss of competto betwee a sexual populato ad a coalto of oe or more cloes, we cosder vasos of a coalto of cloes by a small group of sexually reproducg dvduals. Ths vadg group may orgate from a remat of the orgal sexual populato that s ow allopatrc to the coalto of cloes that compettvely replaced t. For smplcty, the rate at whch asexual populatos are revaded by sexuals s set equal to the rate at whch sexuals are vaded by asexuals, rrespectve of whether the latter s due to mutatos or boa-fde vasos. The results, such as those Table 4, are ot crtcally depedet o these two rates beg equal, or eve close value. Rather, the dstrbuto of sexually reproducg speces wth taxa depeds o the terplay of extcto ad vaso rates, as well as compettve excluso processes ad the degree of stochastcty the system. Table 2. Determstc equlbrum populato szes whe model parameters have the basele determstc values = 10, h = 0.5, ŝb = 20, K = 1, γ = 2, δ = 10, c = 10, d = 100, p = 0.49 (.e., π = 8.75), ad z = 3, = 1,,10. Number of cloal les Equlbrum sze Equlbrum sze of each competg wth of sexual populato ozero cloal le the sexual populato N/A extcto extcto extcto extcto extcto extcto extcto extcto 11.3

9 ANN. ZOOL. FENNICI Vol. 38 Competto, extcto, ad the sexualty of speces 323 Dscusso Equatos 2 4 embody the uderlyg precepts of the froze che varato hypothess (Vrjehoek & Pfeler 1997) ad the closely related froze pheotypc varato hypothess (Jokel et al. 1997). These two hypothess have respectvely bee show to be compatble wth the geetc structure of partheogeetc populatos of molles (Poeclopss spp.) ad sals (Potamopyrgus atpodarum). Whether these competg sexual ad cloal populatos form a traspecfc or cogeerc complex of competg populatos, depeds o how log the populatos have bee reproductvely solated ad o our tedecy to lump or splt taxa. The above two hypotheses apply oly to taxa whch partheogeess s able to arse from mutatos or through terspecfc hybrdzatos. Clearly the rates of such evets are postve taxa where partheogeess s observed, but appear to be zero taxa, such as mammals, where both materal ad pateral geomes are eeded for embryos to develop to term (Sura et al. 1984, Kevere et al. 1996). Reproductve phases The demographc balace hypothess posts a sexual phase should occur wheever sexual to asexual mutato rates are relatvely low or levels of stochastcty, ad hece extctos of asexual speces s relatvely hgh compared wth sexual speces (Table 4). Wth reducg levels of stochastcty, the asexual populatos go extct less ofte (Fg. 3) ad t becomes creasgly lkely that more tha oe asexual le exsts at ay pot tme (Fg. 4). Whe the level of stochastcty has dropped far eough, a coalto of several asexual les overwhelms ad excludes the sexual populato (Fg. 4 ad Table 4) ad we have a asexual phase for the taxoomc group questo. From the determstc results (Table 2), a asexual phase must set at some pot as the level of stochastcty dmshes dow to zero. The breadth of stochastc levels supportg the trasto betwee the sexual ad asexual phases (.e. the mxed phase Table 4) depeds o the vaso rate ε ad o the extcto threshold θ. Edeostgmatd mtes A example of sexual ad asexual phases related Acar geera s see the edeostgmata mtes, whch are a collecto of eleve closely related famles totalg 159 speces explotg edaphc habtats ad moss mats. Almost half the speces are asexual (79 asexual ad 80 sexual), yet fve of the eleve famles are purely asexual (although oe of the famles cossts of oly oe speces) whle oe s purely sexual. Also, wth the remag fve mxed famles, the majorty of the geera are ether purely sexual or asexual (Table 1). Of the twety-four edeostgmata geera cotag more tha oe speces, eght are purely sexual (10.5 speces o average per geus), eleve are purely asexual (6.2 speces o average per geus), leavg oly fve mxed (2.4 sexual ad 6.0 asexual speces o average per geus) (Table 1). Gve the approxmately equal umber of sexual ad asexual Table 3. The coeffcets of varato (CV) the sze of the sexual ad asexual populatos s tabulated here whe evrometal stochastcty s corporated the model ad the level of demographc stochastcty the model s gve by the tabulated value σ 2 as lsted below (see Methods secto for more detals). Parameter σ 2 Sexual CV Asexual CV ) 1) Ths value s calculated oly over the tal terval before the populato goes extct (.e., drops to a level less tha oe thousadth of ts solato equlbrum level) whch varably happes before 1000 geeratos have traspred.

10 324 Getz ANN. ZOOL. FENNICI Vol. 38 Probablty of survval 1 sexual: s 2 = 3.5 ad s 2 = 5.0 asexual: s 2 = 3.5 asexual: s 2 = Geeratos 1000 Fg. 3. Survvorshp curves, obtaed by averagg the results of oe thousad Mote Carlo smulatos over 5000 geeratos for the case of a sgle cloal le competg wth a sexual populato (tal codtos are x (0) = 3, = 1,,10, y 1 (0) = 30, ad y (0) = 0, = 2,,10), are plotted for the two creasgly stochastc cases σ 2 = 3.5 ad σ 2 = 5.0 (.e. creasg levels of demographc stochastcty but the same level of stochastcty the uderlyg resources. Sold les represet the asexual populatos for the two cases (gray s σ 2 = 3.5 ad black s σ 2 = 5.0) ad the dotted le represets the sexual populato whch both cases ever goes extct. purely asexual group of sx speces s approxmately 1/2 6 < Thus the probablty of gettg etee out of twety-four geera at radom to be purely sexual or asexual s exceedgly small. Assumg that mutatos for cloal reproducto arse equally frequetly all sexual speces of edeostgmatd mtes, the demographc balace hypothess posts, as a explaato for the hghly skewed dstrbuto of purely sexual ad asexual geera Table 4 (c.f. Table 1), that purely asexual geera are foud evromets that are less varable tha the mxed geera whch tur are foud evromets that are less varable tha the purely sexual geera. Thus, the edeostgmatd mtes provde a group of speces that could be use to test the demographc balace hypothess. To do so would requre measurg the sze of competg sympatrc populatos of the 27 speces over eough geeratos to obta relable estmates of meas ad varaces, ad the comparg the coeffcets of varato for sexual versus asexual speces. Also, the coeffcets of varato averaged across a group of sympatrc speces could also be compared wth a spatally dstct group of speces that represets a dfferet reproductve phase. speces the group, the probablty of assemblg at radom a purely sexual group of te speces s approxmately 1/2 10 < or a Dstrbuto of partheogeess The exstece of sexual ad asexual phases Table 4. The mea umber of extat cloal les ad the lkelhood of the sexual populato beg preset (cf. Table 2) are lsted for 3 separate smulatos of each of the three dfferet levels of stochastcty tabulated here (the vaso parameter s ε = ad the extcto threshold s θ = 0.01) a. Also see Fg. 4 for some typcal plots of the results. CV: sexual/asexual (σ 2 ) 0.20/0.24 (0.5) 0.24/0.25 (2.0) 0.36/0.33 (10) Phase Asexual Mxed Sexual Smulato ru Sexuals b Mea umber of cloal les a Mea values are calculated by samplg whether the sexual populato exsts ad how may asexual cloes exst every 100 geeratos durg geeratos to of smulatos wth tal codtos x (0) = 3, = 1,,10, y 1 (0) = 30, y (0) = 0, = 2,,10. b These etres represet the probabltes of the sexual populato beg preset ad are rouded to 1 decmal place wth stadard devatos omtted for clarty. Thus the 0 s ad 1 s are ot ecessarly exactly 0 or 1.

11 ANN. ZOOL. FENNICI Vol. 38 Competto, extcto, ad the sexualty of speces 325 s 2 = 0.5 (CV: 0.20/0.24) s 2 = 2.0 (CV: 0.24/0.25) Fg. 4. The sze of the sexual populato (top graph) ad oe of the te asexual cloes (mddle graph), as well as the umber of extat cloal les (bottom graph) are plotted every 100 geeratos over geeratos for oe ru of each of the two stochastc cases σ 2 = 0.5 ad σ 2 = 2.0 (parameters: basele determstc values, vaso parameter ε = , extcto parameter θ = 0.01, stochastcty of resources as descrbed the Methods; tal codtos: x (0) = 3 for = 1,,10, y 1 (0) = 30 ad y (0) = 0 for = 2,,10). Relatve populato sze 10 Number Sexual populato (sum of 10 pheotypes) Example of 1 of 10 asexual pheotypes Number of extat asexual populatos Geeratos Geeratos 5 mples that the dstrbuto of cloal les should be clumped wth geera whe sexual populatos are prevalet, or clumped wth hgher taxa f sexual populatos have bee excluded by coaltos of cloal les for suffcetly log perods of tme o evolutoary scales. The bdellod rotfers represet the hghest kow taxoomc level that s exclusvely asexual. I the bdellod rotfers, however, the level of stochastcty that ca be tolerated before trasto to a mxed phase occurs may be greatly elevated by the fact that these rotfers have reduced extcto rates through adaptatos that allow ther eggs to survve dry or freezg codtos ahydrobotc forms ad successfully rehydrate oce the eggs are aga a aquaeous evromet (Örsta 1998). May examples of exclusvely partheogeetc famles or geera exst other taxa, cludg more complex vertebrates, partcularly the orders Acar (mtes) ad Hymeoptera (ats, bees, wasps, ad sawfles). Haplodplod geetc systems (dplod females develop from fertlzed eggs ad haplod males develop from ufertlzed eggs) are commo Acar ad oblgate Hymeoptera. Ths fact leads oe to speculate that rates of trasto from sexual to partheogeetc reproducto are ehaced by haplodplody because these geetc systems permt the developmet of adults from ufertlzed eggs. A comprehesve aalyss evaluatg the merts of ths speculato s, however, beyod the scope of ths paper. Partheogeess occurs at least 22 older vertebrata geera (Vrjehoek et al. 1989), cludg the molles (Vrjehoek 1994) metoed above, the salamaders Ambstoma, ad whptal lzards, Cemdophorus. I the vertebrates, the orgs of partheogeess appear to be through terspecfc hybrdzato (Dawley 1989, Vrjehoek 1989) rather tha drect mutato. The model predcts that we should fd

12 326 Getz ANN. ZOOL. FENNICI Vol. 38 partheogeetc speces wheever extcto rates of partheogeetc speces do ot overwhelm the rates at whch partheogeetc les arse. The reaso for the absece of partheogeetc leages brds s ot clear, but as mammals, whch we recall requres both paretal geomes for successful embryogeess (Sura et al. 1984, Kevere et al. 1996), may be related more to compatbltes betwee partheogeess ad embryogeess tha to rates at whch cloal les go extct. Further, males have greater value speces where they play a crtcal role rearg youg, whch appears to be more mportat brds tha ay other class of vertebrates. Cocluso We should ot expect ay sgle factor to adequately expla the dstrbuto of reproductve systems across all the eukaryotes. Rather, may processes ad costrats at the geomc, physologcal, behavoral, ad ecologcal levels are lkely to play a role. The most eglected process explag the curret dstrbuto of sexual ad asexual speces, however, s that asexual speces have hgher extcto rates relatvely stochastc evromets. The power of ths smple explaato s that t also predcts asexually domated taxa relatvely stable evromets. We see such domato at the geus ad famly taxoomc levels edeostgmatd mtes ad at the class level rotfers. The ablty of populato models to predct observed frequeces of asexual speces varous taxa wll always be lmted by the extraordary complexty of real systems. Despte ths, the aalyss preseted here provdes for the frst tme a explaato of why, uder a spectrum of codtos leadg to dfferet levels of stochastcty populatos, we mght expect to see asexual, mxed, ad sexual phases exstg sde by sde related taxoomc groups where mutatos from sexual to partheogeetc reproducto occur. Ths explaato represets a testable hypothess the case of the edeostgmatd mtes, provded coeffcets of varato ca be estmated for the sze of populatos of the 27 speces of mtes costtutg ths taxoomc group. Ackowledgemets I thak Smo Lev, Sr Robert May, ad Mchael Rosezweg for supportve commets o a tal draft much dfferet from ths fal verso, George Barlow, Phlp Starks, ad Davd Wake for sghtful commets ad leads to the lterature, ad Peter Baxter, Paul Cross, Era Karmo, James Lloyd-Smth, Ja Washbur, Stuart West, ad Chrs Wlmers for commets o varous versos of the mauscrpt. Refereces Beddgto, J. R Mutual terferece betwee parastes or predators ad ts effect o searchg effcecy. J. Am. Ecol. 44: Bell, G. 1982: The Masterpece of ature: the evoluto ad geetcs of sexualty. Uversty of Calfora Press. Berkeley. Burt, A Perspectve: sex, recombato, ad the effcacy of selecto was Wesma rght? Evoluto 54: Case, T. J. & Taper, M. L. 1986: O the coexstece ad coevoluto of asexual ad sexual compettors. Evoluto 40: Caswell, H Matrx populato models: costructo, aalyss, ad terpretato (2d ed.). Sauer, Suderlad, Massachusetts. Charlesworth, B. 1993: The evoluto of sex ad recombato a varyg evromet. J. Heredty 84: Dawley, R. M. 1989: A troducto to usexual vertebrates. I: Dawley, R. M. & Bogart, J. M. (eds.), Evoluto ad ecology of usexual vertebrates: Bullet 466, New York State Museum, Albay, New York. DeAgels, D. L., Goldste, R. A. & O Nell, R. V. 1975: A model for trophc teracto. Ecology 5: Docaster, C. P., Poud, G. E. & Cox, S. J. 2000: The ecologcal cost of sex. Nature 404: Ebert, D. & Hamlto, W. D. 1996: Sex agast vrulece: the coevoluto of parastc dseases. Treds Ecol. Evol. 11: Getz, W. M Populato dyamcs: a per capta resource approach. J. Theor. Bol. 108: Getz, W. M A ufed approach to multspeces modelg. Natural Resource Modelg 5: Getz, W. M. 1993: Metaphysologcal ad evolutoary dyamcs of populatos explotg costat ad teractve resources: r-k selecto revsted. Evol. Ecol. 7: Getz W. M. 1996: A hypothess regardg the abruptess of desty depedece ad the growth rate of populatos. Ecology 77:

13 ANN. ZOOL. FENNICI Vol. 38 Competto, extcto, ad the sexualty of speces 327 Hamlto, W. D. 1980: Sex versus osex versus parastes. Okos 35: Hurst, L. D. & Peck J. R. 1996: Recet advaces uderstadg of the evoluto ad mateace of sex. Treds Ecol. Evol. 11: Jokel, J., Lvely, C. M., Fox, J. A. & Dybdahl, M. F. 1977: Flat reacto orms ad froze pheotypc varato cloal sals (Potamopyrgus atpodarum). Evoluto 51: Kevere, E. B. Fudele, R., Narasmha, M., Barto, S. C. & Sura, M. A. 1996: Geomc mprtg ad the dfferetal roles of paretal geomes bra developmet. Dev. Bra Res. 92: Keghtley, P. D. & Eyre-Walker, A Deleterous mutatos ad the evoluto of sex. Scece 290: Kodrashov, A. S. 1993: Classfcato of hypotheses o the advatage of amphmxs. J. Heredty 84: Kodrashov, A. S. 1986: The asexual plody cycle ad the org of sex. Nature 370: Ladled, J. L. Johstoe, R. A. & Judso, O. P Coevolutoary dyamcs of sex a metapopulato: escapg the Red Quee. Proc. Royal Soc. Lod. B 253: Lvely, C. M. 1985: Host-paraste coevoluto ad sex. BoScece 46: Lych, M., Buerger, R., Butcher, D. & Gabrel, W. 1993: The mutatoal meltdow asexual populatos. J. Heredty 84: May, R. M Models for two teractg populatos. I: May, R. M. (ed.) Theoretcal ecology (2d ed.): Sauer Assocates, Suderlad, MA. Mayard Smth, J. 1986: Cotemplatg lfe wthout sex. Nature 324: Mayard Smth, J. 1978: The evoluto of sex. Cambrdge Uversty Press, Cambrdge, Eglad. Norto, R. A., Kethley, J. B., Johsto, D. E. & O Coer, B. M. 1993: Phylogeetc perspectves o geetc systems ad reproductve modes of mtes. I: Wresch, D. L. & Ebbert M. A. (eds.), Evoluto ad dversty of sex rato sects ad mtes: Chapma ad Hall, New York. Örsta, A. 1998: Factors affectg log-term survval of dry bdellod rotfers: a prelmary study. Hydrobologa 387/388: Schoombe, S. W. & Getz, W. M Evolutoary stable desty-depedet strateges a geeralzed Beverto ad Holt model. Theor. Popul. Bol. 53: Sura, M. A. H., Barto, S. C. & Norrs, M. L. 1984: Developmet of recosttuted mouse eggs suggests mprtg of the geome gametogeess. Nature 308: Va Vale, L. 1973: A ew evolutoary law. Evolutoary Theory 1: Vrjehoek, R. C. 1989: Geetc ad ecologcal costrats o the orgs ad establshmet of usexual vertebrates. I: Dawley, R. M. & Bogart, J. M. (eds.), Evoluto ad ecology of usexual vertebrates: Bullet 466, New York State Museum, Albay, New York. Vrjehoek, R. C. 1993: The org ad evoluto of cloes versus the mateace of sex Poeclopss. J. Heredty 84: Vrjehoek, R. C. 1994: Usexual fsh: model systems for studyg ecology ad evoluto. Au Rev. Ecol. Syst. 25: Vrjehoek, R. C., Dawley, R. M, Cole, C. J. & Bogart, J. P. 1989: A lst of the kow usexual vertebrates. I: Dawley, R. M. & Bogart, J. M. (eds.), Evoluto ad ecology of usexual vertebrates: Bullet 466, New York State Museum, Albay, New York. Vrjehoek, R. C. & Pfeler E. 1997: Dfferetal survval of sexual ad asexual Poeclopss durg evrometal stress. Evoluto 51: Welch, D. M. & Meselso, M. 2000: Evdece for the evoluto of bdellod rotfers wthout sexual reproducto or geetc exchage. Scece 288: Wesma, A The sgfcace of sexual reproducto the theory of atural selecto. I: Poulto, E. B., Schoelad, S. & Shpley, A. E. (eds.), Essays upo heredty ad kdred bologcal problems: Claredo Press, Oxford. West, S. A., Lvely, C. M. & Read A. F. 1999: A pluralst approach to sex ad recombato. J. Evol. Bol. 12: West, S. A. & Peters, A. D. 2000: Payg for sex s ot easy. Nature 407: 962. Wllams, G. C. 1975: Sex ad evoluto. Prceto Uversty Press, Prceto, New Jersey.

14 328 Getz ANN. ZOOL. FENNICI Vol. 38 Appedx Determstc Model Let x(t) 0 represet the desty (umbers or bomass) of a populato at tme t. Assume populato growth s regulated through traspecfc competto for a lmtg resource z(t) 0. If φ(x,z) s the amout of resources that each dvdual s able to extract from ts evromet durg ts reproductve lfe ad R(φ)s the reproductve value of each dvdual at brth, the the model we develop here s based o elaboratg the structure of R ad φ fudametal growth equato x(t + 1) = R(φ(x(t),z(t))x(t), (A1) ad the structurg the populato to competg pheotypes, each of whch may or may ot terbreed wth other pheotypes. I model A1, the uts of tme are geeratos. For a semelparous populato, R(φ) = b(φ)s(φ), where b(φ) s the per capta brth rate (average umber of progey per adult) ad s(φ) s the proporto of progey that survve to reproduce whle for a age-structured teroparous populato, the value of R ad the legth of a geerato ca be calculated from a statoary lfe table, assumg the populato s at ts statoary age dstrbuto (Caswell, 2001). I the cotext of a homogeeous resource z, the otos of resource-depedet ad rato-depedet extracto rates are geeralzed by the Beddgto-DeAgels resource extracto fucto (Beddgto 1975, DeAgels et al. 1975; also Getz 1984, 1991, 1993) δz φ( xz, ) = (A2) d + cx+ z where d > 0, c > 0, ad δ > 0 respectvely are extracto effcecy, terferece competto, ad maxmum extracto rate parameters. Other forms for ths fuctos ca ad have be used (May, 1981). For smplcty, we assume R(φ) = bs(φ), where b > 0 s costat, ad for costats 0< ŝ < 1, K > 0, ad γ 1, sa ˆ s( φ)=. (A3) 1 + ( K / φ) γ The costat a s expressed as a = 1 + (K/δ) γ (recall the δ s the maxmum extracto rate coeffcet fucto A2), whch mples that the maxmum survval rate s s( δ )= sˆ. The parameter K scales the abscssa ad the parameter γ 1 determes how rapdly survvorshp decles the eghborhood of K (Getz 1996). Note that at φ = 0 mples s(0) = 0 mplyg that all dvduals de whe o resources are cosumed (from Eq. A2 holds wheever resources are abset,.e., z = 0). Ths formulato ca be geeralzed may ways to model several populatos competg for a spatally, temporally, utrtoally, ad physcally ( terms of how easly they are hadled ad processed by the cosumer) structured resource. Perhaps the most relevat way to characterze the resources s from the perspectve of the cosumers competg. For example, the cotext of competg populatos, wth desty of populato deoted by x, = 1,,, oe perspectve s to assume that we ca structure the resources to compoets z, = 1,,, such that speces favors compoet over all other compoets j, j = 1,,. If we use p j to deote the proporto of tme

15 ANN. ZOOL. FENNICI Vol. 38 Competto, extcto, ad the sexualty of speces 329 that dvduals of type sped explotg resource compoet j, the t follows for each = 1,, that p p j 0, j = 1,, ad p j = 1. If we further assume that the degree to whch dvduals of j = 1 type k terfere wth a dvdual of type explotg resource compoet j s proportoal to the relatve amout of tme dvduals of type k sped explotg resource compoet j, the the extracto fucto A2, the cotext of a dvduals of type explotg all resource compoets, geeralzes to (usg the vector otato x = (x 1,,x ) ad z = (z 1,,z ), where prme deotes vector traspose) φ ( xz, )= δ j 1 pz j j. (A4) d + c p x + pkz k = j j k k k = 1 k = 1 Here δ I > 0 s the maxmum resource extracto rate (we mplctly assume here that resources are substtutable so that o resource s lmtg o ts ow), c j > 0 s a costat that scales relatve levels of terferece wth respect to the partcular resource compoet beg exploted, ad the costats d j 0 are the relatve effceces wth whch dvduals of pheotype are able to explot resource compoet j. Geeralzg Eq. A1 to a system of -competg cloal populatos s straghtforward ad takes the form x (t +1) = b s (φ (x,z))x (t). However, the cotext of terbreedg pheotypes, the model s complcated by the fact that we eed to accout for the frequecy of matg types ad hertablty of pheotype. Let q jk = Pr{pheotypes j ad k have a progey of type }. The, f we do ot dstgush betwee the order whch we lst males ad females, t follows that q jk = q kj. Now assume pamxs (.e. radom matg) ad a 1:1 sex rato. The females of pheotype j, of whch there are x /2, wll breed wth males of pheotype k proporto to ther represetato x k /x T (where x T = k = 1 x k ) to produce a total of bq x j x j jk 2 x k T of progey of type, where b j s the fecudty of females of pheotype j. Now, assumg the brth rates are depedet of the amout of resources extracted, but survval rates are ot, Eq. 1 geeralzes to where x t s q b b x t x t j() k() ( + 1) = ( φ( xz, )) jk( j + k ), (A5) 2x () t j = 1 k = 1 T sa ˆ s( φ)= 1 + ( K / ) φ γ ad a = 1 + ( K / δ) γ For the purposes of obtag some sghts to the how easly sexual ad asexual populatos are able to vade oe aother terms of a few key parameters relatg to the effcecy of explotato, (A6)

16 330 Getz ANN. ZOOL. FENNICI Vol. 38 the degree of overlap resource utlzato, ad the hertablty of pheotype the sexual populato, the followg reduced parameter case ca be vestgated. Frst, assume all pheotypes h have the same hertablty factor represeted by a parameter h [0,1] such that q = h+ 1, h 1 h qj = qj = + h for j ad q 2 = 1 jk for j ad k. (Note that these probabltes satsfy the requremet that q jk = = 1 1). Secod, assume all pheotypes the sexual populato are demographcally detcal other tha the fact that each s λ > 1 tmes more effcet explotg ts preferred resource compoet, as well as havg a preferece π > 1 for ts preferred resource compoet. I ths case, t follows that there exst a set of postve parameters c, b, s, δ, γ, a, ad d such that for = 1,,, we have c = c, b = b, s = s, δ = δ,, γ = γ (whch mples a = a), d = d, ad π 1 p =, whle for j, j = 1,,, we set d j = d/l, ad pj = 1 + π 1 + π. (Note π j = 1, as j = 1 requred). Usg the fact that h h h pjkxj() t xk() t = 1 h+ x x() t xk() t 2 j = 1 k = 1 k = 1 k h h + 1 x j() t xk() t = hx() t xt() t + 1 x j = 1 k = 1 j k models A5 ad A6 smplfy to yeld Eq. 3 the ma text. Smlarly, after defg z A4 smplfes to yeld Eq. 4. For the case of oterbreedg pheotypes s the parameters q jk Eq. A5 clearly are q = 1, q j = q j = 0 for j ad q jk = 0 for j ad k. Oce we take to accout that partheogeetc females produce females at twce the rate of sexually reproducg females, ad chage the ames of the varables from x to y to dcate we are dealg the asexual rather tha sexual pheotypes, Eq. A5 reduces to Eq. 2 the ma text. T () t 2 T = k = 1 z k, Eq.