### Introduction

- Top of page
- Summary
- Introduction
- A cowbird community: generalist–specialist–host
- Population dynamics model
- Impact of a generalist on a host–specialist community
- Invasion of a specialist into a host–generalist community
- Discussion
- Acknowledgements
- References
- Appendix

Brood parasites include generalists that use a wide range of hosts, and specialists that use very few. Specialists are rarer both in species number and abundance within a species (Rothstein & Robinson 1998; Davies 2000). Here we present a study of the population dynamics of a system of three species of South American cowbirds, two brood parasites, and a host for the former two.

The existence of systems with more than one parasite attacking the same host raises problems for both ecological and evolutionary stability. If a specialist and a generalist attack one host, it is of immediate interest to study whether one parasite may displace the other, or drive the host and hence the specialist to extinction. The ecological analysis may also throw light on the evolution path from generalism to specialization or vice versa.

Specialism at species level occurs in most African indigobirds (*Vidua* spp.) that parasitize different species of close relative grassfinches (Payne *et al*. 2000; Sorenson *et al*. 2003). Within a species, the common cuckoo *Cuculus canorus* is divided into host-specific races (gentes) that specialize in different hosts (Davies 2000). Females of each race lay a distinctive egg type that tends to match the host's egg. Gentes are restricted to female lineages, with cross-mating by males maintaining the common cuckoo genetically as one species (Gibbs *et al*. 2000). In the brown-headed cowbird *Molothrus alter* some females use more than one host species, both within and between breeding seasons, while other females parasitize specific host species regardless of their availability. This raises the possibility that brown-headed cowbird populations may consist of combinations of females some of whom are host generalists and others host specialists (Alderson, Gibbs & Sealy 1999; Woolfenden *et al*. 2003).

Brood parasitism poses a threat to some host populations (Robinson *et al*. 1995a,b; Trine, Robinson & Robinson 1998). Generalists are particularly threatening because their populations may be uncoupled from that of their relatively uncommon hosts. In contrast, specialists are less likely to drive hosts to extinction because their population dynamics are coupled to their host's (May & Robinson 1985; Takasu *et al*. 1993). The impact of generalist and specialist parasites depends on the characteristics of the system each forms with their hosts.

The topic is particularly timely because some brood parasites species have expanded their geographical range (Robinson *et al*. 1995a; Cruz *et al*. 1998) and are attacking new host populations and species. From a conservation point of view it is crucial to understand the dynamics of invasions and under which conditions a parasite persists in a new host or host community and/or leads to its decline and eventual extinction.

The population dynamics of host–brood parasite systems were studied in detail by May & Robinson (1985), who examined generalized and specialized parasitism in separate models. In their models, the consequences of parasitism for the host population are given in terms of the probability of parasitism and mortality rates. These authors represented the number of offspring fledged in parasitized and unparasitized nests by their average, regardless of the distribution of parasitic eggs per nest, and included minimum known biological properties of specific brood parasite systems. We introduce a degree of realistic complexity and develop new models including a generalized May and Robinson model that combines both parasitic species, using these extended models to test for the robustness of May and Robinson's predictions.

Our system is formed by the shiny cowbird *Molothrus bonariensis*, the screaming cowbird *M. rufoaxillaris*, and the nonparasitic bay-winged cowbird *Agelaiodes badius*. Note that *A. badius* has been treated as a species of *Molothrus* until recently (Lanyon 1992; Lanyon & Omland 1999). Shiny cowbirds parasitize at least 214 species (Ortega 1998), while screaming cowbirds parasitize almost exclusively the bay-winged cowbird, which is also parasitized by shiny cowbirds (Fraga 1998). This system has characteristics that can be expected to influence population dynamics. We pay attention to the following facts: (i) females lay at most one egg per day; (ii) when the total clutch (host plus parasite) exceeds a maximum, the host deserts (Fraga 1998); and (iii) parasites peck and puncture host eggs (Fraga 1998). These properties are not rare: they are present in most avian host–brood parasite systems. We study the importance of these properties by confronting our model to the model of May & Robinson (1985) that does not include them. We also study the persistence of a host species that is attacked by two parasitic species by comparing the behaviour of a three-species model with the host–specialist model.

### Population dynamics model

- Top of page
- Summary
- Introduction
- A cowbird community: generalist–specialist–host
- Population dynamics model
- Impact of a generalist on a host–specialist community
- Invasion of a specialist into a host–generalist community
- Discussion
- Acknowledgements
- References
- Appendix

We consider the density of the generalist constant (henceforth noted *G*). This is justified because the generalist attacks many species, and hence its density is not coupled to that of any specific host. We do not include density dependence on the host population in order to highlight the influences of the interactions with other species. This is further justified because the impact of parasitism appears to be extremely strong for this host (Fraga 1998), probably dominating other regulatory forces. The dynamics of the system are described by two equations, one for the number of host females (*N*_{t}), which equals the number of nests, and the other for the number of specialist females (*P*_{t}).

Cowbirds have a relatively discrete reproduction period, so that their dynamics are naturally described on a yearly basis. Therefore, the model uses discrete-time difference equations. It assumes that host females have one brood each year, and that host females as well as parasite females hatched in year *t* become fully mature in year *t* + 1. The host and parasite adult mortality rates (µ and ν, respectively) are age-independent. The survival rate of chicks (from hatching to first year) in unparasitized nests (*s*) is assumed to be independent of their own number. Similarly, the survival rates of host chicks in parasitized nests (*s*′) and of the specialist chicks (*s*_{p}) are independent of the nest mates. Survival for the host and the parasite during the first year is much lower than for subsequent years (see Table 1).

Table 1. Notation and some values of parameters that come into play in the model developed in this paper (Fraga 1998) Host parameters |

*s* | Survival from hatching to first year | 0·39 (unparasitized nests) |

*s*′ | | 0·34 (parasitized nests) |

*b* | Number of host eggs at the time of hatching | 3·75 (unparasitized nests) |

*b*′ | | 2 (parasitized nests) |

σ | Number of female hosts reared to reproductive | |

σ′ | adulthood σ = (1/2)*sb*, σ′ = (1/2)*s*′*b*′ | |

µ | Adult probability of dying during the year | 0·24 |

*D*_{A} | Lifetime of a nest available for parasitism | ≤ 16 days |

*D*_{S} | Duration of the breeding season | 120 days |

*x* | Fraction of nests in laying-period among available nests | ≤ 0·25 |

Specialist parasite parameters |

*T* | Number of eggs produced each year by one female parasite | Presumably in [20–50] |

*s*_{p} | Survival of parasite chicks from hatching to first year | 0·275 |

*p* | Number of parasitic eggs in a nest | 2·54 |

σ_{p} | Number of female parasites reared to reproductive adulthood | |

ν | Adult probability of dying during the year | 0·62–0·84 |

*t* | Egg-production time in days | 1 |

*P*_{a}(0) = *e*^{−λ} | Probability that a nest escapes parasitism | 0·2 |

*a* | Searching efficiency and *A* ≡ *D*_{A}/*D*_{S} | *a* is unknown |

*A* |

Generalist parasite parameters |

*Ḡ* | Population density | Unknown |

| Probability that a nest escapes parasitism | 0·76 |

*T*_{G} | Number of eggs laid each year in nests of the host under consideration | unknown |

*a*_{G} | Searching efficiency and *A*_{G} ≡ (*a*_{G}D_{A})/*D*_{S} | *a*_{G} is unknown |

*A*_{G} |

The model assumes that a parasitic event, also called a nest attack, always results in the laying of a parasitic egg in the nest, and may be accompanied by a puncture with a specified probability. Both parasitic species distribute attacks at random and independently of each other. The probability that a nest is attacked *r* times by each of the parasite species is given by a Poisson distribution with mean λ or λ_{G}, respectively, the mean number of attacks by specialist and generalist. λ = *N*_{a}*/N*_{t}, where *N*_{a} is the total number of specialist attacks.

Females cannot lay more than one egg in a day and *T* eggs per season. This can be taken into account by assuming that the number of attacks per female parasite follows a type II functional response (Hassell & May 1973) in which the traditional handling time term represents the time to mature an egg (*t*_{m})

- ( eqn 1 )

*D*_{S} is the duration of the breeding season (in days). Each nest is available for parasitism (in its laying and incubation period) for *D*_{A} days, so that there are in total *ND*_{A}/*D*_{S} nests available for parasitism each day. Henceforth *t*_{m} will be set either to 1 (day) or to 0 (representing a case in which there is no limit to egg production). Finally, *a* is the searching efficiency of the parasite.

An expression similar to eqn 1 is used for the generalist with *a*_{G} being the searching efficiency and *T*_{G} being the number of eggs produced by a generalist female laid in nests of the host under consideration. In the following, we use

- ( eqn 2 )

where *G* is the total number of eggs laid in host nests by the generalist population. The fact that the breeding season of host and parasite may not overlap completely (Table 1), can be taken into account within this parameter combination. *G* increases with the overlap of seasons.

Whenever the total number of eggs in a nest exceeds *MBS* (Maximum Brood Size), the nest is deserted, and deserting parents do not re-nest (see the section on the impact of the generalist).

At each attack, the parasite female punctures or removes a host egg with frequency *Pun*.

Table 1 lists symbols and their meaning. The dynamics of host and specialist in the presence of a generalist is described by the following equations

- ( eqn 3 )

The equation for the number of nests at year *t* + 1 (first equation in eqn 3) results from the number of nests at year *t*, subtracting the adult females that died during the year and adding yearling females. The latter are described by the last four terms in this equation that correspond to yearling females emerging from nests that have: (i) not been attacked; (ii) been attacked exclusively by the specialist; (iii) been attacked exclusively by the generalist; and (iv) been attacked by both parasites. Each of the four terms is the product of the sex ratio (1 : 2), the survival rates of the chicks, and the number of chicks hatching weighted by the probability that the nest be parasitized and not deserted. The joint probability that the nest is parasitized and not deserted is given by functions *g*(λ, λ_{G}), *g*_{G}(λ, λ_{G}), and *ĝ*(λ, λ_{G}), respectively, for the last three categories of nests, see (ii) to (iv) above. In fact *g*_{G}(λ, λ_{G}) = *g*(λ_{G}, λ) as generalist and specialist parasites are assumed to have identical parasitic behaviour.

The number of specialist females at year *t* + 1 is the sum of the number of adults that survive the year plus the female yearlings from nests attacked exclusively by the specialist and from nests attacked by both parasites (second equation in eqn 3). The functions *f*(λ, λ_{G}) and *f̂*(λ, λ_{G}) represent, respectively, the number of specialist eggs hatching in the latter two categories of nests weighted by the probability that the nest be parasitized and not deserted. Finally, *x* is the fraction of available nests in their laying period.

We now address in detail the four functions determining the joint probability that the nest is parasitized and not deserted. In a nest that received *r* attacks by the specialist and *q* by the generalist, the total number of eggs is the sum of *b* − (*r* + *q*) *Pun* host eggs plus *r* + *q* parasitic eggs. When all host eggs have been punctured, there remain *r* + *q* eggs in the nest. Whenever *MBS* is exceeded the brood is lost. The function *f* and *g* give the number of host and specialist chicks, respectively. As already stated, *r* and *q* are drawn from independent Poisson distributions. This yields

- ( eqn 4 )

where *U*(*x*) = 1 if *x* positive, 0 otherwise expresses the *MBS* condition, and Θ(*x*) = *x* if *x* positive, 0 otherwise expresses the puncture effect.

We now introduce a generalized version of the May and Robinson model where specialist and generalist host uses are combined into a single set of equations. These authors assumed that each parasitized nest gives rise to a fixed number of hosts (*b*′) and adult parasites (*p*′) in the next generation, independently of the number of attacks.

- ( eqn 5 )

Note that here neither the puncture effect nor nest desertion are included, but this new model differs from the *Pun*0, *MBS*∞ limit of eqn 4 in the fact that the number of chicks reared to maturity is now set to a constant. Substituting eqn 5, eqn 3 simplifies to:

- ( eqn 6 )

where σ = (1/2)*sb*, σ′ = (1/2)*s*′*b*′ and σ_{p} = (1/2)*s*_{p}p (see Table 1 for notations). May & Robinson (1985) studied this model in two particular cases, i.e. when one or the other parasite is not present in the system (note that in their notations, *sb* = γ, *s*_{p}*p* = Γ, *s*′*b*′ = γ′ and *x* = 1).

### Impact of a generalist on a host–specialist community

- Top of page
- Summary
- Introduction
- A cowbird community: generalist–specialist–host
- Population dynamics model
- Impact of a generalist on a host–specialist community
- Invasion of a specialist into a host–generalist community
- Discussion
- Acknowledgements
- References
- Appendix

Does the presence of a generalist parasite change the conditions of stability of the host–specialist system? To answer this question, the stability boundaries of the host–specialist system (eqn 3 with *G* = 0) were compared with those of the three-species system (*G* > 0), mainly numerically by iterating eqn 3. Stability criteria are computed in the Appendix. We present most of the model's results as stability boundaries in terms of µ and *s* (the host's survival to first year and mortality rate), which have a crucial influence on stability. Similar conclusions to those presented here could have been drawn from a plot in the (µ, *s*′)-plane. Stability plots for various µ- and *s*-values were generated keeping the other parameters constant. Steady states were usually rapidly reached (*t* ≈ 100). In some cases, the simulations resulted in stable limit cycles.

Figure 1 shows the stability boundaries for *T* = 50 (the largest realistic value for the specialist cowbird species), and *G* = 0 (i.e. in the absence of the generalist) or 20 eggs laid by the generalist in host nests (other parameter values as in Table 2). In the case with *G* = 0, we show the region of small-amplitude limit cycles defined arbitrarily as the region where both populations oscillate at a stable amplitude smaller than 100 individuals. This region can be combined with the stable equilibrium region when studying the persistence of a real biological system. Then the persistence of the three-species system occurs for µ broadly in [0·1; 0·45] and *s* in [0·34; 0·5]. These are realistic values for the cowbird system (Table 1). Further simulations have shown that the conditions for coexistence shrink if *MBS* or *Pun* increases or if *T* decreases.

Table 2. Parameter values used in numerical simulations unless otherwise specified *A = A*_{G} | 0·2 |

*s*_{p} | 0·27 |

*s*′ | 0·34 |

*b* | 4 |

*b*′ | 2 |

*p* | 2·54 |

*x* | 0·2 |

ν | 0·7 |

*Pun* | 0·6 |

*MBS* | 7 |

*T* | 50 |

*t*_{m} | 1 |

For comparison, Fig. 2 shows the stability boundaries for a simplified model excluding nest desertion, egg puncture and egg-maturation time (*MBS* = ∞, *Pun* = 0, and *t*_{m} = 0). This is in fact the generalized May & Robinson model (1985) of eqn 6. We ran the model with *G* = 0, 5, or 20. As *G* increases from 0 to 5 the stability boundary is displaced towards larger *s*. Increasing generalist parasitism further has a dramatic impact on coexistence compared with the previous model, reducing the stability region to a very small area. Under this model, stability is impossible for *G* > 20 eggs.

Parasitic cowbirds have a very high fecundity. Scott & Ankney (1980) estimated that the annual fecundity of brown-headed cowbirds was 40 eggs (but see Alderson *et al*. 1999; Woolfenden *et al*. 2003). Similarly, Kattan (1993) estimated that shiny cowbirds could lay up to 120 eggs during the breeding season, but his study was conducted in the tropics, where the breeding season is twice as long as in temperate regions. We assumed that values for the number of eggs laid by the specialist female (*T*) were in the range of 20–50 eggs. In our model (Fig. 1), apart from the largest values in this interval, the three-species system cannot persist (simulations give a lower limit of 39 eggs). Moreover, the stability is restricted to a rather limited range of µ- and *s*-values; the smaller *T*, the smaller this range.

Consider now a population of specialist parasites that lay *T* = 50 eggs per female each year and a population of generalist parasites that lay *G* = *T*_{G}*G* = 3·8 eggs each year. The latter is fixed at a low value that leads to a percentage of host nests parasitized by the generalist of 25%, which is what Fraga (1998) observed for cowbirds in natural conditions (Table 1). The most important result is that the stability diagram obtained by simulation differs little from Fig. 1 when *G* = 0. Furthermore, the resulting percentage of host nests parasitized by the specialist is 98·5% (as observed for the cowbird system by Fraga 1998). The simulation predicts that the fraction of over-crowded nests that are deserted is 0·03, and thus the number of successful nests per host female is very close to one. This suggests that further assumptions on the occurrence of second nest attempt would not have a large impact on stability.

We have shown in this section that the higher the generalist parasitism (*G*), the more restrictive the conditions of persistence of the community. Generalist parasitism soon reaches a level at which the host and the specialist are driven to extinction whatever their parameters. This occurs when the generalist lays a total of 36 eggs or more (when a single specialist female lays up to 50 eggs), according to our simulation, to be compared with 20 eggs when none of the three factors under consideration in this paper are included. The generalist parasite population persists in any case as it attacks other hosts. Note that there is no constraint on invasions of the host–specialist community by a generalist, because the generalist population density does not depend on the two others. The three-species system resulting from such an invasion persists or not depending on parameter values. In our model, each of the three factors, namely egg-maturation time, brood size limitation, and egg puncture, decreases the impact of generalist parasitism.

### Discussion

- Top of page
- Summary
- Introduction
- A cowbird community: generalist–specialist–host
- Population dynamics model
- Impact of a generalist on a host–specialist community
- Invasion of a specialist into a host–generalist community
- Discussion
- Acknowledgements
- References
- Appendix

We analysed the dynamics of a host–brood parasite community formed by a generalist and a specialist brood parasite with a host parasitized by both. We include realistic features of a known system that seem intuitively to be important for population dynamics. We are conscious that perhaps the most important contribution of theoretical modelling to biological thinking is extracting general principles from streamlined, simple models of great generality, and that adding complexity has a cost in terms of the heuristic contribution models make, but we trade this cost off against working with models that cannot make reliable predictions because crucial constraints of the real systems are missing.

We used as a benchmark a series of models developed by May & Robinson (1985), and built more complex models. The features of the real system that we added are: (i) parasite females produce no more than one egg per day; (ii) parasite females puncture (or remove) host eggs; and (iii) hosts desert overcrowded nests. While May and Robinson's model did not include nest abandonment and egg puncturing explicitly, their model was sensitive to these two parameters through their impact on survival data. In the present paper, other potentially significant properties were ignored. Among these, we assumed that the generalist population is entirely decoupled of the other two and did not implement a safe fraction of the host's nests in the temporal refuge resulting from nonoverlapping of reproductive seasons.

Our main findings were that (1) persistence is much restricted when the three features mentioned above are included in the model, the influence of the specialist's maximum rate of egg production and maximum seasonal fecundity being more important in constraining the conditions for persistence than the other two (M. Ney-Nifle unpublished). As these aspects had been ignored by pre-existent brood parasitism models, the dynamics for invasions and stability suggested by previous models do not give an accurate picture. It would be impossible to understand, and hence predict and intervene in ecological invasions such as those currently on course by shiny cowbirds in North America using models that exclude these crucial properties.

Comparing the three-species system with a two-species specialist–host model shows that (2) normally, the invading generalist parasite has a moderate impact on the coexistence of host and specialist, but at high density of the generalist, specialist and host can be driven to extinction. The analysis of the invasion of the host–generalist system by the specialist suggests that (3) invasion is successful only in the (restricted) stability boundaries of the three-species system. The dynamics of invasions were examined by comparing the three-species system with the two possible parasite–host systems. The results are intuitively obvious: Introduction of a generalist has only moderate effect on the persistence of a specialist–host system. When instead a specialist is introduced to a generalist–host system, the specialist often does not prosper, unless all the parameters are within the limited region where the three-species system is stable.

Interestingly, the range of parameter values where the coexistence of the three species is possible includes those that have been measured in field studies on cowbirds (Fraga 1998). The system is only stable in a small region. This suggests that the real system may be very vulnerable to perturbations: particularly any factor affecting host chick survivorship in unparasitized nests or host adult mortality could destabilize the system leading to local extinctions.

A factor that determines the persistence of systems, including the specialist, is the number of eggs that females can lay in available nests. Two parameters control this number: rate of egg production (1/*t*_{m}) and maximum seasonal fecundity (*T*). Together these parameters set the asymptote of the type II functional response. The destabilizing influence of type II functional responses in host–parasitoid systems is well known (Hassell & May 1973). Nevertheless, it is customary to assume that the asymptote is sufficiently high as to allow simplifying the models by assuming linear functional responses. The present study suggests that in brood parasitism, for realistic values of *T* and *t*_{m} the destabilizing influence of type II functional responses could be so important that ignoring it would lead to misleading results.

Our model includes some simplifications on the biology of the three cowbird species. First, in most of this work we ignored intraspecific density-dependent processes. Clearly, the inclusion of such factors would expand the stability boundaries of the model. Density dependence was ignored on the grounds that little information is available on intraspecific competition in cowbirds and that including it would obscure the influence of interspecific interactions. The fact that independently measured field estimations of parameter values fall inside the stability boundaries suggests that intraspecific competition is not a major feature of the three-species interaction. Secondly, we simplified the egg puncture process by assuming that, when visiting an already parasitized nest, parasite females puncture only host eggs. In fact, parasites are known sometimes to peck parasite eggs (Fraga 1998; Mermoz *et al*. 1999). The inclusion of punctures on parasite eggs would result in an increase in the number of host eggs and a decrease in the number of specialist and generalist eggs. However, most likely, this will have a moderate influence in the dynamics of the system because: (i) while host eggs are punctured in all parasitized nests, parasite eggs are only affected in multiply parasitized nests, and (ii) parasite eggs have a thicker eggshell than host eggs resulting in a lower probability of being destroyed (Mermoz *et al*. 1999). We also assumed that hosts deserting an over-crowded nest do not re-nest. In natural conditions this is not always the case, especially when nest desertion occurs early in the breeding season. This difference between the real system and the model assumptions is also likely to have a moderate impact. In our model, the predicted frequency with which nests were abandoned was quite low. On the other hand, our model did not consider nest predation. This over-estimation of nest success should counterbalance, at least partially, the under-estimation incurred by ignoring re-nesting. Finally, we assumed that all the females and males breed. This assumption could be invalid because it is known that this host has individuals that do not breed and help with parental duties to reproductive pairs (‘helpers at the nest’, Fraga 1991). There is little information about the sex and proportion of helpers in the host population (Fraga 1991). One of the favoured hypotheses for explaining helping behaviour is that helpers do not breed independently because there are no territories available (habitat saturation, Hatchwell & Komdeur 2000). If this is also the case in the host, our assumption holds for low population densities.

Cowbird species range from partial specialists to broad generalists. This has broad consequences for the establishment of two-species systems: while the stability characteristics of the host–generalist system will depend mostly on the characteristics of the generalist species, stable host–specialist systems are only possible for a restricted set of parameter values. Invasion of the host–generalist system by the specialist is only possible under the restricted conditions that lead to the stability of the three-species systems. Invasion by a generalist is always possible, but outside the three-species stability boundaries it will lead to the instability of the host–specialist system. This might have little impact on the generalist population while leading the other two species to extinction.

The model presented here can be applied to the study of a three-species community consisting of two generalist brood parasites sharing a host. This situation has recently arisen in North America, where generalist brown-headed cowbirds have started to compete for the use of hosts with the invading generalist shiny cowbird (Cruz *et al*. 1998). According to our model, the crucial parameter for coexistence is the sum of the number of eggs laid by each parasite species in the host nests that they share. The larger this number, the more likely the extinction of the host. Because the impact of the different generalist parasite species on host population is additive in the model, the arrival of a new generalist parasite will have a strong and immediate impact on host populations.