renesting determines seasonal fecundity in songbirds - BioOne

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to renest a er a nest failure or a er successfully fledging a brood. Recent analyses have o en assumed set maximum numbers of nesting a empts and successful.
The Auk 122(1):280–291, 2005 © The American Ornithologists’ Union, 2005. Printed in USA.

RENESTING DETERMINES SEASONAL FECUNDITY IN SONGBIRDS: WHAT DO WE KNOW? WHAT SHOULD WE ASSUME? J  A. G1,3  C M. P  2 1

College of Mathematics and Science, University of Central Oklahoma, Edmond, Oklahoma 73034, USA; and 2 Environmental Law Center, Vermont Law School, South Royalton, Vermont 05068, USA

A.—Because of the difficulty of following female songbirds through an entire breeding season, field ornithologists are seldom able to directly measure seasonal fecundity (defined as number of offspring produced per female during an entire breeding season). Instead, it is more commonly inferred from some measure of nest-productivity data (e.g. average number of offspring fledged per nesting a empt) using algorithms that make assumptions about the propensity of females to renest a er a nest failure or a er successfully fledging a brood. Recent analyses have o en assumed set maximum numbers of nesting a empts and successful broods, and that all females breed up to those maxima. However, whereas data from songbirds intensively followed for an entire breeding season show that they are capable of up to 4–8 nesting a empts, many authors, in estimating seasonal fecundity, assume a maximum of only 1–4 nesting a empts. We applied a model to a Prairie Warbler (Dendroica discolor) data set (Nolan 1978) that allowed direct comparisons of (1) seasonal-fecundity estimates obtained assuming fixed maximum numbers of renestings and broods with (2) estimates obtained assuming that numbers of renesting a empts and successful nests are constrained only indirectly by length of breeding season. Although results under the la er assumption are concordant with Nolan’s (1978) direct empirical measure of Prairie Warbler seasonal fecundity, estimates under assumptions of fixed maxima of renestings or broods are in serious error for many parameter choices. As such, our analyses disclose that essentially all estimates of seasonal fecundity in the literature derived by assuming a limited maximum number of nesting a empts or of successful broods are biased. Most commonly, when nest mortality is high, seasonal fecundity is underestimated; in some cases where nest mortality is low, seasonal fecundity is overestimated. We recommend that researchers estimating seasonal fecundity from nest-productivity data use a model that explicitly sets breeding-season length and thereby only indirectly constrains the possible number of nesting a empts and successful broods. Received 5 September 2003, accepted 1 October 2004. Key words: fecundity models, nest success, renesting, seasonal fecundity, songbirds.

La Nidificación Repetida Determina la Fecundidad Estacional en las Aves Canoras: ¿Qué Sabemos? ¿Qué Deberíamos Suponer? R  .—Debido a la dificultad de seguir aves canoras hembras a lo largo de una estación reproductiva completa, pocas veces los ornitólogos de campo están en condiciones de medir directamente la fecundidad estacional (definida como el número de crías producidas por hembra durante una estación reproductiva completa). En cambio, la fecundidad estacional es más comúnmente inferida a

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partir de alguna medida de productividad del nido (e.g. el número promedio de crías que llegaron a dejar el nido por evento de nidificación) usando algoritmos que hacen suposiciones en cuanto a la disponibilidad de las hembras a nidificar nuevamente luego de que un nido fracasa o de que crían exitosamente una nidada. Análisis recientes han supuesto generalmente que existe un número fijo máximo de intentos de nidificación y de nidadas exitosas y que todas las hembras se reproducen para alcanzar dichos máximos. Sin embargo, mientras que los datos provenientes de aves canoras que han sido seguidas intensamente a lo largo de una estación reproductiva completa muestran que son capaces de emprender hasta 4 a 8 intentos de nidificación, al estimar la fecundidad estacional muchos autores suponen un máximo de sólo 1 a 4 intentos de nidificación. Aplicamos un modelo a un conjunto de datos de Dendroica discolor (Nolan 1978) que nos permitió hacer comparaciones directas de (1) las estimaciones de fecundidad estacional obtenidas suponiendo números máximos fijos de nidificaciones repetidas y nidadas con (2) las estimaciones obtenidas suponiendo que los números de intentos de nidificaciones repetidas y de nidos exitosos están limitados sólo indirectamente por la duración de la estación reproductiva. Aunque los resultados basados en este último supuesto concuerdan con la medida empírica directa de la fecundidad estacional de D. discolor de Nolan (1978), las estimaciones basadas en los supuestos de un máximo fijo de nidificaciones repetidas o de nidadas presentan errores serios en la selección de múltiples parámetros. De este modo, nuestros análisis revelan esencialmente que todas las estimaciones de fecundidad estacional presentes en la literatura que han supuesto un número máximo limitado de intentos de nidificación o de nidadas exitosas están sesgadas. Más comúnmente, cuando la mortalidad de los nidos es alta, se subestima la fecundidad estacional; en algunos casos en los que la mortalidad de los nidos es baja, se sobrestima la fecundidad estacional. Recomendamos que los investigadores que estiman la fecundidad estacional a partir de datos de productividad de nidos usen un modelo que determine explícitamente la duración de la estación reproductiva y por ende que limite sólo de modo indirecto el número posible de intentos de nidificación y de nidadas exitosas.

S      songbirds is measured as the number of offspring produced by a female in an entire breeding season. It is a key parameter in assessing fitness and population viability (Perrins et al. 1991). However, few field ornithologists measure seasonal fecundity directly, because following and recording all the breeding activities of a cohort of females through a breeding season is generally difficult. More frequently, seasonal fecundity is inferred from some measure of nest productivity (such as average number of offspring fledged per nesting a empt). Such algorithms necessarily make assumptions about renesting a er nest failure or successful fledging. In recent years, we have seen a series of methods for estimating seasonal fecundity from nest-productivity data (vide infra), each with its own inherent assumptions about renesting. One straightforward method is to estimate seasonal fecundity as the product of some measure of

average female productivity (clutch size or offspring per nesting a empt) and average number of nesting a empts (e.g. Ricklefs and Bloom 1977, Farnsworth and Simons 2001, Fauth 2001). That approach is intuitive, but it requires complete empirical data on nesting a empts—data that are frequently not available. It does not require explicit assumptions about renesting, except that the level of renesting observed is the actual level. When an assumption about renesting is required, the simplest approach is to assume that females nest only once in a season and do not renest following nest failure; in this case, nest productivity would equal seasonal fecundity (Ward and Smith 2000). The obvious flaw in that approach was highlighted by May and Robinson (1985) when they applied their model calculating a parasitism threshold, using that assumption, to nest-success statistics for Song Sparrows (scientific names in Tables 1 and 2).

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They found that Song Sparrows, if they never renested, would be unable to maintain stable populations even with no brood parasitism, and concluded that female sparrows “must nest more than once [per season] to replace themselves” (May and Robinson 1985). With incomplete sampling of nesting effort, some authors have assumed that songbirds can nest, at most, only one, two, three, or four times in each breeding season, and that each female can have, at most, one or two successful broods in a year (e.g. Donovan et al. 1995, Schmidt and Whelan 1999). Such analyses also implicitly assume that all females continue breeding until one or the other of those two limits is met. However, those assumptions are inconsistent with observations of renesting in the most thoroughly studied songbirds (e.g. Nolan 1978, Payne 1989, Kershner et al. 2001), in which the maximum empirically observed number of nesting a empts per female varies between four and eight. Some authors justify their simplifying assumptions by arguing that more extended renesting would involve only a very small number of ultimately successful broods and would thus have trivial effects on estimates of seasonal fecundity or population growth rates (McCoy et al 1999, Schmidt and Whalen 1999). However, Duguay et al.’s (2001) analysis demonstrates the potential for error with that approach; they arrived at a seasonal-fecundity estimate of 2.74 by assuming that the average number of Wood Thrush nesting a empts per female per season, as obtained from Fauth’s (2000) data, was equal to the maximum possible. That estimate is substantially less than the 3.19 young per female one obtains using the more direct, intuitive approach of Fauth (2001)—that is, essentially multiplying the average number of nesting a empts by the average number of young per nesting a empt. There has not been, heretofore, a systematic investigation of the potential biases inherent in renesting assumptions for inferring seasonal fecundity. As an alternative, our previous model (Pease and Grzybowski 1995) assumes that the number of nesting a empts and successful broods is limited by the amount of time available in the breeding season and by how that time is spent on successful and unsuccessful nesting a empts (Grzybowski and Pease 2000). We have previously tested that model on several data sets, and the approach has been applied by others (vide infra). For some parameter values

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(e.g. high levels of nest predation and brood parasitism), it can allow the level of renesting to extend beyond the commonly applied fixed maxima; conversely, it can force renesting to end before a set maximum is reached. Here, we quantify the extent to which assumptions that fix the maximum possible number of renestings or broods alter estimated seasonal fecundity. We use Nolan’s (1978) Prairie Warbler data to test our approach and quantify potential biases by comparing the seasonal fecundity he measured directly in the field with that inferred indirectly under various assumptions about renesting. Our method is to analyze exactly the same nest-productivity data set using different sets of assumptions about renesting. By comparing the seasonal fecundity estimated from a single unchanging data set under different sets of assumptions, we are able to isolate the extent to which an assumption about renesting affects estimated seasonal fecundity. M  The computer model employed here is a slight modification of the simple case of our previous seasonal-fecundity model (Pease and Grzybowski 1995). It predicts seasonal fecundity using demographic parameters describing nest success, breeding season and nesting cycle, and intensity of nest predation and brood parasitism. It assumes that the length of the breeding season is fixed as the average number of days during which females engage in breeding activity. Specifically, it assumes that the average female will always renest if the prior nesting a empt fails before day ss of the breeding season (the average number of days in which females a empt nestings), but will never initiate a nest a er day ss of the breeding season. We thus define the breeding season as the period in which nests can be initiated. (Our complex extension allows different females to breed for different numbers of days, an elaboration that produces mathematical results similar to those of our simple case, but that also contains additional parameters that can seldom be observed empirically.) Our model places no constraint on the number of nesting a empts possible beyond that indirectly imposed by breeding-season length, average life expectancy of a nest, and amount of time lost to nests that failed because of predation

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or brood parasitism. It does, however, more directly constrain the maximum number of successful broods: if it takes tr days to successfully fledge a brood and initiate another nest, no more than 1 + floor(ss/tr) successful broods are possible, where floor(x) is the function giving the greatest integer not greater than x. We previously tested this approach to estimating seasonal fecundity using data collected on the Prairie Warbler and Black-capped Vireo (Pease and Grzybowski 1995). We found that seasonal fecundity estimated from nest-productivity data under our model and its assumptions closely matched seasonal fecundity measured directly in the field, which suggests that this approach to estimating seasonal fecundity from nest-productivity data produces reliable estimates. Our original model (Pease and Grzybowski 1995) did not separately track the fraction of the breeding population having any given number of prior nesting a empts, or the fraction having any given number of prior successful broods. In the Appendix, we generalize the original model by developing a book-keeping system (employing partial differential equations) that allows us to determine, for each day in the breeding season, the fraction of the breeding females with 0, 1, 2, or 3 (etc.) prior nesting a empts, and the fraction that have previously produced 0, 1, or 2 (etc.) successful nests. The generalized model in the Appendix contains no additional parameters that must be estimated from demographic data, beyond those required by the underlying model of Pease and Grzybowski (1995). Rather, it computes, explicitly and separately, certain quantities previously not available as model output.

The altered model thus explicitly tracks the fraction of breeding females having any arbitrary number of prior nesting a empts or prior successful broods. We also modified it so that the user can prevent all females with more than a given number of prior nesting a empts and successful nests from breeding—for example, to prevent any female with ≥3 renesting a empts or ≥1 successful brood from renesting. We also modified it to allow the user to permit females to breed beyond the end of the breeding season if they have not yet achieved the set maximum number of nesting a empts or number of successful broods. Those modifications allow us to compare (1) seasonal-fecundity estimates calculated from nest-success data without directly limiting the number of renesting a empts with (2) those calculated when the number of renesting a empts and successful broods is directly limited to a set maximum. Direct empirical measures of renesting.—In the more thorough field studies, the maximum number of nesting a empts observed per female varied from four to eight (Table 1). We incorporated that range of nesting a empts in our analysis. Note that the maximum nesting capacities would not be seen in many studies with lower sampling intensities. Also, note that Holmes et al. (1992) had the lowest maximum number of nesting a empts (four total a empts) and the lowest levels of nest failure, a point developed below. What do other authors assume about renesting?— Many authors, in estimating seasonal fecundity from nest-history data, assume that individual females a empt to nest a maximum of only two, three, or four times during a breeding

T 1. Maximum number of nesting a empts by female songbirds observed during a single breeding season in intensive field studies. Maximum number of nesting a empts observed

Author

Species

Nice (1937) Nolan (1978, table 55) Payne (1989) Major (1991) Holmes et al. (1992) Woodworth (1997) Kershner et al. (2001) Brown and Roth (2002)

Song Sparrow (Melospiza melodia) Prairie Warbler (Dendroica discolor) Indigo Bunting (Passerina cyanea) White-fronted Chat (Epthianura albifrons) Black-throated Blue Warbler (D. caerulescens) Puerto Rican Vireo (Vireo latimeri) Blue-gray Gnatcatcher (Polioptila caerulea) Wood Thrush (Hylocichla mustelina)

4 8a 7 ≥5 4 6 7 5

a Natural a empts. Nolan (1978) induced one female to make nine nesting a empts by artificially depredating some of her nests.

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season (Table 2), and that females produce, at maximum, only one or two successful broods per year (Table 2). We incorporated that lower range of nesting a empts in our analyses. Such analyses also implicitly assume that all females continue breeding until one or the other of those maxima is obtained. Comparing seasonal fecundity estimates made using different assumptions.—Our model assumes only that (a) numbers of nesting a empts and of successful broods are limited indirectly by total breeding-season length (herea er “assumption set 1”). Several authors have used those assumptions in estimating seasonal fecundity (Table 2). In contrast are assumptions that set (b) the maximum number of nesting a empts allowed by each adult female in a breeding season and (c) the maximum number of successful broods possible for each female in a breeding season, and implicit assumptions that (d) all females will achieve either the maximum number of nesting a empts or the maximum number of successful broods (herea er “assumption set 2”; see Table 2). A byproduct of some analyses that employ assumption set 2 is that, for certain parameter values, the breeding season will be longer than that for assumption set 1. As an aid to understanding the problems that this creates, it is useful to estimate seasonal fecundity under assumptions (a), (b), and (c), but not (d) (hereafter “assumption set 3”). That sets the maximum number of nesting and breeding a empts possible, subject to the constraint that females cannot breed beyond the last day of a breeding season of fixed length. Assumption set 2 differs from assumption set 3 in that, for certain parameter values, it allows females to breed beyond the last day of the breeding season. To our knowledge, assumption set 3 has never been used in the literature to estimate seasonal fecundity. We employ it as a tool to help disentangle the rather unintuitive and complicated factors that influence estimated seasonal fecundity. Conceptual approach.—Our basic approach is to compare seasonal fecundity estimated under the three different assumption sets. By comparing seasonal-fecundity estimates obtained using models that make or omit a particular assumption, we can isolate the effect of that assumption on the estimated seasonal fecundity. This reasoning is analogous to that employed in an empirical experiment where one compares the dependent

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variable (e.g. estimated seasonal fecundity) in the control (e.g. model without the assumption of interest) with that in the experimental manipulation (e.g. model with the assumption of interest), thereby isolating the effects of the experimental manipulation (i.e. of the assumption). Computer implementation of our model.—The computer code implementing the present model is essentially identical to that of the Pease and Grzybowski (1995) model, inasmuch as the internal structure of the present model is essentially unaltered. See the Appendix and Pease and Grzybowski (1995) for details. The computer code is available from J.A.G. on request. R   Using Nolan’s (1978) estimates for model parameters from his Prairie Warbler data, our model came to the same estimate of seasonal fecundity as he observed directly (2.2 offspring per adult female per year; Pease and Grzybowski 1995). That estimate relies on assumption set 1, which suggests that assumption set 1 produces unbiased estimates of seasonal fecundity, and providing a standard against which to compare seasonal-fecundity estimates obtained under other assumption sets. In looking at those comparisons, however, note that nest mortality and, hence, renesting is relatively high in Nolan’s (1978) data set. Estimates of seasonal fecundity obtained under assumptions that artificially constrain the number of nesting a empts or of successful broods (assumption set 2) are widespread in the literature (summarized in Table 2). To investigate the potential bias in those estimates, we compared seasonal fecundity as estimated under assumption set 2 with that estimated under assumption set 1 (percentages in bold in Table 3, where “100%” indicates no bias). Under assumption set 2, seasonal fecundity can be biased low (when the fixed renesting maximum is set too low) or biased high (when the renesting maximum effectively allows breeding beyond the end of the breeding season). Those biases (deviations from 100%), both negative and positive, can be quite large, depending on the specific fixed renesting limit applied (Table 3). They occur when the maximum number of broods is set at one or at two. One suspects, a priori, that the overestimates shown in bold in Table 3 are a consequence of

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T 2. Assumptions (in bold) about the maximum number of broods (Bmax), maximum number of nesting a empts (Nmax), and length of breeding season (ss) made in estimating seasonal fecundity. Bmax 1 1 1 2 2 2 1 1–2 1 2j 1 2 1

– – – – – – – – – 1 – – a

Nmax

ss (days)

Species

Exactly one nesting aempt a 1 – Theoretical model b 1 – Warbling Vireo (Vireo gilvus) c Fixed maximum numbers of nesting aempts and successful broods d 2 – Ovenbird (Seiurus aurocapilla) e 3 – Red-eyed Vireo (V. olivaceus) e 3 – Wood Thrush e 3 – Wood Thrush f 2 – Rose-breasted Grosbeak (Pheucticus ludovicianus) f 2–4 – 7 species total g 3 or 4h – Theoretical models; single-brooded i 4k – Theoretical models; double-brooded i 2 – Ovenbird l 3 – Wood Thrush m 2 – Ovenbird n Maximum numbers of nesting aempts and successful broods set by breeding-season length o – 68 Black-capped Vireo (V. atricapilla) p – 50 Prairie Warbler p – 63–104 Puerto Rican Vireo q – 38 Red-eyed Vireo r – 73 Blue-headed Vireo (V. solitarius)r – 60 Wood Thrush r – 47 Worm-eating Warbler (Helmitheros vermivorum) r – 90 Wood Thrush s – 17–35 Red-eyed Vireo t Miscellaneous assumptions – – Ovenbird u – – Wood Thrush v – – Literature review w

Special case of assumption set 2 (see text). May and Robinson 1985. c Ward and Smith 2000. d Assumption set 2 (see text). e Donovan et al. 1995. f Friesen et al. 1999. g McCoy et al. 1999. h Two renesting a empts allowed a er depredation events, three a er abandonment following parasitism. i Schmidt and Whelan 1999. j Assumes that only a fraction of the females that could potentially renest a empt to do so. k Up to maximum nesting a empts allowed; all individuals renest a er abandonment following parasitism, but only a fraction renest following depredation. l Flaspohler et al. 2001. m Duguay et al. 2001. n Manolis et al. 2002. o Assumption set 1 (see text). p Pease and Grzybowski 1995. q Woodworth 1999. r Dececco et al. 2000. s Simons et al. 2000. t Marshall et al. 2002. u Porneluzi and Faaborg 1999. Assume only one brood per year, and that seasonal fecundity is product of clutch size and observed fraction of females with a successful nest. v Fauth 2001. Assume that seasonal fecundity is product of average number of nesting a empts per female (Nave) and average number of offspring per nesting a empt (Nave = 3.1). w Martin 1995. Assume that seasonal fecundity is product of average number of successful nesting a empts per female and clutch size. b

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T 3. Percentage of empirically observed seasonal fecundity (2.2 young per female; Nolan 1978) inferred from renesting assumptions. Percentages in bold are those computed using assumption set 2 (see text) compared with those using assumption set 1 (assumed to be the true value; see text). Percentages in regular type are those computed using assumption set 3 (see text) compared with those using assumption set 1 (see text). Unbiased estimates are indicated by 100% (equals a seasonal-fecundity value of 2.2). A bias of 52%, for example, would reflect a seasonal-fecundity estimate of 1.14. Maximum number of successful broods 1 2a

Maximum number of nesting a empts 2

4

6

8

52% 52% 58% 58%

84% 81% 111% 93%

104% 88% 155% 100%

117% 88% 189% 100%

a Because the nesting cycle is 38 days and the breeding season is 50 days, only 2 successful broods are possible.

the fact that assumption set 2 sometimes allows females to breed beyond the end of the breeding season (keeping in mind that 100% is an unbiased estimate obtained from assumption set 1). To investigate that hypothesis, we asked whether the overestimates disappear under assumption set 3, which sets renesting limits but never allows a female to breed beyond the end of the breeding season. That is indeed the case; the seasonal fecundity estimated under assumption set 3 is always less than or equal to the true value (percentages presented in regular type in Table 3 are always ≤100%). Because most limits on nesting in the literature are set at 1.0, rather than 1.0 (see Schmidt and Whalen 1999, appendix 1, footnote a). For Nolan’s (1978) Prairie Warbler parameters, we calculate Schmidt and Whalen’s (1999) summed probabilities for all nest fates as 1.08. That effect cascades through their model calculations and leads to their estimated seasonal fecundity being biased high (were it not otherwise compensated for). We conjecture that the compensating bias arises because they did not set the maximum number of nesting a empts

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at a sufficiently high number, which, without the compensating error, would cause their estimated seasonal fecundity to be biased low. Thus, the correspondence they obtained between their model’s predicted seasonal fecundity and Nolan’s (1978) empirical value is not evidence that their assumptions are correct. As an exemplar of the importance of measuring renesting accurately and properly accounting for it in estimating seasonal fecundity, consider how small differences in breeding-season length and, hence, in the potential for renesting, can lead to small but real differences in seasonal fecundity. In an extraordinarily careful study, Marshall et al. (2002) compared seasonal fecundity of Red-eyed Vireos in forest tracts sprayed with Bacillus thuringiensis to reduce gypsymoth (Lymantria dispar) abundance with that in unsprayed tracts. They found that Red-eyed Vireos in the sprayed tracts delayed the start of their breeding season by about three to five days, which resulted in an estimated decrease in seasonal fecundity of 0.15 to 0.25 offspring. To detect an effect that small, one must carefully account for renesting and breeding-season length, as Marshall et al. (2002) did. Our analyses demonstrate that, in analyzing songbird fecundity data, the assumptions that one makes about renesting can have significant effects on estimated seasonal fecundity. Inasmuch as one needs estimates of seasonal fecundity to assess the health of songbird populations, faulty renesting assumptions jeopardize such estimates and any consequent decisions regarding conservation or future research, as well as inaccurately portraying songbird population biology. Using assumptions and analyses that fully and properly account for renesting (e.g. Marshall et al. 2002) is much be er than the costly prospects of misguided management and research. A   We thank B. Dunning and four anonymous reviewers for providing critical comments on earlier versions of this paper. L   C  B, W. P.,  R. R. R. 2002. Temporal pa erns of fitness and survival in the Wood Thrush. Ecology 83:958–969. C  , E. D.,  K. Y . 1999.

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G  P 

290 A 

Pease and Grzybowski (1995) should be consulted for an intuitive description of the following partial differential equations; here, we closely follow, and in some respects generalize, the model and notation presented in that study. More specifically, except where explicitly stated otherwise, all variables and parameters here have the same definitions as in Pease and Grzybowski (1995). Define ux,y(t,s) so that

is the fraction of all adult females that are unparasitized between days t1 and t2 of the nesting cycle, between days s1 and s2 of the breeding season, have x prior successfully fledged broods (x ≥ 0 always), and have y prior nesting a empts (y ≥ 0 always). Define px,y(t,s) analogously, as the density of females that currently have one or more parasite eggs in their nest; note that, because every female must have at least as many prior nesting a empts as she has successful nests, x ≤ y. Following the simple case of the Pease and Grzybowski (1995) model, denote the instantaneous nest predation and brood parasitism rates at times t of the nesting cycle and s of the breeding season as d(t,s) and ρ(t,s), respectively. Also, define a(t,s), analogously, as the probability that a breeding female will abandon a nest immediately a er a parasitism event (0 ≤ a ≤ 1). Then, for x ≥ 0, y ≥ 0, and x ≤ y:

[Auk, Vol. 122

of all adult females initially entering the breeding pool on day s of the breeding season. The remaining two boundary conditions describe females that reinitiate the nesting cycle during the course of the breeding season. The first of these considers females initiating a new nesting cycle within the breeding season a er a nest failure and without having yet successfully fledged a brood (i.e. x = 0):

y≥1 where the first integral accounts for females that renest a er losing a brood to nest predation and the second accounts for females that abandon a nest a er brood parasitism, and where tr is the day of the nesting cycle on which a female can renest a er successfully fledging a brood, and where g(s) is the probability that a female will renest on day s of the breeding season if her nest fails or she successfully fledges a brood. The last boundary condition describes those females initiating a new nesting cycle with at least one prior successful nest (x ≥ 1). In contrast to the previous equation, in this case we must account not only for renesting a er nest failure (two integrals in the following equation), but also for renesting a er successfully fledging a brood (first term in the following equation):

x ≥ 1 and y ≥ 1

The first boundary condition specifies the date(s) on which females initially enter the breeding season: u0,0(0,s) = f(s) where the density function f(s) gives the fraction

Our computer simulations specialized the above equations. We assumed that the function f(s) is a delta function centered at s = 0; this corresponds to a pulse of all adult females starting to breed simultaneously on day 0 of the breeding season. Furthermore, we assumed that g(s) = 1.0 for 0 ≤ s ≤ ss and 0.0 otherwise; this is a mathematical statement of the assumption that

January 2005]

Renesting and Seasonal Fecundity

all individuals a empt an additional brood when given the opportunity, up until the end of the breeding season. As in Pease and Grzybowski (1995), our computer simulations assume additionally that the instantaneous nest-predation and brood-parasitism rates are positive constants over windows (te , tf) and (te , ti) of the nesting cycle, respectively, and 0 otherwise:

291

using a step-size of 0.05 days in both the nesting cycle and breeding season dimensions. We used Nolan’s (1978) Prairie Warbler demographic data, as summarized in Pease and Grzybowski (1995). This analysis starts with a pulse (delta function) of females at the start of the nesting cycle and breeding season, and then follows those females and their fates. We computed seasonal fecundity, F, from:

d(t,s) = d for te < t < tf d(t,s) = 0 otherwise ρ(t,s) = ρ for te < t < ti

x≤y

ρ(t,s) = 0 otherwise.

where tf , the end of the window of susceptibility to predation, is typically taken as the fledging date, and where fu and fp are the number of offspring fledged from successful unparasitized and parasitized nests, respectively.

Similarly, we assume that a(t,s) = a, a constant. To obtain the results presented in Table 3, we simulated the above equations numerically