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.