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Here we use a variant of the second approach to address the importance of the components to total reproductive output. We contrast results for three populations, selected because they represent a variety of life histories and geographic distributions and because some pertinent information is available. The populations are Mallards in the Prairie Pothole Region, Snow Geese in the Arctic, and Wood Ducks in the southern United States. To perform the comparison we 1) develop a rudimentary model; 2) use available information on parameter values or, lacking any, make our best guesses; 3) propose "extreme" values of parameters, approximately the smallest and largest values to be expected during a 10-year period (a more detailed approach was taken for the Snow Goose); 4) determine the reproduction anticipated under each combination of parameter values; 5) perform a regression analysis relating the reproduction to each parameter; and 6) compare the relative importance of each parameter for the three populations. We then discuss the environmental or other factors that cause variation in the parameters for each population.

1. A Model

where there are K distinguishable age classes, and R_i is the production rate for females in age class i. This value can be further decomposed according to nesting attempts, giving

where D_i is the proportion of females of age class i that attempt to breed, and Q_ij is the production from the j^th nesting attempt of females in age class i. This allows a maximum of J nesting efforts in a breeding season.

The production from a particular nesting attempt itself involves several factors and can be expressed as the product:

where A_ij is the probability that a female aged i will make the j^th nesting attempt in a breeding season, C_ij is the average clutch size of the j^th nesting attempt by a female of age i, H is the nest success rate, E is the survival rate of eggs in successful nests, and B is the survival rate of young.

Notice that most parameters are indexed by age of the female and nesting attempt, because age and attempt were shown to influence them. Effects of age and attempt on nest success rate (H), survival rate as eggs (E), and survival rate as young (B) have not been demonstrated clearly, although the model could easily be generalized to accommodate those effects.

2. Influence of the Components

Most components enter multiplicatively into the equation for production. For example, the total young produced by individuals in age class i can be expressed as

The components F_i, D_i, Σ A_ijC_ij, H, E, and B are, in a sense, equally influential; a 10% increase in one will produce the same change as a 10% increase in another. There are two subtle variations, however. One is the nature of age-related changes, particularly in the proportion attempting to breed (D_i), nesting persistence (A_ij), and clutch size (C_ij). The other departure from multiplicity involves changes with nesting attempt for nesting persistence (A_ij) and clutch size (C_ij).

3. Example -- Mallards in the Prairie Pothole Region

Nesting probabilities (A_ij) are more complicated to calculate. Consider first the rates for adults: A₂₁ = 1.00; all hens that breed will obviously make the first nesting attempt. Subsequent values depend on the outcome of previous attempts. Because a hen cannot attempt a j^th nest unless she tried the (j-1)^st, we can decompose the probability of attempting the j^th clutch as follows, conditional on the outcome of the (j-1)^st attempt:

		A2j = Pr {make nesting attempt j}
		    = Pr {make nesting attempt j | attempt j-1 failed}
		      × Pr {attempt j-1 made and failed}
		      + Pr {make nesting attempt j | attempt j-1 succeeded}   	
		      × Pr {attempt j-1 made and succeeded}
		    = rj (1 - H) A2,j-1 + qj H A2,j-1,    (5)

where r_j is the conditional probability of attempting a j^th clutch given the previous attempt failed, and q_j is the probability of attempting a j^th clutch given the previous attempt succeeded. We take r₁ = 1 by convention. Since we assume for Mallards that double brooding does not occur, q_j = 0 for all j. Solving equation 5 recursively, we get

Following Cowardin and Johnson (1979), who proposed values for r_j based on the criteria that these probabilities should decrease with j (subsequent renesting efforts become less likely) and vary inversely with H (if clutch survival is high, destruction is likely to occur later in the nesting cycle, and renesting is less probable), we suggest for wet years

			r1 = 1
			rj = α(1 - H)/(j - 1)  for j > 1,

where α is an index to nesting intensity, about one in wet or normal years and lower in dry years. Using these values in equation (6) yields

	A21 = 1
	A2j = αj-1 (1 - H)2(j-1)/(j - 1)!	for j = 2,3,4,5.

We take α = 1.1 in a wet year, α = 0.9 in an average year, and α = 0.5 in a dry year (Cowardin, Gilmer, and Shaiffer 1985). As an illustration, consider an average year (α = 0.9) with nest success H = 0.12. Then A₂₁ = 1.00, A₂₂ = 0.70, A₂₃ = 0.24, A₂₄ = 0.06, and A₂₅ = 0.01. These values total 2.01, indicating an average of about two nest attempts per breeding adult female.

Yearling Mallards are less persistent in renesting than older hens. This difference can be accommodated simply by multiplying the A_ij (j > 1) values of adults by, say, 80%. We then obtain for an average year: A₁₁ = 1.00, A₁₂ = 0.56, A₁₃ = 0.19, A₁₄ = 0.05, A₁₅ = 0.01. These total 1.81.

Clutch size is larger for older Mallards than yearlings and decreases with nesting attempt. We propose the following clutch sizes under average conditions: C₁₁ = 10, C₁₂ = 9, C₁₃ = 8, C₁₄ = 7, C₁₅ = 6; C₂₁ = 11, C₂₂ = 10, C₂₃ = 9, C₂₄ = 8, C₂₅ = 7. Combining these values with the {A_ij} above give average clutch sizes in an average year of 9.38 for yearlings and 10.30 for adults. If wetland and feeding conditions are excellent, clutch sizes will be increased by 1; if those conditions or weather is very unfavorable, or nest parasitism considerable, clutch sizes will be reduced by 1.

Nest success rates of Mallards in the prairie pothole region are highly variable (Klett, Shaffer, and Johnson 1988), but we take H = 0.12 as a typical value (Greenwood et al. 1987), H = 0.40 as a typical high value, and H = 0.05 for a not-uncommon low value.

Survival of eggs in successful nests is generally high; we use E = 0.91, as the average of six studies reported in Table 14-2. Variation does not seem appreciable, so we take E = 0.96 and 0.86 as extreme points. Brood survival is lower, however, and we take the average of three studies included in Chapter 12 of this volume, plus the 0.68 obtained by Lokemoen, Duebbert, and Sharp (1990) to be B = 0.47. Extreme values were B = 0.28 and B = 0.68, which we will use.

To assess the influence of variation in the parameters described above, we computed equation 4 for each combination of three levels (high, typical, and low) of each of the six parameters (D, α, C, H, E, B). We assumed a population consisting of 44% yearlings and 56% older hens; these fractions represent a population with the stable age distribution based on survival rates given in Table 14-1. We subjected the resulting 729 observations to a regression analysis (Proc REG of SAS Institute 1985). The relative influence of each parameter was assessed by comparing standardized regression coefficients. Results indicated that nest success (H) and brood survival (B) were the most influential components of production, with standardized regression coefficients of 0.76 and 0.40, respectively (Table 14-4). Breeding incidence (D) was the next most influential, followed by nesting intensity (α), clutch size (C), and survival of individual eggs (E).

For Mallards in the Prairie Pothole Region, nest success is determined primarily by predation, with wetland conditions or precipitation also playing a role. The determinants of brood survival are largely unknown but are thought to include predation, weather, food supplies in wetlands, and possibly disease; some effects may be density-dependent. Breeding incidence in Mallards may vary according to wetland conditions early in the breeding season. Thus predation and wetland conditions likely exert the greatest influence on productivity of prairie Mallards.

4. Example -- Snow Geese in the Arctic

From Prevett (1972), Rockwell, Findlay, and Cooke (1983), and Cooke and Rockwell (1988), we estimate the proportion in each age class that breeds in a typical year to be D₁ = 0, D₂ = 0.35, D₃ = 0.60, D₄ = 0.90, and D₅ = 0.95. Breeding incidence may be diminished by adverse weather on the breeding grounds (Uspenski 1965), drought in the prairies during spring migration (Davies and Cooke 1983), and possibly high density or a lack of nest sites (Hanson et al. 1972). We took these D values as random deviates from beta distributions with parameters p and q. Values of the parameters were D₂: p = 3.5, q = 6.5; D₃: p = 6, q = 4; D₄: p = 18, q = 2; D₅: p = 38, q = 2. These gave deviates averaging 0.35, 0.60, 0.90, and 0.95 for the respective age classes, as well as appropriate variation.

Snow Geese ordinarily make but one nesting attempt in a breeding season, but limited continuation nesting may occur if the first clutch is lost early in laying (Barry 1967; F. Cooke, pers. commun.). We accordingly permit J = 2 attempts, and set nesting probabilities to be A_i1 = 1 for all i and, following the reasoning we used for Mallards, A_i2 = 0.5D_i(1 - H)². This formulation allows slightly more continuation nesting for older geese and when breeding conditions are favorable (the inclusion of D_i), when clutch destruction is likely to occur earlier (the dependence on H), but lower renesting overall (the 0.5).

Clutch size of Snow Geese is most markedly affected by the seasonal decline and age of the female. In addition, clutches may be smaller in years of poor conditions along the spring migration route or when breeding densities are high (Findlay and Cooke 1983, Cooch et al. 1989). For initial clutch sizes in a typical year we use averages from Rockwell, Findlay, and Cooke (1983; Table 1): C₂₁ = 3.4, C₃₁ = 3.8, C₄₁ = 4.1, and C₅₁ = 4.4. The standard deviation among years was 0.28, irrespective of age, and clutch sizes had an approximately normal distribution. Thus we took clutch size to be normal with mean appropriate for age and standard deviation 0.28. Clutch sizes of continuation nests will average much smaller, we assume by an average of two eggs.

Nest success rates of the La Perouse Bay Snow Geese were remarkably high. From data provided by E. Cooch and F. Cooke (pers. commun.), the average was about 0.73 (standard deviation = 0.04), and the yearly values were such that log(0.80 - nest success) was distributed roughly normally with mean -2.81 and standard deviation 0.62.

Egg loss at La Perouse Bay was minor, egg survival averaging 0.95 (standard deviation = 0.015) during 1973-85 (E. Cooch and F. Cooke, pers. commun.). We found that log(egg survival) was approximately normally distributed with mean -0.055 and standard deviation 0.0155. Additionally, an average of 0.91 (standard deviation = 0.032) of the eggs hatched. We found that log(hatch rate - 0.85) was nearly normally distributed with mean -2.98 and standard deviation 0.52.

Gosling survival in the La Perouse Bay study averaged 0.75 (standard deviation 0.056) during 1973-85 (E. Cooch and F. Cooke, pers. commun.). We generated gosling survival rates from a normal distribution with those parameters.

We computed equation 4 for 1000 random sets of variates (D, C, H, E, B) independently generated from distributions described above. We assumed a population consisting of 21% yearlings, 16% two-year-olds, 13% three-year-olds, 10% four-year-olds, and 40% older hens. These values were based on a population with stable age distribution derived from survival rates of 40.7% for young-of-the-year and 79.5% for older birds (based on Richards 1986). A comparison of standardized regression coefficients indicated that brood survival, clutch size, and nest success were the most influential variables, followed by breeding incidence and egg survival (Table 14-4).

For Snow Geese, brood survival is influenced by predation (Bousfield and Syroechkovskiy 1985, Cooke and Rockwell 1988) and possibly starvation (Cooke and Rockwell 1988). Clutch size is reduced in years with delayed nesting and when conditions along the migration route are poor (Cooke and Rockwell 1988). Nest success is mostly affected by predation and abandonment, the latter possibly due to high densities of birds (Owen and Wells 1979), extreme weather conditions (Harvey 1971, Cole 1979, Cooke and Rockwell 1988), and reduced nutrient reserves of females (Harvey 1971, Ankney and MacInnes 1978). Breeding incidence is affected by weather conditions at the onset of the breeding season, drought along the spring migration path (Davies and Cooke 1983), and possibly high density or lack of nest sites (Hanson et al. 1972). Thus, productivity of the Snow Goose appears to be most markedly affected by weather and predation.

Cooke and Rockwell (1988) concluded that survival, especially during the first year of life, was the major source of variation in lifetime reproductive success of Snow Geese at La Perouse Bay. Most mortality occurs during the hunting season, so their results are not directly comparable to ours, which are restricted to the breeding season.

5. Example -- Southern Wood Ducks

Nesting probabilities differ from those of the Mallard in two regards: a longer nesting season and ample food supplies may encourage repeated renesting (Fredrickson and Hansen 1983), and renesting after a successful breeding effort occurs fairly regularly (Fredrickson and Hansen 1983). Because renesting rates will not decline as fast as those of the Mallard, in Equation 5 we take

			A21 = 1
			A2j = {α(1 - βH)2}j-1/(j - 1)!	   for j = 2,3,4,5.

Again we can view α as a measure of nesting effort, and β essentially indicates how much greater nesting effort would be for Wood Ducks than for Mallards under similar circumstances. We take α = 1.10, 0.90, and 0.50 for excellent, average, and poor conditions, respectively, and set β = 0.70. Alpha is likely to vary mostly with density of breeding birds, relative to the number of available nest sites. Under average conditions (α = 0.90 and nest success H = 0.60) we get the following values: A₂₁ = 1.00, A₂₂ = 0.30, A₂₃ = 0.046, A₂₄= 0.0046, A₂₅ = 0.00035.

Yearling Wood Ducks begin nesting much later than older birds (Grice and Rogers 1965, Hansen 1971), so would be able to make fewer attempts during a season. We accommodate this difference by multiplying nesting persistence values of adults by 70%. We then obtain in a typical year: A₁₁ = 1, A₁₂ = 0.21, A₁₃ = 0.032, A₁₄ = 0.0032, A₁₅ = 0.00024.

Wood Ducks, especially in situations with artificial nest structures, are subject to considerable intra- and interspecific parasitism. This dump nesting is more common early in the breeding season (Grice and Rogers 1965), when nest sites are limited (Clawson, Hartman, and Fredrickson 1979) and possibly when nest structures are congregated and highly visible (Semel and Sherman 1986). We consider two levels of dump nesting, none and high. Yearling Wood Ducks initiate nesting later than older birds, so are less likely to be parasitized. We suggest the following probabilities that a nest under the high dump nesting scenario will be parasitized: P₁₁ = 0.70, P₁₂ = 0.50, P₁₃ = 0.30, P₁₄ = 0.10, P₁₅ = 0.00; P₂₁ = 0.90, P₂₂ = 0.70, P₂₃ = 0.50, P₂₄ = 0.30, and P₂₅ = 0.10. Values for late clutches are inconsequential because very few such clutches are laid. Intraspecifically parasitic birds are considered members of the breeding population.

Clutch sizes of Wood Ducks are confused by dump nesting. Normal clutches of Wood Ducks average larger than those of Mallards and probably decline more slowly with repeated nesting attempts (Clawson, Hartman, and Fredrickson 1979). We take for normal clutches: C₁₁ = 11, C₁₂ = 10.5, C₁₃ = 10, C₁₄ = 9.5, C₁₅ = 9; C₂₁ = 14, C₂₂ = 13.5, C₂₃ = 13, C₂₄ = 12.5, and C₂₅ = 12. Dump clutches are taken to be 1.75 times as large as normal. We assess response to variation in clutch size by changing averages for normal clutches by ± 1 egg.

Success rates of normal nests are generally high; when Bellrose (1980:190) summarized apparent rates from 22 studies; the median was 73% and the range from 32% to 95%. Because these values may be biased high (Mayfield 1961), we take H = 0.60 as a typical value, and H = 0.40 and 0.80 as extremes likely to be encountered during a 10-year period. Dump nests are more likely to be abandoned, so for them we take H to be 80% of the value of normal nests.

Egg survival in successful nests tends to be somewhat lower than among Mallards, especially in dump nests. We take the values of Clawson, Hartman, and Fredrickson (1979) for averages, E = 0.78 for normal nests and E = 0.63 for dump nests (81% of the normal value). Based on values they recorded during nine years of study, we use extremes of E = 0.64 and E = 0.86 for normal nests and assume survival of eggs in dump nests is 81% of the value for normal nests.

We found 10 estimates of brood survival rate (Grice and Rogers 1965, McGilvrey 1969, Baker 1971b, Brown 1973, Ball et al. 1975, and Haramis and Thompson 1984), which, except for one outlier, ranged from 41% to 65%, with a median of 54%. We use these as extremes and typical value in our analysis.

We considered a population consisting of 49% yearlings and 51% older hens, the stable age distribution of a stationary population with adult female survival rate estimates obtained from F. A. Johnson et al. (1986). We computed equation 4 with two or three levels of each of seven parameters; dump nesting was the additional parameter not included for the Mallard. A comparison of standardized regression coefficients indicated that brood survival and nest success were the most influential parameters, followed by breeding incidence and survival of individual eggs (Table 14-4).

The major identified cause of brood mortality in southern Wood Ducks is predation; survival rates may vary substantially from early- to late-hatched broods (Grice and Rogers 1965). Nest success is largely dependent on predation, which in managed situations may increase with the breeding population, and on abandonment, which also is density-dependent. Breeding incidence appears to be influenced by population density, and survival of individual eggs is depressed in dump nests, which are most common in high-density breeding situations. Thus, the dynamics of southern Wood Ducks breeding in managed areas seem most strongly influenced by predation and some density-dependent processes.

6. The Three Populations Contrasted

Predation and wetland conditions have the greatest impact on important parameters of prairie Mallards. Snow Geese are mostly affected by weather and predation and managed southern Wood Ducks by predation and density-dependence.

Although the concept underlying our comparative approach is similar to that of key factor analysis, we emphasize that our intention here is illustration only. Our analyses and comparisons are entirely dependent on our assumed model and the three values (one typical and two extreme, except for Snow Geese) that we used for each component. Whenever the extreme values differed substantially from the typical value, then the variation associated with that variable had to be large. In many instances our typical and extreme values were little more than educated guesses, and even when estimates were available, we did not incorporate associated estimates of sampling variation into the exercise (Manly 1977). Perhaps most importantly, we know nothing of the true joint distribution of the component variables over time. Certainly such a distribution would look far different from that created by our approach based on all possible combinations of variables. Data needed to conduct accurate analyses of the type illustrated here could come only from long-term studies in which each component is carefully estimated each year, such as that of F. Cooke.

Because of these caveats, we do not claim that most of the variation in reproductive output is actually associated with the components identified by this exercise. Nonetheless, we believe that the indicated components are likely to be important. In addition, we strongly suspect that different variables are important for different species and populations, as the analyses suggested.