 ## Boundary Estimation

Delineation of population boundaries is important for:

Real boundaries are irregular (Fig.1); they have "islands", "lakes", and "folds". Irregular boundaries are very difficult to analyze. For example, it is impossible to estimate the distance between boundaries in two consecutive years. Fig.1. Regular and irregular boundaries

A regular boundary can be defined as a boundary that crosses any perpendicular to the general boundary direction only once (Fig.1). In a mathematical sense, the regular boundary is a function in the rotated coordinate system with the X-axis parallel to the general boundary direction. It is possible to use a circular general boundary in a polar coordinate system.

The problem is finding a regular boundary which fits the empiric boundary in the best way. Sharov et al. (1995) suggested a method of boundary estimation called "Best Cell Classification Method" (BC):

• Use a raster map (grid) of population presence/absence. If data are quantitative, then population is considered present if population numbers are above some threshold, T.
• Subdivide the area into thin strips perpendicular to the general boundary direction (Fig. 2).
• In each strip, select a point at which the number of misclassified grid cells as occupied vs. non-occupied is minimal (Fig. 3).
• If there is no minimum or if the minimum point is too close to the end of the strip, then this strip is ignored (treated as a "missing point").
• Connect best boundary points in adjacent strips to form a boundary line. Fig. 2. Best cell classification method for estimating boundaries. Red circles are occupied grid cells, and empty circles are non-occupied. The blue line is the boundary. Fig. 3. Finding a point in a strip that minimizes the number of misclassified grid cells as occupied vs. non-occupied.

This algorithm is not sensitive to the exact orientation of strips. That is why it is possible to draw them perpendicular to the general boundary direction determined by eye. General boundary direction can be estimated more accurately using the following iterative procedure:

1. Select preliminary boundary direction
2. Estimate boundary points using the BC-method described above
3. Estimate regression from boundary points in the rotated coordinate system with X-axis coinciding with preliminary general boundary direction
4. Use regression line as a new general boundary direction in the next iteration
5. Stop iteration when regression slope is smaller than required precision level.
Several modifications of this method are possible. First, different weights can be assigned to misclassified grid cells that are occupied compared to non-occupied. This may be needed if the population is highly fragmented in space. Second, the BC method can be used with a polar coordinate system (this may be needed if boundaries are approximately circular or semi-circular). In a polar coordinate system, thin sectors are used instead of strips as shown in Fig. 4. Fig. 4. Best cell classification method for estimating boundaries in polar coordinates. Red circles are occupied grid cells, and empty circles are non-occupied. The blue line is the boundary.

The BC method has been used for more than six years in the Slow the Spread project that is targeted at reducing the rate of population expansion of the forest pest insect called gypsy moth (Lymantria dispar, Lepidoptera: Lymantriidae). We used male moth captures in pheromone traps to monitor the progression of the population front. Population boundaries were estimated for thresholds of 1, 3, 10, 30, 100, and 300 moths/trap.
See: maps of gypsy moth distribution.

#### References

Sharov A.A., E.A. Roberts, A.M. Liebhold, and F.W. Ravlin. 1995. Gypsy moth (Lepidoptera: Lymantriidae) spread in the Central Appalachians: Three methods for species boundary estimation. Environ. Entomol. 24: 1529-1538. Get a reprint (PDF)
Sharov, A. A., B. C. Pijanowski, A. M. Liebhold, and S. H. Gage. 1999. What affects the rate of gypsy moth (Lepidoptera: Lymantriidae) spread in Michigan: winter temperature or forest susceptibility? J. Agric. and Forest Entomol. 1: 37-45.

You can download software here. It contains executables for DOS and the C-code. Also it has several tools that may be helpful. If you use this program, please, acknowledge Alexei Sharov and reference the paper indicated above: