MURIEL NEY-NIFLE*, CARLOS BERNSTEIN*, JUAN C. REBOREDA and ALEX KACELNIK

Ecology 005 74, Populaion dynamics and avian brood parasiism: Blackwell Publishing, Ld. persisence and invasions in a hree-species sysem MURIEL NEY-NIFLE*, CARLOS BERNSTEIN*, JUAN C. REBOREDA and ALEX KACELNIK *Laboraoire Biomérie e Biologie Evoluive, Universié Claude Bernard, 696 Villeurbanne cedex, France; Deparameno de Ecolog a, Genéica y Evolución, Faculad de Ciencias Exacas y Naurales, Universidad de Buenos Aires, Pabellón II Ciudad Universiaria, C48EHA Buenos Aires, Argenina; and Deparmen of Zoology, Souh Parks Road, Oxford OX 3PS, UK Summary. Avian brood parasies include species ha are hos specialiss and ohers ha are generaliss. The impac of each kind of parasie on he persisence of he hos populaion is sudied by means of a populaion dynamics model.. Our model examines condiions for coexisence and invasions in a communiy of hree Souh American cowbirds, he shiny cowbird Molohrus bonariensis (a generalis parasie), he screaming cowbird M. rufoaxillaris (a specialis parasie), and he baywinged cowbird Agelaiodes badius (a nonparasie ha hoss he oher wo). 3. Three biologically realisic characerisics no previously included in brood parasiism models are explored and shown o be crucial for he sabiliy of he sysem. These characerisics are: (i) female parasies ake a leas a day o produce an egg and canno sore eggs for delayed laying his is modelled by means of a ype II funcional response; (ii) parasies ofen remove or puncure (desroy) hos eggs when visiing a nes; and (iii) hoss deser ness when he oal cluch (hos plus parasie) exceeds some hreshold. 4. These characerisics have a direc impac on parasie populaion renewal and reduce dramaically he sable coexisence condiions. 5. Comparing he sabiliy condiions of he hos specialis sysem wih hose of he hree-species sysem shows he impac of he arrival of a generalis parasie on he persisence of he hos specialis sysem when he hree characerisics are presen. The sabiliy boundaries are resriced when he generalis is absen and change lile for a realisic densiy of generalis. 6. The sudy of invasion by a specialis ino a hos generalis sable communiy shows ha he parameer region for coexisence and invasion coincide. 7. Comparison of our model agains is precursors using field daa for he parameers when available, shows ha he hree-species cowbird model sysem is sable for empirically realized parameer values, unlike a previous model by May & Robinson (985; American Nauralis, 6, 475 494) where none of he hree characerisics were included. Key-words: brood parasiism, cowbirds, populaion dynamics, hos specialism, Molohrus sp. Ecology (005) 74, doi: 0./j.365-656.004.009.x Ecological Sociey Inroducion Brood parasies include generaliss ha use a wide range of hoss, and specialiss ha use very few. Specialiss Correspondence: M. Ney-Nifle, Tel.: +33 (0)4 7 43 9 9; Fax: +33 (0)4 7 43 3 88; E-mail: ney@biomserv.universiy-lyon.fr are rarer boh in species number and abundance wihin a species (Rohsein & Robinson 998; Davies 000). Here we presen a sudy of he populaion dynamics of a sysem of hree species of Souh American cowbirds, wo brood parasies, and a hos for he former wo. The exisence of sysems wih more han one parasie aacking he same hos raises problems for boh ecological and evoluionary sabiliy. If a specialis and a

75 Specialis and generalis cowbirds generalis aack one hos, i is of immediae ineres o sudy wheher one parasie may displace he oher, or drive he hos and hence he specialis o exincion. The ecological analysis may also hrow ligh on he evoluion pah from generalism o specializaion or vice versa. Specialism a species level occurs in mos African indigobirds (Vidua spp.) ha parasiize differen species of close relaive grassfinches (Payne e al. 000; Sorenson e al. 003). Wihin a species, he common cuckoo Cuculus canorus is divided ino hos-specific races (genes) ha specialize in differen hoss (Davies 000). Females of each race lay a disincive egg ype ha ends o mach he hos s egg. Genes are resriced o female lineages, wih cross-maing by males mainaining he common cuckoo geneically as one species (Gibbs e al. 000). In he brown-headed cowbird Molohrus aler some females use more han one hos species, boh wihin and beween breeding seasons, while oher females parasiize specific hos species regardless of heir availabiliy. This raises he possibiliy ha brown-headed cowbird populaions may consis of combinaions of females some of whom are hos generaliss and ohers hos specialiss (Alderson, Gibbs & Sealy 999; Woolfenden e al. 003). Brood parasiism poses a hrea o some hos populaions (Robinson e al. 995a,b; Trine, Robinson & Robinson 998). Generaliss are paricularly hreaening because heir populaions may be uncoupled from ha of heir relaively uncommon hoss. In conras, specialiss are less likely o drive hoss o exincion because heir populaion dynamics are coupled o heir hos s (May & Robinson 985; Takasu e al. 993). The impac of generalis and specialis parasies depends on he characerisics of he sysem each forms wih heir hoss. Alhough hos brood parasie sysems have received less aenion han hos parasie or hos parasioid sysems, several auhors have examined he problem (May & Robinson 985; Takasu e al. 993; Haraguchi & Seno 995; Grzybowski & Pease 999; Woodworh 999). Work on sysems formed by a hos, a generalis, and a specialis parasioid has shown ha he condiions for he persisence of hese sysems are very resriced (Wilson, Hassell & Godfray 996). Wheher his applies for sysems formed by brood parasies and heir hoss needs ye o be explored. The opic is paricularly imely because some brood parasies species have expanded heir geographical range (Robinson e al. 995a; Cruz e al. 998) and are aacking new hos populaions and species. From a conservaion poin of view i is crucial o undersand he dynamics of invasions and under which condiions a parasie persiss in a new hos or hos communiy and/or leads o is decline and evenual exincion. The populaion dynamics of hos brood parasie sysems were sudied in deail by May & Robinson (985), who examined generalized and specialized parasiism in separae models. In heir models, he consequences of parasiism for he hos populaion are given in erms of he probabiliy of parasiism and moraliy raes. These auhors represened he number of offspring fledged in parasiized and unparasiized ness by heir average, regardless of he disribuion of parasiic eggs per nes, and included minimum known biological properies of specific brood parasie sysems. We inroduce a degree of realisic complexiy and develop new models including a generalized May and Robinson model ha combines boh parasiic species, using hese exended models o es for he robusness of May and Robinson s predicions. Our sysem is formed by he shiny cowbird Molohrus bonariensis, he screaming cowbird M. rufoaxillaris, and he nonparasiic bay-winged cowbird Agelaiodes badius. Noe ha A. badius has been reaed as a species of Molohrus unil recenly (Lanyon 99; Lanyon & Omland 999). Shiny cowbirds parasiize a leas 4 species (Orega 998), while screaming cowbirds parasiize almos exclusively he bay-winged cowbird, which is also parasiized by shiny cowbirds (Fraga 998). This sysem has characerisics ha can be expeced o influence populaion dynamics. We pay aenion o he following facs: (i) females lay a mos one egg per day; (ii) when he oal cluch (hos plus parasie) exceeds a maximum, he hos desers (Fraga 998); and (iii) parasies peck and puncure hos eggs (Fraga 998). These properies are no rare: hey are presen in mos avian hos brood parasie sysems. We sudy he imporance of hese properies by confroning our model o he model of May & Robinson (985) ha does no include hem. We also sudy he persisence of a hos species ha is aacked by wo parasiic species by comparing he behaviour of a hree-species model wih he hos specialis model. A cowbird communiy: generalis specialis hos The bay-winged cowbird (he hos) occupies regions of cenral and souhern Souh America. The screaming cowbird (he specialis) overlaps exensively wih he hos, while he shiny cowbird (he generalis) overlaps wih boh species everywhere (Fraga 998; Orega 998). Bay-winged cowbirds breed from lae Ocober o mid March (Fraga 998). Their breeding season overlaps oally wih ha of he specialis, bu only parially wih he generalis, which breeds from lae Sepember o lae January (Orega 998). Fraga (998) found ha he majoriy of bay-winged cowbird ness in his sudy area were parasiized by screaming cowbirds while less han 5% were parasiized by shiny cowbirds (see also Hoy & Oow 964; Mason 980), and ha more han 80% of he ness were muliply parasiized. There is no evidence for deparures from randomness in he disribuion of parasiic aacks. Birds ake a leas a day o produce an egg ha if no laid canno be sored. This reduces he maximum number of eggs laid. Toal egg laying also depends on he duraion of he breeding season and he maximum number of eggs a parasie female can produce in a season.

76 M. Ney-Nifle e al. Beyond a oal (hos plus parasie) cluch size of abou eigh eggs hoss abandon he nes (Hoy & Oow 964; Fraga 998). Boh screaming and shiny cowbirds puncure hos eggs when hey parasiize ness (Fraga 998; Massoni & Reboreda 998; Mermoz & Reboreda 999). Screaming cowbirds puncure on average 0 63 hos eggs (Fraga 998) and shiny cowbirds 0 65 hos eggs, per parasiic even (Massoni & Reboreda 998; Mermoz & Reboreda 999). In ness conaining eggs of wo or hree species, parasiic females peck eggs indiscriminaely. However, because parasiic eggs have a more rounded shape and a hicker eggshell han hos eggs (Spaw & Rower 987; Rahn, Curran-Evere & Booh 988; Mermoz & Ornelas 004) hey have a lower probabiliy of puncure (Mermoz e al. 999). We assume ha he number of parasie eggs los by puncures was negligible. Populaion dynamics model We consider he densiy of he generalis consan (henceforh noed G). This is jusified because he generalis aacks many species, and hence is densiy is no coupled o ha of any specific hos. We do no include densiy dependence on he hos populaion in order o highligh he influences of he ineracions wih oher species. This is furher jusified because he impac of parasiism appears o be exremely srong for his hos (Fraga 998), probably dominaing oher regulaory forces. The dynamics of he sysem are described by wo equaions, one for he number of hos females (N ), which equals he number of ness, and he oher for he number of specialis females (P ). Cowbirds have a relaively discree reproducion period, so ha heir dynamics are naurally described on a yearly basis. Therefore, he model uses discreeime difference equaions. I assumes ha hos females have one brood each year, and ha hos females as well as parasie females hached in year become fully maure in year +. The hos and parasie adul moraliy raes (µ and ν, respecively) are age-independen. The survival rae of chicks (from haching o firs year) in unparasiized ness (s) is assumed o be independen of heir own number. Similarly, he survival raes of hos chicks in parasiized ness (s ) and of he specialis chicks (s p ) are independen of he nes maes. Survival for he hos and he parasie during he firs year is much lower han for subsequen years (see Table ). The model assumes ha a parasiic even, also called a nes aack, always resuls in he laying of a parasiic egg in he nes, and may be accompanied by a puncure wih a specified probabiliy. Boh parasiic species disribue aacks a random and independenly of each oher. The probabiliy ha a nes is aacked r imes by each of he parasie species is given by a Poisson disribuion wih mean or G, respecively, he mean number of aacks by specialis and generalis. = N a / N, where N a is he oal number of specialis aacks. Females canno lay more han one egg in a day and T eggs per season. This can be aken ino accoun by assuming ha he number of aacks per female parasie follows a ype II funcional response (Hassell & Table. Noaion and some values of parameers ha come ino play in he model developed in his paper (Fraga 998) Hos parameers s Survival from haching o firs year 0 39 (unparasiized ness) s 0 34 (parasiized ness) b Number of hos eggs a he ime of haching 3 75 (unparasiized ness) b (parasiized ness) σ Number of female hoss reared o reproducive σ adulhood σ = (/)sb, σ = (/)s b µ Adul probabiliy of dying during he year 0 4 D A Lifeime of a nes available for parasiism 6 days D S Duraion of he breeding season 0 days x Fracion of ness in laying-period among available ness 0 5 Specialis parasie parameers T Number of eggs produced each year by one female parasie Presumably in [0 50] s p Survival of parasie chicks from haching o firs year 0 75 p Number of parasiic eggs in a nes 54 σ p Number of female parasies reared o reproducive adulhood ν Adul probabiliy of dying during he year 0 6 0 84 Egg-producion ime in days P a (0) = e Probabiliy ha a nes escapes parasiism 0 a Searching efficiency and A D A /D S a is unknown A Generalis parasie parameers G Populaion densiy Unknown e G Probabiliy ha a nes escapes parasiism 0 76 T G Number of eggs laid each year in ness of he hos under consideraion unknown a G Searching efficiency and A G (a G D A )/D S a G is unknown A G

77 Specialis and generalis cowbirds May 973) in which he radiional handling ime erm represens he ime o maure an egg ( m ) N P a ATN A ad = where + A N D m eqn D S is he duraion of he breeding season (in days). Each nes is available for parasiism (in is laying and incubaion period) for D A days, so ha here are in oal ND A /D S ness available for parasiism each day. Henceforh m will be se eiher o (day) or o 0 (represening a case in which here is no limi o egg producion). Finally, a is he searching efficiency of he parasie. An expression similar o eqn is used for he generalis wih a G being he searching efficiency and T G being he number of eggs produced by a generalis female laid in ness of he hos under consideraion. In he following, we use G T G G eqn where G is he oal number of eggs laid in hos ness by he generalis populaion. The fac ha he breeding season of hos and parasie may no overlap compleely ( Table ), can be aken ino accoun wihin his parameer combinaion. G increases wih he overlap of seasons. Whenever he oal number of eggs in a nes exceeds MBS (Maximum Brood Size), he nes is desered, and desering parens do no re-nes (see he secion on he impac of he generalis). A each aack, he parasie female puncures or removes a hos egg wih frequency Pun. Table liss symbols and heir meaning. The dynamics of hos and specialis in he presence of a generalis is described by he following equaions G N+ = ( µ + sbe e + s g(, G) + sg ( G, ) + s g(, G)) N eqn 3 P+ = ( ν) P + xsp( f(, G) + ƒ (, G)) N The equaion for he number of ness a year + (firs equaion in eqn 3) resuls from he number of ness a year, subracing he adul females ha died during he year and adding yearling females. The laer are described by he las four erms in his equaion ha correspond o yearling females emerging from ness ha have: (i) no been aacked; (ii) been aacked exclusively by he specialis; (iii) been aacked exclusively by he generalis; and (iv) been aacked by boh parasies. Each of he four erms is he produc of he sex raio ( : ), he survival raes of he chicks, and he number of chicks haching weighed by he probabiliy ha he nes be parasiized and no desered. The join probabiliy ha he nes is parasiized and no desered is given by funcions g(, G ), g G (, G ), and g(, G ), respecively, for he las hree caegories of ness, see (ii) o (iv) above. In fac g G (, G ) = g( G, ) as generalis and specialis parasies are assumed o have idenical parasiic behaviour. S A The number of specialis females a year + is he sum of he number of aduls ha survive he year plus he female yearlings from ness aacked exclusively by he specialis and from ness aacked by boh parasies (second equaion in eqn 3). The funcions f(, G ) and ƒ(, G ) represen, respecively, he number of specialis eggs haching in he laer wo caegories of ness weighed by he probabiliy ha he nes be parasiized and no desered. Finally, x is he fracion of available ness in heir laying period. We now address in deail he four funcions deermining he join probabiliy ha he nes is parasiized and no desered. In a nes ha received r aacks by he specialis and q by he generalis, he oal number of eggs is he sum of b (r + q) Pun hos eggs plus r + q parasiic eggs. When all hos eggs have been puncured, here remain r + q eggs in he nes. Whenever MBS is exceeded he brood is los. The funcion f and g give he number of hos and specialis chicks, respecively. As already saed, r and q are drawn from independen Poisson disribuions. This yields r G g(, G) = e e Θ[ b rpun] U[ MBS Θ[ b rpun] r] r r! G g(, G) = e e Θ[ b ( r + q) Pun] U [ MBS r q q r G Θ[ b ( r + q) Pun] r q] q! r! r G f(, G) = e e U[ MBS Θ[ b rpun] r] r ( r )! G ƒ (, G) = e e U[ MBS Θ[ b ( r + q) Pun] r q q r G r q] q!( r )! eqn 4 where U(x) = if x posiive, 0 oherwise expresses he MBS condiion, and Θ(x) = x if x posiive, 0 oherwise expresses he puncure effec. We now inroduce a generalized version of he May and Robinson model where specialis and generalis hos uses are combined ino a single se of equaions. These auhors assumed ha each parasiized nes gives rise o a fixed number of hoss (b ) and adul parasies (p ) in he nex generaion, independenly of he number of aacks. G gmr(, G ) = b ( e ) e G gmr(, G ) = b ( e )( e ) G fmr(, G ) = p( e ) e G ƒ MR(, G ) = p( e )( e ) eqn 5 Noe ha here neiher he puncure effec nor nes deserion are included, bu his new model differs from he Pun 0, MBS limi of eqn 4 in he fac ha he number of chicks reared o mauriy is now se o a consan. Subsiuing eqn 5, eqn 3 simplifies o: G N+ = ( µ + ( σ σ ) e e + σ ) N P+ = ( ν) P + xσp( e ) N eqn 6

78 M. Ney-Nifle e al. where σ = (/)sb, σ = (/)s b and σ p = (/)s p p (see Table for noaions). May & Robinson (985) sudied his model in wo paricular cases, i.e. when one or he oher parasie is no presen in he sysem (noe ha in heir noaions, sb = γ, s p p = Γ, s b = γ and x = ). Impac of a generalis on a hos specialis communiy Does he presence of a generalis parasie change he condiions of sabiliy of he hos specialis sysem? To answer his quesion, he sabiliy boundaries of he hos specialis sysem (eqn 3 wih G = 0) were compared wih hose of he hree-species sysem (G > 0), mainly numerically by ieraing eqn 3. Sabiliy crieria are compued in he Appendix. We presen mos of he model s resuls as sabiliy boundaries in erms of µ and s (he hos s survival o firs year and moraliy rae), which have a crucial influence on sabiliy. Similar conclusions o hose presened here could have been drawn from a plo in he (µ, s )-plane. Sabiliy plos for various µ- and s-values were generaed keeping he oher parameers consan. Seady saes were usually rapidly reached ( 00). In some cases, he simulaions resuled in sable limi cycles. Figure shows he sabiliy boundaries for T = 50 (he larges realisic value for he specialis cowbird species), and G = 0 (i.e. in he absence of he generalis) or 0 eggs laid by he generalis in hos ness (oher parameer values as in Table ). In he case wih G = 0, we show he region of small-ampliude limi cycles defined arbirarily as he region where boh populaions oscillae a a sable ampliude smaller han 00 individuals. This region can be combined wih he sable equilibrium region when sudying he persisence of a real biological sysem. Then he persisence of he hree-species sysem occurs for µ broadly in [0 ; 0 45] and s in [0 34; 0 5]. These are realisic values for he cowbird sysem (Table ). Furher simulaions have shown ha he condiions for coexisence shrink if MBS or Pun increases or if T decreases. For comparison, Fig. shows he sabiliy boundaries for a simplified model excluding nes deserion, egg puncure and egg-mauraion ime (MBS =, Pun = 0, and m = 0). This is in fac he generalized May & Robinson model (985) of eqn 6. We ran he model wih G = 0, 5, or 0. As G increases from 0 o 5 he sabiliy boundary is displaced owards larger s. Increasing generalis parasiism furher has a dramaic impac on coexisence compared wih he previous model, Table. Parameer values used in numerical simulaions unless oherwise specified A = A G 0 s p 0 7 s 0 34 b 4 b p 54 x 0 ν 0 7 Pun 0 6 MBS 7 T 50 m Fig.. Condiions for coexisence of hos and specialis parasie (hin line) and hos, specialis and generalis parasies (G = 0, hick line). This is he resul of numerical ieraion of eqn 3 wih parameer values of Table. The x-axis is he survival of hos chicks o mauriy in unparasiized ness. This canno be lower han he corresponding survival rae in parasiized ness, namely 0 34. There is only a small region of sabiliy delimied by he wo hin lines ha ges even smaller when generalis parasies come ino play (hick lines). A low µ or high s, boh hos and specialis parasie populaions grow exponenially. A high µ hey boh go exinc. Beween hese wo exremes, as moraliy increases and s says lower han 0 78, he sysem goes from he sabiliy region o a region of small and hen large sable limi cycles. This region is shown on he figure in case wih G = 0. Fig.. The May and Robinson model leads o a region of sable coexisence of hos and specialis parasie lying beween he lower horizonal line (µ = s = 0 34) and he hinner line. In presence of he generalis parasie he lower bound does no change, bu he upper bound moves righ and down, see he inermediae line G = 5. For G = 0 (hicker line) coexisence is hardly possible and he specialis and hos populaions are driven o exincion by he generalis parasie.

79 Specialis and generalis cowbirds reducing he sabiliy region o a very small area. Under his model, sabiliy is impossible for G > 0 eggs. Parasiic cowbirds have a very high fecundiy. Sco & Ankney (980) esimaed ha he annual fecundiy of brown-headed cowbirds was 40 eggs (bu see Alderson e al. 999; Woolfenden e al. 003). Similarly, Kaan (993) esimaed ha shiny cowbirds could lay up o 0 eggs during he breeding season, bu his sudy was conduced in he ropics, where he breeding season is wice as long as in emperae regions. We assumed ha values for he number of eggs laid by he specialis female (T) were in he range of 0 50 eggs. In our model (Fig. ), apar from he larges values in his inerval, he hree-species sysem canno persis (simulaions give a lower limi of 39 eggs). Moreover, he sabiliy is resriced o a raher limied range of µ- and s-values; he smaller T, he smaller his range. Consider now a populaion of specialis parasies ha lay T = 50 eggs per female each year and a populaion of generalis parasies ha lay G = T G G = 3 8 eggs each year. The laer is fixed a a low value ha leads o a percenage of hos ness parasiized by he generalis of 5%, which is wha Fraga (998) observed for cowbirds in naural condiions (Table ). The mos imporan resul is ha he sabiliy diagram obained by simulaion differs lile from Fig. when G = 0. Furhermore, he resuling percenage of hos ness parasiized by he specialis is 98 5% (as observed for he cowbird sysem by Fraga 998). The simulaion predics ha he fracion of over-crowded ness ha are desered is 0 03, and hus he number of successful ness per hos female is very close o one. This suggess ha furher assumpions on he occurrence of second nes aemp would no have a large impac on sabiliy. We have shown in his secion ha he higher he generalis parasiism (G), he more resricive he condiions of persisence of he communiy. Generalis parasiism soon reaches a level a which he hos and he specialis are driven o exincion whaever heir parameers. This occurs when he generalis lays a oal of 36 eggs or more (when a single specialis female lays up o 50 eggs), according o our simulaion, o be compared wih 0 eggs when none of he hree facors under consideraion in his paper are included. The generalis parasie populaion persiss in any case as i aacks oher hoss. Noe ha here is no consrain on invasions of he hos specialis communiy by a generalis, because he generalis populaion densiy does no depend on he wo ohers. The hree-species sysem resuling from such an invasion persiss or no depending on parameer values. In our model, each of he hree facors, namely eggmauraion ime, brood size limiaion, and egg puncure, decreases he impac of generalis parasiism. Invasion of a specialis ino a hos generalis communiy So far we gave prioriy o he specialis hos sysem and examined how a generalis alers he resuling condiions for sabiliy. Now we urn o he opposie quesion: given a wo-species sysem formed by a generalis and one of is hoss, can a specialis expanding is range from a region where i survives in sympary wih he hos join in and coexis wih he previous wo species? STABILITY OF THE HOST GENERALIST SYSTEM The criical level of generalis parasiism ha a hos populaion can wihsand before being driven o exincion can easily be compued. Recall ha in he absence of a specialis parasie he model is described by N+ = NF( N) r G G G FN ( ) = µ + sbe + s e Θ[ b rpun] r r! eqn 7 where F(N ) is he hos reproducive rae. Noe ha F(N ) depends on N hrough G. Below a criical level of generalis parasiism, he hos grows exponenially, and above his level i is driven o exincion. The criical level of generalis parasiism can be expressed eiher as he number of eggs a generalis female lays in G he hos ness, or as a probabiliy of aack pg = e. [p G ] criical and [G] criical can be ploed as funcions of adul hos moraliy µ (Fig. 3). I is ineresing o compare hese resuls wih he basic hos generalis model wih no egg puncure, and no limiaions in eiher egg-mauraion ime or brood size (May & Robinson 985). Figure 3 illusraes ha hos populaion persisence depends on generalis parasiism in boh cases bu i does so in a less sensiive way when more biological realism is included. INVASION CONDITIONS The goal of his secion is o analyse he oucome of an invasion of a hos generalis sysem by he specialis. This could be assessed by seing ha he hos populaion is a equilibrium in he presence of he generalis, and ha invading specialiss arrive in very low numbers. An invasion can be considered successful whenever he specialiss increase in numbers, i.e. heir growh rae is larger han uniy (Wilson e al. 996). In models where here is no preinvasion equilibrium hos densiy, invasion succeeds when Γ + lim ln Γ Γ P > 0 P = 0 eqn 8 In a consan environmen, i.e. in absence of sochasiciy, inequaliy 8 amouns o checking ha he long-erm growh rae Γ (ln P Γ ln P 0 ) akes a finie posiive value as ime, Γ, approaches infiniy (e.g. in Mez, Nisbe & Geriz 99). We firs sudy invasion wih neiher egg puncures nor nes deserions included (eqn A). This shows he impac of he biological process ha, according o he resuls in he previous secion, has he sronges

80 M. Ney-Nifle e al. Fig. 3. The criical parasiism level G and he criical probabiliy of parasiism, p G (inse) as a funcion of adul hos moraliy µ. Criical means ha above his hreshold he hos falls o exincion. Upper curves (hick lines) are obained numerically from our model, which is compared wih he May and Robinson model (lower hin lines calculaed analyically). For he laer, he quaniies ploed here are no defined when moraliy is lower han survival in parasiized ness (0 34), a limiaion ha does no exis in our model. If hos moraliy rae is, say, 0 4 he criical level of generalis parasiism is 8 eggs laid (and only five in May & Robinson (985); our model hence predics a relaively low impac of he generalis on he hos communiy. impac on he persisence of he sysem namely, eggmauraion ime (leing m = ). Linearizing he equaions for very small P, one ges P xσ ATN + p = ν + P + AN eqn 9 We have shown earlier ha in he absence of specialiss and above he criical level of generalis parasiism (inequaliy A4), hos densiy crashes, and ha below his level i grows wihou bounds. This unchecked growh depends on excluding densiy dependence of he hos populaion oher han he effec of specialis parasiism, ha is he variable we wish o explore. The expression in eqn 9 ends o ν when he hos densiy ends o zero and o ν + xσ p T when he hos densiy increases wihou bounds. Combining hese limis and inequaliy 8 shows ha he hos generalis sysem canno be invaded above he criical level of generalis parasiism, and ha invasion succeeds below his level if he reproducive opporuniies for he specialis, xσ p T are high enough (recall ha x is he fracion of ness available for reproducion, σ p. is he number of chicks reared o adulhood, and T is he number of eggs produced per female each year). The same endency according o parasie moraliy and reproducive rae is obained including he hree facors described earlier. This resul is obained by compuing condiion 9 numerically. Firs some hos densiy dependence is added so ha he hos populaion does no rise wihou bound if he specialis goes exinc. Assuming ha densiy-dependen moraliy acs a he end of he season a a rae δ, he equaion for he hos populaion (eqn 7) becomes Fig. 4. Condiions for invasion and sabiliy of communiies consising of indicaed combinaions of hos (H), specialis parasie (S), and generalis parasie (G). This is he resul of numerical ieraion of eqn 3 and 4 wih parameer values of Table. The doed line is he boundary above which he hos is driven o exincion by he generalis parasie. N + = N F(N )Exp( δn ) eqn 0 wih F(N ) in eqn 7. We use δ = 0 0 (i.e. very large carrying capaciy). The specialis is inoculaed a very low populaion level (e.g. P 0 = 0 8 ) o a hos populaion in equilibrium. The coupled equaions for hos and specialis parasies are ieraed for a mos 0 000 years. The densiy-dependen moraliy rae does no change he main conclusions on invasion (comparing δ = 0 and δ = 0 0 ) and holds for larger values of δ. The inroducion of he densiy-dependen moraliy, δ, has he rivial effec of sabilizing a hos populaion ha would oherwise grow exponenially, a he carrying capaciy. For hese reasons his facor has no been included sooner in he model. Numerical resuls show ha he hos generalis sysem can be invaded by he specialis and he hreespecies communiy is dynamically sable. Figure 4 shows he region of ineres (lower lef quarer of Fig. ) excep ha he boundary in he hos generalis sysem (beween hos expansion and hos exincion) is added. Similarly, our model implies ha he boundary depends lile on he generalis parasie efficiency (more eggs laid, G, or larger searching efficiency, a G ), in he region of parameer space of ineres. To conclude, he invasion region by he specialis parasie coincides wih he sabiliy region of hreespecies communiy. This region is small, which suggess ha he esablishmen of such a communiy as observed in he Molohrus cowbirds is a rare even. Discussion We analysed he dynamics of a hos brood parasie communiy formed by a generalis and a specialis brood parasie wih a hos parasiized by boh. We include realisic feaures of a known sysem ha seem inuiively o be imporan for populaion dynamics.

8 Specialis and generalis cowbirds We are conscious ha perhaps he mos imporan conribuion of heoreical modelling o biological hinking is exracing general principles from sreamlined, simple models of grea generaliy, and ha adding complexiy has a cos in erms of he heurisic conribuion models make, bu we rade his cos off agains working wih models ha canno make reliable predicions because crucial consrains of he real sysems are missing. We used as a benchmark a series of models developed by May & Robinson (985), and buil more complex models. The feaures of he real sysem ha we added are: (i) parasie females produce no more han one egg per day; (ii) parasie females puncure (or remove) hos eggs; and (iii) hoss deser overcrowded ness. While May and Robinson s model did no include nes abandonmen and egg puncuring explicily, heir model was sensiive o hese wo parameers hrough heir impac on survival daa. In he presen paper, oher poenially significan properies were ignored. Among hese, we assumed ha he generalis populaion is enirely decoupled of he oher wo and did no implemen a safe fracion of he hos s ness in he emporal refuge resuling from nonoverlapping of reproducive seasons. Our main findings were ha () persisence is much resriced when he hree feaures menioned above are included in he model, he influence of he specialis s maximum rae of egg producion and maximum seasonal fecundiy being more imporan in consraining he condiions for persisence han he oher wo (M. Ney-Nifle unpublished). As hese aspecs had been ignored by pre-exisen brood parasiism models, he dynamics for invasions and sabiliy suggesed by previous models do no give an accurae picure. I would be impossible o undersand, and hence predic and inervene in ecological invasions such as hose currenly on course by shiny cowbirds in Norh America using models ha exclude hese crucial properies. Comparing he hree-species sysem wih a wospecies specialis hos model shows ha () normally, he invading generalis parasie has a moderae impac on he coexisence of hos and specialis, bu a high densiy of he generalis, specialis and hos can be driven o exincion. The analysis of he invasion of he hos generalis sysem by he specialis suggess ha (3) invasion is successful only in he (resriced) sabiliy boundaries of he hree-species sysem. The dynamics of invasions were examined by comparing he hreespecies sysem wih he wo possible parasie hos sysems. The resuls are inuiively obvious: Inroducion of a generalis has only moderae effec on he persisence of a specialis hos sysem. When insead a specialis is inroduced o a generalis hos sysem, he specialis ofen does no prosper, unless all he parameers are wihin he limied region where he hree-species sysem is sable. Ineresingly, he range of parameer values where he coexisence of he hree species is possible includes hose ha have been measured in field sudies on cowbirds (Fraga 998). The sysem is only sable in a small region. This suggess ha he real sysem may be very vulnerable o perurbaions: paricularly any facor affecing hos chick survivorship in unparasiized ness or hos adul moraliy could desabilize he sysem leading o local exincions. A facor ha deermines he persisence of sysems, including he specialis, is he number of eggs ha females can lay in available ness. Two parameers conrol his number: rae of egg producion (/ m ) and maximum seasonal fecundiy (T). Togeher hese parameers se he asympoe of he ype II funcional response. The desabilizing influence of ype II funcional responses in hos parasioid sysems is well known (Hassell & May 973). Neverheless, i is cusomary o assume ha he asympoe is sufficienly high as o allow simplifying he models by assuming linear funcional responses. The presen sudy suggess ha in brood parasiism, for realisic values of T and m he desabilizing influence of ype II funcional responses could be so imporan ha ignoring i would lead o misleading resuls. Our model includes some simplificaions on he biology of he hree cowbird species. Firs, in mos of his work we ignored inraspecific densiy-dependen processes. Clearly, he inclusion of such facors would expand he sabiliy boundaries of he model. Densiy dependence was ignored on he grounds ha lile informaion is available on inraspecific compeiion in cowbirds and ha including i would obscure he influence of inerspecific ineracions. The fac ha independenly measured field esimaions of parameer values fall inside he sabiliy boundaries suggess ha inraspecific compeiion is no a major feaure of he hree-species ineracion. Secondly, we simplified he egg puncure process by assuming ha, when visiing an already parasiized nes, parasie females puncure only hos eggs. In fac, parasies are known someimes o peck parasie eggs (Fraga 998; Mermoz e al. 999). The inclusion of puncures on parasie eggs would resul in an increase in he number of hos eggs and a decrease in he number of specialis and generalis eggs. However, mos likely, his will have a moderae influence in he dynamics of he sysem because: (i) while hos eggs are puncured in all parasiized ness, parasie eggs are only affeced in muliply parasiized ness, and (ii) parasie eggs have a hicker eggshell han hos eggs resuling in a lower probabiliy of being desroyed (Mermoz e al. 999). We also assumed ha hoss desering an over-crowded nes do no re-nes. In naural condiions his is no always he case, especially when nes deserion occurs early in he breeding season. This difference beween he real sysem and he model assumpions is also likely o have a moderae impac. In our model, he prediced frequency wih which ness were abandoned was quie low. On he oher hand, our model did no consider nes predaion. This over-esimaion of nes success should counerbalance, a leas parially,

8 M. Ney-Nifle e al. he under-esimaion incurred by ignoring re-nesing. Finally, we assumed ha all he females and males breed. This assumpion could be invalid because i is known ha his hos has individuals ha do no breed and help wih parenal duies o reproducive pairs ( helpers a he nes, Fraga 99). There is lile informaion abou he sex and proporion of helpers in he hos populaion (Fraga 99). One of he favoured hypoheses for explaining helping behaviour is ha helpers do no breed independenly because here are no erriories available (habia sauraion, Hachwell & Komdeur 000). If his is also he case in he hos, our assumpion holds for low populaion densiies. Cowbird species range from parial specialiss o broad generaliss. This has broad consequences for he esablishmen of wo-species sysems: while he sabiliy characerisics of he hos generalis sysem will depend mosly on he characerisics of he generalis species, sable hos specialis sysems are only possible for a resriced se of parameer values. Invasion of he hos generalis sysem by he specialis is only possible under he resriced condiions ha lead o he sabiliy of he hree-species sysems. Invasion by a generalis is always possible, bu ouside he hree-species sabiliy boundaries i will lead o he insabiliy of he hos specialis sysem. This migh have lile impac on he generalis populaion while leading he oher wo species o exincion. The model presened here can be applied o he sudy of a hree-species communiy consising of wo generalis brood parasies sharing a hos. This siuaion has recenly arisen in Norh America, where generalis brown-headed cowbirds have sared o compee for he use of hoss wih he invading generalis shiny cowbird (Cruz e al. 998). According o our model, he crucial parameer for coexisence is he sum of he number of eggs laid by each parasie species in he hos ness ha hey share. The larger his number, he more likely he exincion of he hos. Because he impac of he differen generalis parasie species on hos populaion is addiive in he model, he arrival of a new generalis parasie will have a srong and immediae impac on hos populaions. Acknowledgemens We hank Miguel Rodríguez Gironés, Sco Robinson, Emmanuel Desouhan, and Isabelle Ama for helpful commens on a previous version of his manuscrip. JCR is Research Fellow of he Consejo Nacional de Invesigaciones Cieníficas y Técnicas (CONICET). This work was suppored by he Cenre Naional de la Recherche Scienifique (MNN and CB), he Universiy of Buenos Aires (gran TW88 o JCR), CONICET (gran PID 0798/98 o JCR), and Agencia Nacional de Promoción Cienífica y Tecnológica (gran 0-0937 o JCR). AK was parly suppored by a Fellowship of he Insiue for Advanced Sudies in Berlin. References Alderson, G.W., Gibbs, H.L. & Sealy, S.G. (999) Deermining he reproducive behaviour of individual brown-headed cowbirds using microsaellie DNA markers. Animal Behaviour, 58, 895 905. Cruz, A., Pos, W., Wiley, J.W., Orega, C.P., Nakamura, T.K. & Praher, J.W. (998) Poenial impacs of cowbird range expansion in Florida. Parasiic Birds and Their Hoss: Sudies in Coevoluion (eds S.I. Rohsein & S.K. Robinson), pp. 33 336. Oxford Universiy Press, New York. Davies, N.B. (000) Cuckoos, Cowbirds and Oher Cheas. Oxford Universiy Press, Oxford. Edelsein-Keshe, L. (988) Mahemaical Models in Biology. McGraw-Hill, Boson. Fraga, R.M. (99) The social sysem of a communal breeder, he bay-winged cowbird Molohrus badius. Ehology, 89, 95 0. Fraga, R.M. (998) Ineracions of he parasiic screaming and shiny cowbirds (Molohrus rufoaxillaris and M. bonariensis) wih a shared hos, he bay-winged cowbird (Molohrus badius). Parasiic Birds and Their Hoss: Sudies in Coevoluion (eds S.I. Rohsein & S.K. Robinson), pp. 73 93. Oxford Universiy Press, New York. Gibbs, H.L., Sorenson, M.D., Marchei, K., de Brooke, M.L., Davies, N.B. & Nakamura, H. (000) Geneic evidence for female hos-specific races of he common cuckoo. Naure, 407, 83 86. Grzybowski, J.A. & Pease, C.M. (999) A model of he dynamics of cowbirds and heir hos communiies. Auk, 6, 09. Haraguchi, Y. & Seno, H. (995) Mahemaical model of he populaion dynamics of brood parasiism: Wha is he benefi for he hos? Journal of Theoreical Biology, 74, 8 97. Hassell, M.P. & May, R.M. (973) Sabiliy in insec hosparasie models. Ecology, 4, 693 76. Hachwell, B.J. & Komdeur, J. (000) Ecological consrains, life hisory rais and he evoluion of cooperaive breeding. Animal Behaviour, 59, 079 086. Hoy, G. & Oow. J. (964) Biological and zoological sudies of he molohrine cowbirds (Iceridae) of Argenina. Auk, 8, 86 03. Kaan, G.H. (993) Reproducive sraegy of a generalis brood parasie, he shiny cowbird, in he Cauca Valley, Colombia. PhD Disseraion, Universiy of Florida, Gainesville, USA. Lanyon, S.M. (99) Parasiism in Blackbirds Icerinae a phylogeneic perspecive. Science, 55, 77 79. Lanyon, S.M. & Omland, K.E. (999) A molecular phylogeny of he blackbirds (Icerinae): five lineage revealed by cyochrome-b sequence daa. Auk, 6, 69 639. Mason, P. (980) Ecological and evoluionary aspecs of hos selecion in cowbirds. PhD Disseraion. Universiy of Texas, Ausin, USA. Massoni, V. & Reboreda, J.C. (998) Coss of brood parasiism and he lack of defenses on he yellow-winged blackbird-shiny cowbird sysem. Behavioral Ecology and Sociobiology, 4, 73 80. May, R.M. & Robinson, S.K. (985) Populaion dynamics of avian brood parasiism. American Nauralis, 6, 475 494. Mermoz, M.E. & Ornelas, J.F. (004) Phylogeneic analysis of life-hisory adapaions in he parasiic cowbirds. Behavioral Ecology, 5, 09 9. Mermoz, M.E. & Reboreda, J.C. (999) Egg-laying behaviour by shiny cowbirds parasiizing brown-and-yellow marshbirds. Animal Behaviour, 58, 873 88. Mez, J.A.J., Nisbe, R.M. & Geriz, S.A.H. (99) How should we define finess for general ecological scenarios? Trends in Ecology and Evoluion, 7, 98 0.

83 Specialis and generalis cowbirds Orega, C. (998) Cowbirds and Oher Brood Parasies. Universiy of Arizona Press, Tucson, AZ. Payne, R.B., Payne, L.L., Woods, J.L. & Sorenson, M.D. (000) Imprining and he origin of parasie hos species associaions in brood-parasiic indigobirds, Vidua chalybeaa. Animal Behaviour, 59, 69 8. Rahn, H., Curran-Evere, L. & Booh, D.T. (988) Eggshell differences beween parasiic and nonparasiic Iceridae. Condor, 90, 96 964. Robinson, S.K., Rohsein, S.I., Briingham, M.C., Pei, L.J. & Grzybowski, J.A. (995a) Ecology and behaviour of cowbirds and heir impac on hos populaions. Ecology and Managemen of Neoropical Migraory Birds (eds T.E. Marin & D.M. Finch), pp. 48 460. Oxford Universiy Press, New York. Robinson, S.K., Thomson,F.R., IIIDonovan, T.M., Whiehead, D.R. & Faaborg, J. (995b) Regional fores fragmenaion and he nesing success of migraory birds. Science, 67, 987 990. Rohsein, S.I. & Robinson, S.K. (998) The evoluion and ecology of avian brood parasiism. Parasiic Birds and Their Hoss: Sudies in Coevoluion (eds S.I. Rohsein & S.K. Robinson), pp. 3 56. Oxford Universiy Press, New York. Sco, D.M. & Ankney, C.D. (980) Fecundiy of he brownheaded cowbird Molohrus-aer. Souhern Onario Canada. Auk, 97, 677 683. Sorenson, M.D., Sefc, K.M. & Payne, R.B. (003) Speciaion by hos swich in brood parasiic indigobirds. Naure, 44, 98 93. Spaw, C.D. & Rohwer, S. (987) A comparaive sudy of eggshell hickness in cowbirds and oher passerines. Condor, 89, 307 38. Takasu, F., Kawasaki, K., Nakamura, H., Cohen, J.E. & Shigesada, N. (993) Modelling he populaion-dynamics of a cuckoo hos associaion and he evoluion of hos defences. American Nauralis, 4, 89 839. Trine, Ch.L., Robinson, W.D. & Robinson, S.K. (998) Consequences of brown-headed cowbird parasiism for hos populaion dynamics. Parasiic Birds and Their Hoss: Sudies in Coevoluion (eds S.I. Rohsein & S.K. Robinson), pp. 73 95. Oxford Universiy Press, New York. Wilson, H.B., Hassell, M.P. & Godfray, H.C.J. (996) Hosparasioid food web: dynamics, persisence, and invasion. American Nauralis, 48, 787 806. Wolfram, S. (999) The Mahemaica Book, Version 4. Cambridge Universiy Press, Cambridge. Woodworh, B.L. (999) Modelling populaion dynamics of a songbird exposed o parasiism and predaion and evaluaing managemen opions. Conservaion Biology, 3, 67 76. Woolfenden, B.E., Gibbs, H.L., Sealy, S.G. & McMaser, D.G. (003) Hos use and fecundiy of individual female brownheaded cowbirds. Animal Behaviour, 66, 95 06. Received 30 Sepember 003; acceped 30 July 004 Appendix The firs secion of his Appendix deals wih wospecies (specialis and hos) communiies and he second wih hree species (specialis, generalis, and hos) communiies. HOST SPECIALIST MODELS Throughou his sudy we use he wo-species model proposed by May & Robinson (985; our eqn 6) as a reference, and sudy he effecs of adding hree differen realisic rais ha are known o apply in real communiies. The wo-species sysem is obained by seing he influence of he generalis as nil (G = 0) in eqn 3. The resuling simplificaion is shown below. N+ = ( µ + sbe + s g( ) N P+ = ( ν) P + x spf( ) N eqn A (see noaion of Table ) where funcions g and f (eqn 4) are r g( ) = e ( b rpun) r r! r f( ) = e r ( r )! eqn A Firs we proceed o he sabiliy analysis in wo seps by geing he equilibrium poins and hen he condiions for sabiliy agains small (linear) perurbaions. In parallel o hese analyical derivaions, we also performed exensive numerical sudies and verified ha we ge he same sabiliy diagram. Calculaions were done wih he sofware Mahemaica (Wolfram 999). Sabiliy when parasies canno lay more han an egg a day Seing {N* = N + = N ;P* = P + = P } in eqn A wih, gmr( ) = b ( e ) fmr( ) = p( e ) where ATP = + A N one ges and m σ σ * = Ln µ σ ATP* = *( + N*) ν* N* = * ATxσp( e ) νa* eqn A3 eqn A3 eqn A4 eqn A5 I mus be noed ha he necessary condiion 0 < e <, imposes σ < µ < σ eqn A6 where we used he fac ha he number of hos chicks reared o mauriy mus be lower in an parasiized han in a unparasiized nes, i.e. σ < σ. The inequaliies in eqn A6 are necessary a equilibrium because if he

84 M. Ney-Nifle e al. righ-hand side of (eqn A6) is violaed, he hos s reproducive rae in unparasiized ness is lower han he moraliy rae, and he populaion collapses even in he absence of parasiism, while if he lef side of (eqn A6) is no verified moraliy rae is lower han he reproducive rae in parasiized ness, and he hos populaion grows wihou bounds. Following sandard echniques (Edelsein-Keshe 988), we now compue he four elemens of he jacobian marix, J, afer linearizaion of eqn A. Sabiliy condiions are given by > De(J) > Trace(J), which yields µ σ ( + ) + ( )( ) 0 > σ µ σ µ m ν σ σ Txσp ( ) ( ) > 0 eqn A7 Txσp σ µ mν σ σ When m = 0 in eqn A7, one recovers he May and Robinson model (985), for which he second inequaliy is rivially saisfied. The only parameers involved in he remaining sabiliy condiion are µ, σ, and σ. In he general case, in which he unrealisic abiliy of generaing eggs insananeously is excluded and hen m > 0, here are addiional parameers involved, alhough he parasie searching efficiency A is noably absen. eqn A7 show ha here is a criical value of he parasie s oal fecundiy, T, below which none of he inequaliies are verified and hence equilibrium is no possible. The sabiliy froniers in he (µ, σ)-plane are obained from he inequaliies in eqn A6 and A7, he laer being solved numerically when no analyical expressions are available. Sabiliy when brood size is limied or when parasies puncure hos eggs In his case he equilibrium sae P* is he soluion of he firs equaion in A (afer N* is eliminaed), ha is, µ = σe ATP + σ g(atp) eqn A8 which can be solved numerically. The sabiliy condiions resuling from he inequaliies > De(J) > Trace(J) lead o he resuls given in he ex. A HOST SPECIALIST GENERALIST MODEL For general forms of he funcions f and g (eqn 4) here are no analyical soluions a equilibrium. Therefore he wo coupled equaions have o be ieraed numerically. The sabiliy boundaries, however, can be inferred analyically in a paricular case ha is used in he secion on invasion of he hos generalis communiy by a specialis parasie. The sabiliy of he hree-species sysem can also be explored aking ino accoun he consrain in egg-producion ime bu leaving ou puncure and limied brood size facors. In his case, he sysem simplifies o G G N+ = ( µ + σe e + σ ( e e )) N P+ = ( ν) P + xσp( e ) N eqn A9 where and G are funcions of N and P (see eqn A4) and ATG = + A N G G G G m eqn A0 Leing N = N* and P = P* in eqn A9 in order o derive he equilibrium poins, leads o eqn A8 (obained in absence of generalis parasiism) wih e µ σ σ σ * = e G * eqn A The second equaion in A9 can be solved in N* numerically. The firs equaion in A9 is solved in P* by sraighforward subsiuion of he soluion for N*. The sabiliy condiions (see above) are hen checked numerically.