Northern Prairie Wildlife Research Center

We required two types of estimates: totals (e.g., number of ponds) and ratios of totals (e.g., average pond size = area of ponds per number of ponds). Where possible, we also wanted variances of these estimates.

To estimate totals, we treated our sample of 10.4-km² plots as a random sample and stratified by waterfowl management district and U.S. Fish and Wildlife Service landownership. The amount of area in a given landownership varied from plot to plot. We employed a combined ratio estimator (Cochran 1977) that made use of the fact that the totals that we wanted to estimate positively correlated with the amount of area in that landownership. Estimation was the computation of an average density by landownership from the sample plots in each waterfowl management district and the multiplication of that value by the total amount of area of that landownership in each waterfowl management district. The overlay of plots on 1:250,000 landownership maps, described under stratification, provided estimates of the amount of land by landownership in each stratum and in each waterfowl management district. The combined totals of all landownerships were estimated by summing the landownership totals. Totals by state and their estimated variances were calculated by summing estimates from constituent waterfowl management districts.

Variances of the landownership totals were calculated with the Formula 6.51 from Cochran (1977). An upper bound to the variance of the combined total landownership of all ownership was estimated from Formula 5.10 (Cochran 1977:93).

We divided appropriate estimates of population totals of one parameter by totals of another parameter to arrive at estimates of population ratios (e.g., pairs per pond). We do not report estimated variances of population ratios because exact formulas do not exist and formulas for approximations are complex (Mood et al. 1974).

**Estimates of Habitat Parameters**

Number of wetland basins, area of wetland habitat, and amount of upland nesting cover were obtained from the maps of 10.4-km² plots prepared by the National Wetland Inventory. The number of ponds and the area of wetland covered by water varied from year to year and were obtained from the classified video scenes.

**Number of Breeding Pairs**

To estimate the size of the breeding duck population, the area of each pond
(from the video scenes) was entered into the baseline regression of each species
to estimate the number of the breeding pairs. These by-pond estimates were summed
by plot and species and then expanded to waterfowl management districts as explained
under *Estimation Procedures*. The resulting estimates were based on the
assumption that the densities of ducks on ponds of the same size remains constant
from year to year and from area to area. An adjustment of the differences among
areas and years in each waterfowl management district was based on counts of
the sample wetland basins. A correction (** G**) was defined as:

To determine the number of predicted pairs in this equation, we required the
area of each counted pond. The video data did not identify each pond by number.
Therefore, we used the product of the estimated proportion of the basin that
contained water and the mapped area of the basin to estimate the area of each
counted pond. Finally, the estimated number of predicted pairs in the waterfowl
management district was multiplied by ** G** to correct for differences
in pair densities among years and among waterfowl management districts.

**Estimation of Production**

The number of young ducks recruited to the population in fall cannot be counted and, therefore, we used a model to calculate estimates. The model was deterministic and produced a result similar to that obtained from a stochastic mallard productivity model (Johnson et al. 1987). For our purposes, the deterministic technique was simpler and permitted more rapid calculation of estimates than the stochastic model. An example illustrates the method. To estimate the number of recruits in each landownership class, we essentially worked backwards from the general relation:

where 2 is a constant based on the assumption of equal sex ratio at hatch, R is the recruitment rate as defined by Cowardin and Johnson (1979), and n is the number of breeding pairs.

The recruitment rate was broken down into the components described by Cowardin and Johnson (1979) in the equation:

where H is hen success, 2 is a constant used because recruitment is defined in terms of only females, Z is the proportion of entire broods that survived to time of census, and B is the average brood size at fledging. For mallards, we used Z = 0.74 and B = 4.9 from Cowardin and Johnson (1979). Brood survival (Z) of 0.74 was also used for gadwalls, blue-winged teals, northern shovelers, and pintails. The mallard brood size of 4.9, adjusted for broods of size zero (Cowardin and Johnson 1979), was used as baseline to estimate Class-III brood (Callop and Marshal 1954) size of other species (Table 4).

Species |
Clutch Size^{a} |
Average Brood Size^{b} |

Mallard | ||

Gadwall | ||

Blue-winged Teal | ||

Northern Shoveler | ||

Northern Pintail |

Year |
Total Basins |
Ponds |
% |
A |

1977 | 224 | 11.3 | 0.5918 | |

1978 | 880 | 44.4 | 0.9108 | |

1979 | 1527 | 77.0 | 1.0444 | |

1980 | 507 | 25.6 | 0.8700 |

Hen success was determined from the relation:

where P is clutch success, * A* is an index to nesting effort (Cowardin
and Johnson 1979), and E is approximately 2.718. A relation between

where W is the percentage of basins that contained water in a waterfowl management district, estimated from the video data and divided by total number of basins as mapped by the National Wetland Inventory.

The number of produced recruits on a plot can be determined from the previous two equations if clutch success in each plot is available. Clutch success and the allocation of nests to the landownership classes were computed as in the following simplified example.

**Fig. 3** Hypothetical distribution of duck (Anatinae) habitats and
nests on a 10.4-km² plot with two landownerships and two habitats. Landownership
classes include land owned by the U.S. Fish and Wildlife Service (service)
and privately owned land (private) in the prairie pothole region of the United
States, 1987-90. **A** and **B** denote arbitrary land cover type.

We assumed that the distribution of habitat and landownership (Fig. 3) was
two habitats A and B and two landownerships, 259.2 ha of service landownership
and 777.6 ha of private landownership. We assumed that the ducks' preference
for habitat A was twice that for habitat B. If n nests were on 64.8 ha of habitat
B. the number of nests on the other tracts of land could be calculated as *area
of the tract* ÷ 64.8 x *preference Value* (Fig. 3). If clutch success
was assumed to be P_{A} = 0.50 and P_{B} = 0.10, we could calculate
the number of successful nesting attempts in habitats A and B (Fig. 4).

**Fig. 4** Numbers of successful nesting attempts by ducks (Anatinae)
on a hypothetical 10.4-km² plot with two landownership classes include
land owned by the U.S. Fish and Wildlife Service (service) and privately owned
land (private) in the prairie pothole region of the United States, 1987-90.

With these assumptions, the total number of nests (Fig.3) is 23n and the number of successful nesting attempts (Fig.4) is 7.9n. Therefore, the weighted clutch success in the entire plot (P) is:

To allocate recruits to the landownership classes, we calculated the proportion of clutches hatched in each landownership class.

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