Line transect detection function fitting solution

Author

Centre for Research into Ecological and Environmental Modelling
University of St Andrews

Modified

November 2024

Solution

Estimation of duck nest density by hand

In this practical, we plot a histogram of line transect data and estimate a detection function. The data were collected during a line transect survey of duck nests in Monte Vista National Wildlife Refuge, Colorado, USA: twenty lines of 128.75 km were specified and a distance out to 2.4m was searched and the perpendicular distances of detected nests were recorded and summarised (Table 1).

Table 1. Frequency of duck nests detected in perpendicular distance bands (metres)
Distance.band Frequency
0.0-0.3 74
0.3-0.6 73
0.6-0.9 79
0.9-1.2 66
1.2-1.5 78
1.5-1.8 58
1.8-2.1 52
2.1-2.4 54
  1. Histogram of detected nests (black) overlaid with the estimated detection function (red) is shown below.

  1. To estimate the area under the curve, I read off the heights of the mid points of my fitted curve (red) as follows: 75, 74, 72, 70, 66, 62, 58, 53. Therefore, my estimate of area under the curve is:

\[ Area_{curve} = (75+74+72+70+66+62+58+53) \times 0.3 = 530 \times 0.3 = 159 \] There are lots of other ways to work out the area under a curve, e.g. counting the number of grid squares under the curve on your graph paper or using the trapezoidal rule.

\[Area_{rectangle} = height \times width = 75 \times 2.4 = 180\]

Hence, my estimate of the proportion of nests detected in the covered region is:

\[\hat P_a = \frac{159}{180} = 0.883\]

  1. How many actual nests were there in the covered area? I saw 534 nests, and I estimate the proportion seen is 0.883, so my estimate of nests in the covered region is:

\[ \hat N_a = \frac{n}{\hat P_a} =\frac{534}{0.883} = 604.7 \textrm{ nests in the covered area}\] This estimate is for a covered area of \(a = 2wL = 2 \times (\frac{2.4}{1000}) \times 2575 = 12.36\) km\(^2\).

  1. I therefore estimate nest density as:

\[\hat D = \frac{\hat N_a}{2wL} = \frac{604.7}{12.36} = 48.9 \textrm{ nests per km}^2\]