Jones et al. (2005a) used data from an intensively studied population of Black-throated Blue Warblers (Dendroica caerulescens) to evaluate the correlation between empirical estimates of seasonal fecundity and estimates derived from a model developed by us (Farnsworth and Simons 2001). They reported that our model substantially underestimated seasonal fecundity. That conclusion was entirely in error. When used as described by Jones et al. (2005a), our model actually estimates substantially higher seasonal fecundity than that derived from empirical data. However, we do not recommend using our theoretical model in this manner to estimate seasonal fecundity or to assess population status. Here, we provide a brief discussion of modeling seasonal fecundity in multibrooded birds, with recommendations on using a modified version of our model for these purposes.
The model is a deterministic mathematical model that estimates seasonal fecundity on the basis of daily survival rates of nests and renesting behavior of breeding females (i.e. how quickly a female initiates a new nest after failed and successful attempts) within a limited breeding season. The model estimates the probability of fledging one or more broods. Jones et al. (2005a) apparently made one or more errors in interpretation or implementation when transcribing the mathematical description into an EXCEL spreadsheet. We reviewed our original published description and discovered a minor typographical error, which we redress here (see Appendix). However, that error alone could not have caused the surprisingly low values reported by Jones et al. (2005a).
Our model (Farnsworth and Simons 2001) was created to assess the constraints and trade-offs that shape the evolution of clutch size in multibrooded bird species. In that analysis, we examined how the allocation of eggs among multiple nesting attempts influenced seasonal fecundity under different conditions. Our theoretical investigation compared relative estimates of seasonal fecundity to find maxima. That analysis did not require an adjustment in brood sizes to account for partial losses before fledging. Such adjustments can easily be incorporated when estimating seasonal fecundity for actual breeding-bird populations (Farnsworth 1998). We recommend using our model to estimate the probability of fledging one brood and the probability of fledging two broods (and when necessary, the probability of fledging three and four broods) and multiplying those estimates by the average realized brood size (nf = number of fledglings per successful nest). See the Appendix for mathematical substitutions to the model description in Farnsworth and Simons (2001).
We are not surprised that corrected estimates of seasonal fecundity from our model are higher than empirical estimates (Table 1 in Jones et al. 2005b). Our model assumes that each female continues to renest as long as sufficient time remains in the breeding season, up to a maximum number of nesting attempts. That renesting behavior may be unrealistic for many species, but our model can be adjusted easily to accommodate more conservative assumptions (see Appendix). By contrast, empirical estimates of seasonal fecundity are necessarily biased low. Even in the intensively studied population of Black-throated Blue Warblers at Hubbard Brook, observations are not perfect. For example, Jones et al. (2005a) described a banded male observed feeding a fledgling from an unobserved nest on their study site. Empirical studies will underestimate fecundity when some nests in the study area are not discovered or when females move into or out of the study area between nesting attempts. Some females that leave the study area during the breeding season may in fact breed elsewhere.
We commend Jones et al. (2005a) for attempting to compare seasonal fecundity estimates from a long-term empirical study with those derived from our modeling approach. We agree with those authors that accurate estimates of seasonal fecundity are vital for answering questions about population viability, source-sink dynamics, and conservation status. We also agree with Grzybowski and Pease (2005) that simple algorithms, assuming all females attempt a fixed number of nests, overestimate fecundity at high levels of nest survival and underestimate it at low levels of nest survival. However, we believe that our modeling approach overcomes that shortcoming by constraining the maximum number of nesting attempts in relation to the length of the breeding season. To make seasonal fecundity estimation more readily available to researchers, we have provided a copy of “Model 1“ from Farnsworth and Simons (2001), including the modifications described here, at staff.xu.edu/∼farnsworth/renest.xls.
Nevertheless, we recognize that our model remains a gross oversimplification of the complex processes governing seasonal fecundity in real populations. Our model requires assumptions that may be unrealistic for some populations. For example, the survival rate of nests may vary throughout the nesting cycle or nesting season. Similarly, the amount of time required to renest may vary throughout the nesting season. Temporal variations such as these are not incorporated in our current modeling framework. Those types of variability are more easily incorporated in alternative modeling strategies, such as individual-based simulations (see Farnsworth 1998).
We correct an error in Farnsworth and Simons (2001). In that paper, equation 11 should be replaced with:
See Farnsworth and Simons (2001) for definitions of symbols.
The original formulation of the model had the assumption that all females renested as long as time remained in the breeding season (up to a maximum number of nesting attempts, m). The model can be amended easily to relax that assumption in populations where the probability of renesting is believed to be less than unity. To accomplish that, we introduce a parameter r, defined as the probability of renesting. Equation 3 in Farnsworth and Simons (2001) thus becomes:
The original model also allowed a female to fledge as many broods as the maximum number of nesting attempts, m. That assumption may be violated in populations such as Black-throated Blue Warblers in New Hampshire, where females routinely engage in three nesting attempts but are not known to fledge three broods (J. Jones pers. comm.). A modification to the model to overcome that assumption is straight-forward. Equation 4 in Farnsworth and Simons (2001) should be replaced with:
where k is the maximum number of broods per female per breeding season, and nf is the realized brood size (number of fledglings per successful nest).