Avian 1995 - A Model to Estimate the Annual Rate of Golden Eagle Population

A Model to Estimate the Annual Rate of Golden Eagle Population
Change at the Altamont Pass Wind Resource Area

Tanya M. Shenk, Alan B. Franklin and Kenneth R. Wilson,
Colorado State University

Golden Eagle (Aquila chrysaetos) deaths caused by wind turbines in the Altamont Pass Wind Resource Area (APWRA), Mt. Diablo Range, Cali-fornia, have been well docu-ment-ed (Hunt 1994). The impact of these wind turbine fatalities on the APWRA Golden Eagle population has not, however, been determined. Determining a population effect and estab-lish-ing a direct causal relationship to the APWRA would require esti-mates of demographic parameters in the APWRA and an appropriate con-trol area both before and after con-struc-tion of the wind turbines (see Eber-hardt and Thomas 1991). Unfortunately, no pre-construc-tion data were collected and no control area was established. However, the concern over a potentially detrimental population impact from the APWRA is still relevant.

In general, impacts on populations can be either positive or negative. Annual rates of population change (

) can be measured to assess such impacts. An annual rate of population change that is positive, i.e. population size increasing, suggests a positive impact. A rate of change that is negative, i.e. population size decreas-ing, suggests a negative impact. Therefore, if we could estimate the annual rate of population change for the Golden Eagle popula-tion in the area surrounding the APWRA, some inference could be made to the effect of the APWRA on the Golden Eagle population. If the population size is stationary or increasing, we would infer that the defined Golden Eagle population is viable despite the presence of the APWRA. If the annual rate of population change is negative, we would be unable to ascribe cause and effect; the population might be declining either because of APWRA or for other reasons. However, if the annual rate of population change were found to be decreasing, there would be reason to initiate a series of more penetrating studies.

To estimate the annual rate of population change, we developed a 3-staged population model for the defined Golden Eagle population around the APWRA. The objective of the model is

The annual rate of population change (

) is a useful metric in that it measures the direction as well as the magnitude of population change:

=1 indicates a stationary population;

<1 a declining population, and

>1 an increasing population. The magnitude of change is

-1. Therefore, we can frame our initial question in terms of null (H_O) and alternate (H_A) hypotheses to be tested:

A key consideration is that model results apply only to conditions during the period of data collection, and not necessarily to the future behavior of the population. Therefore, the model does not attempt to predict the future but is used only to estimate the finite rate of population change (

) within the period of study--a snapshot of the status of the population within a given time. Although this model was not the only way chosen to assess the effects of the APWRA, it does provide a key com-pon-ent in addressing those effects.

We chose a single-sex, stage-based model in the interest of parsimony (Burnham and Anderson 1992). This model represents a tradeoff between funding constraints and number of parameters that can be estimated precisely (i.e., with coefficients of variation <10%). As additional parameters are added to the model, sˆe(

) will increase because sampling variances in the model are essentially additive. This can be rectified by increasing the precision of each parameter, but to do so requires a larger sample size and, hence, increased project costs. In order to maintain high power for H_O:

> 1 (i.e., a high probability of rejecting H_O if it is false), we chose to estimate a smaller number of para-meters with sufficient samples for adequate precision. By restricting the model to a single sex, the number of parameters to be estimated can be reduced by about 50%. Given this rationale for a single-sex model, a female-based model was selected because female fertility can be assessed more directly.

The model follows the life history characteristics of Golden Eagles in the APWRA as described by Hunt (1994). A fledged young has a certain probability of surviving to become a non-territorial "floater" the following year. The bird remains in this stage for an indeter-minate period of time. Each year, it has a certain probability of surviving as a floater or of entering the territorial population if a vacancy occurs. If it becomes a territorial bird, it continues to survive each year with some probability and produces more fledged young at some rate, thus starting the cycle over again. This life cycle can be described in terms of a mathematical model, which in turn can be used to estimate annual rates of population change (McDonald and Caswell 1993).

To describe the life cycle of Golden Eagles at APWRA, we used a standard stage-based model with three stages and five annual loop trans-missions (Fig. 1a; Caswell 1989). The three stages are defined as discrete classifications of individuals. The directional lines (Fig. 1a) connecting the stages represent transmission loops between the stages in the form of survival (P_i), reproduction (F_i), or transition probabilities (G_i). The model is based on a post-breeding survey and has a time step of 1 year (Noon and Sauer 1992). As stated previously, the model only incorporates a single sex. To modify the standard model to represent the Golden Eagle population in the Alta-mont Pass area, we first define the stages to represent three discrete behavioral categories. Stage 1 contains female fledglings alive at time t. Stage 2 contains non-territorial females, which include both subadults (<4 years of age) and floaters (non-territorial adults) at time t. Stage 3 contains territorial females at time t. Fewer transmission loops are necessary to describe the defined population because two of the trans-ition prob-abil-ities are assigned values of zero: (1) P₁ = 0 because fledglings cannot remain in this stage for longer than 1 year, and (2) F₂ = 0 because we assume non-territorial females do not breed. Therefore, a reduced model of three transmission loops adequately describes the Golden Eagle pop-u-la-tion model (Fig. 1b). For clarification of the reduced model, we substitute the theoretical nomenclature with the behavioral stages and parameter estimates to be used for each of the transition probabilities (Fig. 1c, Table 1). Transition from stage 1 to stage 2 (G₁) will be estimated as survival of female fledglings from year t to t+1 (s_F). P2 will be estimated as non-territorial female survival (s_NT). G₂ will be estimated as the prob-ability ( ) of a non-territorial female becoming a territorial female. P₃ represents annual territorial female survival (s_T). Annual reproduction, F₃, is represented as the number of females fledged per surviving territorial female (s_Tb_T).

TABLE 1. Parameter notation and description for a 3-staged population model developed for the defined Golden Eagle population around the Altamont Pass Wind Resource Area.

Parameter
Caswell notation	Eagle Model notation	Estimate (Eagle model)	Definition
P₁			Theoretically the probability of a female fledg-ling in year t remaining a fledgling in year t+1 stage (impossible, therefore eliminated).
P₂	S_NT	s_NT	Probability of a non-territorial female in year t surviving to year t+1.
P₃	S_T	s_T	Probability of a territorial female in year t sur-viving to year t+1.
G₁	S_F	s_F	Probability of a female fledgling in year t surviving to become a non-territorial female in year t+1.
G₂	a	a	Probability of a non-territorial female in year t surviving and transitioning to a territorial female in year t+1.
F₂			Theoretically, the mean number of female young fledged per non-territorial female. However, we assume this always = 0
F₃	S_Tb_T	s_Tb_T	Mean number of female young fledged per sur-viving territorial female
			Annual rate of population change.

FIGURE 1. A graphical representation of (a) a general, theoretical, 3-staged population model, (b) the reduced Golden Eagle theoretical model, and (c) the parameter-based model for the defined Golden Eagle population around the Altamont Pass Wind Resource Area.

Matrix representations of the general theoretical model, the reduced Golden Eagle theoreti-cal model, and the parameter-based Golden Eagle model are as follows:

Substituting the Golden Eagle model parameters, the three loop transmissions, modified from Caswell (1989:102), are as follows:

where all parameters are defined in Table 1. The characteristic equation of the model is

Estimates for s_F, s_NT, s_T, , and b_T obtained from data collected in the field will be substituted into Eq. (4), the characteristic equation, to solve for . The standard error of will be estimated using the delta method, which incorporates the sampling variances for each of the parameter estimates (Oehlert 1992; Alverez-Buylla and Slatkin 1994). The estimate of and its standard error can then be used to test the null hypothesis

All models rely on certain assumptions. The more general the model, the longer the list of assumptions that must be met for model inferences to be unbiased. Two sets of assump-tions are associated with our model. First, three assumptions underlie our use of the basic matrix model (McDonald and Caswell 1993):

Second, there are additional assumptions that are specific to modeling the Golden Eagle population at APWRA. These include

Estimation of parameters for inclusion in the model is key to the results of the model. The model and the parameters used in the model are interlinked in terms of bias and precision. The model to estimate the finite rate of population change was chosen based on which parameters could be estimated accurately and precisely in the field, where precision of the estimates depends on having sufficient sample sizes. For the purposes of our model, we are interested in both (a) a point estimate of each parameter and (b) its standard error as a measure of the precision of that point estimate.

Proposed methods for estimating survival within stages and trans-itions between stages rely heavily on radio-telemetry of individuals. The use of radiotelemetry allows estimation of these parameters using approp-riate statistical models (Pollock et al. 1989; Bunck and Pollock 1993). Estimation of fledgling rate is more problematic in that it requires determin-ing whether territorial pairs nest and whether those nests are successful. To identify whether territorial pairs nest, individuals must be correctly determined to (a) hold a terri-tory and (b) either have a nest or not (G. Hunt, pers. comm.). Once numbers of fledg-lings can be ascribed to each sampled pair of birds (including zeroes for nonbreeding terri-torial pairs), fledging rates can be estimated as arithmetic means with their standard errors.

The estimate of

based on our model measures only changes in the defined Golden Eagle population related to birth and death rates. Immigration or emigration rates are not included in the model. Therefore, inferences from this model will only reflect whether the population within the defined limits around the Altamont Pass area can be sustained solely on birth and death processes. This model represents the first step in an iterative approach to estimating the effects of the APWRA on Golden Eagle populations. Additional steps may include more experimental approaches if the model indicates that the Golden Eagle popula-tion surrounding the APWRA is not self-sustaining, based on birth and death rates.

We recognize that use of this model is not an ideal approach. However, we believe that it is a good initial approach, given the con-straints of time and funding and the fact that the Altamont wind development has been in place for a considerable length of time. How-ever, we would not recommend this approach for proposed projects of a similar nature that have not yet been built. For these types of projects, a more classical experimental approach (see Eberhardt and Thomas 1991) would allow for causal inferences concerning effects of the project on the populations being considered.

Alvarez-Buylla, E.R. and M. Slatkin. 1994. Finding confidence limits on population growth rates: three real examples revisited. Ecology 75:255-260.

Bunck, C.M. and K.H. Pollock. 1993. Estimating survival of radio-tagged birds. p. 51-63 In: J-D. Lebreton and P.M. North (eds.), Marked individuals in the study of bird popula-tions. Birk-h�user Verlag, Basel, Switzerland.

Burnham, K.P. and D.R. Anderson. 1992. Data-based selection of an appropriate biological model: the key to modern data analysis. p. 16-30 In: D.R. McCullough and R.H. Barrett (eds.), Wildlife 2001: populations. Elsevier, London, U.K. 1163 p.

Eberhardt, L.L. and J.M. Thomas. 1991. Designing environmental field studies. Ecol. Monogr. 61:53-73.

Hunt, G. 1994. A pilot Golden Eagle population project in the Altamont Pass Wind Resource Area, California. National Renew-able Energy Laboratory, Golden, CO. 212 p.

McDonald, D.B. and H. Caswell. 1993. Matrix methods for avian demography. p. 139-185 In: D.M. Power (ed.), Current ornithology, vol. 10. Plenum Press, New York.

Noon, B.R. and J.R. Sauer. 1992. Population models for passerine birds: structure, para-meterization and analysis. p. 441-464 In: D.R. McCullough and R.H. Barrett (eds.), Wildlife 2001: popula-tions. Elsevier, London, U.K. 1163 p.

Pollock, K.H., S.R. Winterstein, C.M. Bunck and P.D. Curtis. 1989. Survival analysis in telemetry studies: the staggered entry design. J. Wildl. Manage. 53:7-15.

Introduction.-Ms. Shenk indicated at the outset that model development is an iter-a-tive process. The modeling group believes that the present model is defensible in its present form but can be improved; they solicited suggestions for improvement.

Model Structure.-A participant asked how the "Altamont population" of Golden Eagles is defined. Ms. Shenk indicated that it is defined geographically; there are some "natural" boundaries, including urbanized areas and habitat boundaries that make this defens-ible. Although the intention is to consider only the resident eagles in the model, when an eagle is captured and radio-tagged it is not known whether it is a resident.

The modeling group initially con-sidered distinguishing eight categories of eagles: four groups of females (fledglings, subadults, adult floaters, and adult territory holders) and the corresponding four groups of males. However, it was determined that it would not be pos-sible to obtain adequate sample sizes for all of these groups, and in the end the model considers three groups ("stages"):

One participant expressed concern about the decision to model only the female component of the population, given that

A participant asked whether it is important to focus on the "limiting" sex, or whether that issue is of mainly academic interest. Ms. Shenk responded that, if there is a limiting sex, that sex should be modeled. However, in practice, radio-tags have been placed on both male and female eagles in the Altamont area, given the high cost to catch eagles relative to the tag cost. Hence, data will be obtained for both sexes, and the issue of "which sex(es) to model" may be moot.

It was also indicated that, given overall funding limitations, there is a tradeoff between the funds that can be devoted to the Altamont Golden Eagle population study and other avian - wind power studies.

Concern was expressed that the population study and modeling effort might not be able to obtain a reasonably precise estimate of the annual rate of population change ( ). Ms. Shenk responded that a preliminary power analysis suggested that, to estimate with a 10% coef-ficient of variation, about 80 fledglings, 25 non-territorial birds, and 25 territory holders would need to be studied. Actual survival rate may be higher than was initially assumed, in which case the required sample sizes would be smaller. Dr. Hunt's view is that can be estimated with reasonable precision if the radio-tagged eagles of both sexes can be pooled, but otherwise not. Some attendees noted that pooling of sexes is not justified unless statis-tical analyses based on adequate sample sizes show no significant difference between sexes.

Model Assumptions.-An attendee expressed concern that the model assumes constant

across all categories of eagles. He noted that, in the quite likely event that

is not constant across categories of eagles, the power of the study and model to reject H_O if it is false would be reduced (i.e., the required sample size would be much larger than has been estimated). He recommended that "stage-invariant

" should be listed explicitly among the main model assumptions.

Another attendee felt that, notwithstanding those concerns, the field study should provide a good indication whether or not the Altamont Golden Eagle population is "healthy". In addition to an estimate of

, the study will provide estimates of juvenile survivorship, abundance of "floaters", reproduction data, and data on the occurrence and circumstances of deaths. This information should provide a good indication of population status. If the population does not appear to be doing well, follow-up studies would be needed to evaluate the problem. Also, all of these data will be available for future refinements of the model. Ms. Shenk noted that the present study will itself provide information about the number of deaths of radio-tagged eagles that can be attributed to the Altamont wind facilities, and thus the effect of these deaths on

A participant noted that, even if the field study and model indicate that the Altamont Golden Eagle population is doing well, the occurrence of any eagle deaths attributable to the wind facilities has legal ramifications. He also noted that the Altamont eagle population study was developed in part to be responsive to the concerns of regulatory and environ-mental groups, with the objective of evaluating the key questions that they had identified.

A question was raised about the complications created by immigration and emigration. Ms. Shenk noted that the present study and modeling effort cannot address that issue explicitly, but the study will provide information useful for designing a possible future study of dispersal. She mentioned that the determination of juvenile survival is strongly con-founded by uncertainty about emigration rate.

Estimates of Model Parameters.-A participant suggested that, in attempting to address this and other uncertainties, data from other Golden Eagle populations should be taken into account. Ms. Shenk explained that some members of the modeling group strongly believe that the model should be driven by data from the Altamont area, and are not keen to use data from elsewhere. The questioner noted that data from elsewhere can be relevant in defining the biology of the species, and comparison of parameter values from other locations with those from the Altamont could provide a "reality check" for the present results.

Conclusions.-Another participant asked what would constitute a reasonable balance between study duration and confidence in results. Ms. Shenk indicated that a 2-year study is very short in relation to the lifespan of eagles and in relation to the range of variability in environmental conditions. Dr. Pollock mentioned that any attempt to study the effects of wind development on an eagle population using traditional or BACI approaches would require a much longer study. He said that, if a very long term study were possible, he would want to consider using a BACI approach in addition to modeling.

During the concluding discussion, one participant expressed the view that the resources being devoted to the Altamont eagle study were insufficient to provide definitive answers, but he agreed that--whatever the merits and problems of the present model--the field study would provide useful information on population status and guidance for follow-on work. Another participant mentioned that, in deciding how to allocate limited resources, a single study and issue such as the Altamont eagles should not be considered in isolation from other potentially valuable work, including risk-reduction and monitoring studies.

	The subadults and adult floaters were combined, but there is concern that their population parameters may not be the same.
	In developing the model, there also was much discussion about which sex to model if only a one sex could be considered. Females were chosen because their reproduc-tive contribution is easier to measure.

	both sexes must be present if the young are to be raised successfully, and
	males may be limiting (male raptors are smaller than females, and have higher mortality rates).