Calculation of geographic distance
Because exceptions in the correlation between genetic distance and
geographic distance would indicate the existence of barriers to gene
flow, we tested isolation-by-distance (IBD; Wright 1943) among
populations of the five species. To estimate geographic distance across
the varied landscape of the mountains, we examined the effects of the
rugged montane terrain on dispersal patterns by hypothesizing scenarios
of likely dispersal routes through, over, and around the mountains. Four
dispersal scenarios for birds were hypothesized, resulting in four
variants of dispersal (geographic) distance: (1) spheric distance,
assuming that birds fly over mountains regardless of landscape; (2)
optimized route cost-distance, with a penalty for traveling above or
below the normal range of elevation for each species, assuming birds
prefer to disperse within a specific elevational range; (3) optimized
route cost-distance, with a penalty only for dispersing above a favored
elevational range, assuming birds detour only to avoid high elevation;
and (4) optimized route cost-distance, with a penalty only for
dispersing below the lower threshold of a favored elevational range,
assuming birds detour only to avoid low elevation.
Using the terrain distance function costDdistance of thegdistance package in R (Van Etten 2017), we computed the
geographic cost-distance. This was achieved first by calculating the
three-dimensional distance among adjacent grid points using the
Pythagorean theorem applied to two
points—the elevational difference and the hypotenuse of the right
triangle they formed—and second by using transition to transfer
the raster matrix into a pairwise conductance distance matrix. In the
case of n raster grids, the raster matrix transfers to ann x n matrix in which only conductible grids have
meaningful values. In our customized penalty function, the
conductibility between two grids was recalculated by reassigning
elevational differences with an extra penalty. The penalty was the sum
of the out-of-bounds parts of the elevation of each point multiplied by
a penalty factor. Calculation of the reciprocal of conductance, L,
between two points was as follows:
L = sqrt (D2 + p2) (1)
where p is the distance between two points, and D is the reassigned
elevational difference. The reassignment was computed as:
D = d + f \(\ \times\) (abs (k1) + abs (k2)) (2)
where d is the initial elevational difference between two points, and k1
and k2 are out-of-bounds elevations of the two points. Abs is the
absolute value, and f is the factor used to decide the strength
of the penalty for traveling out of the elevational bounds. The
assignment of f is described below.
For scenarios with no penalties, spheric distances were calculated using
pairwise latitude and longitude with the function pointDistancefrom the raster package in R (Hijmans, 2019, Karney, 2011). For
those distances with applied penalties, the shortest distance across the
landscape was calculated using an elevational raster map from the
GMTED2010 database (Danielson and Gesch 2011). We set the preferred
elevational range of each species based on empirical records from the
literature (Table 1; Zhao 2001), then calculated the detour routes on
the basis of this range. Because a direct route may not conform entirely
to a species’ elevational range, we added a penalty to sections of the
routes outside the species’ range. These are presented as the ffactor in Eq. (2). The f values were set to 0.001, 0.01, 0.1, 1,
and 100 times the deviation from the upper or lower bound of the
preferred elevation before searching for the least costly (most
efficient) dispersal route. Least-cost routes were plotted to see their
displacement as the f factor changes. It should be noted,
however, that the penalized distance cannot be considered as an actual
distance or be compared to values of other scenarios; the cost-distance
was only penalized outside a certain range according to the hypothetical
preference of the bird, which differs among scenarios, so it is not
simply proportional to penalty level.