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This process can be represented as follows:

Number Fledged                                                       
                  Pairs    Nests   Hatched Nests     Fledged        
    = Wetlands x ------- x ----- x ------------- x ------------  (1)
                 Wetland   Pairs       Nest        Hatched Nest     

We used data for the mallard for this exercise, because it was of special interest in other research that our study supported (Brace et al. 1987) and was generally the most numerous duck species on all area-years. We lacked information on the survival of ducklings on areas we studied, so the final component could not be determined. This left

Hatched nests                                         
                  Pairs    Nests   Hatched Nests     
    = Wetlands x ------- x ----- x -------------  (2)
                 Wetland   Pair        Nest          

By estimating, for each half-area-year, the number of Hatched Nests and the 4 components on the right-hand side of equation (2), we determined the relative importance of each of them in relation to the variability of Hatched Nests.

An alternative formulation is

Hatched Nests                           
             Nests   Hatched Nests     
    =Pairs x ----- x -------------  (3)
             Pair        Nest          

Not all of these variables were directly estimated in our field study; some had to be calculated from other variables. The number of Hatched Nests was determined by taking the number of successful mallard nests found on the area searched within each half-area-year and scaling it upwards to account for the amount of nesting habitat on that half-area-year that was not searched.

Pairs and Wetlands were measured directly. For this analysis we used the total number of temporary, seasonal, and semipermanent wetlands that were wet in May.

The Nest-per-Pair value required an estimate of the total number of nests initiated on a half-area-year, as follows:

1. Nest success rate estimates the number of successful nests divided by the total number of nests initiated.
2. Nest success was independently estimated by the Mayfield method (Johnson 1979).
3. The total number of nests initiated thus can be estimated as the number of successful nests divided by the nest success rate (Miller and Johnson 1978).
4. The resulting value is divided by the number of Pairs to yield the Nests-per-Pair value.

Some derived values were clearly out of line because mallard populations on each half-area-year were not closed (i.e., birds could freely move into and out of each area) and, possibly, because of errors in estimating the quantities. For example, on the west half of the Ceylon Study Area in 1984, only 1 successful mallard nest was observed. The estimated hatch rate of mallard nests was only 0.002, so the single successful nest is estimated to represent 1 ÷ 0.002 = 500 initiated nests. This is far too many for the 20 mallard pairs estimated to be on that area. Accordingly, we constrained the number of nests per pair to be no greater than 4. This constraint was imposed for only a few (n = 7) half-area-years.

We determined the relations between Hatched Nests and each component by examining bivariate plots and calculating correlation coefficients. Because of the multiplicative form of equation (2), logarithms were taken to yield an additive model:

log(Hatched nests)                           
    = log(Wetlands) + log(Pairs/Wetland)     
      + log(Nests/Pair)                      
      + log(Hatched Nests/Nest)           (4)

log(Hatched Nests) = log(Pairs) + log(Nests/Pair)    
                     + log(Hatched Nests/Nest)    (5)