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<br />densities are locations that fall in the top 10% of density for the species being considered). This ' <br />analysis was performed by first measuring the nearest distance from high density backwater <br />locations to stocking locations (or high density pond locations), then measuring the nearest , <br />distance from backwater locations with the species absent to the same potential source locations. <br />Theoretically, if the high density backwater locations are closer to pond or stocking locations <br />than locations where the species was absent, this lends evidence to stocking events or pond fish <br />populations being sources of a particular species to critical habitat (Figures 2 and 3). Of course, <br />if there is no difference between these two proximity measurements, this does not mean that the <br />ponds or stocking events are not potential sources. Rather, the ponds and stocking events simply <br />are not spatially related in the way we may expect, and therefore are not likely to be identifiable <br />sources using spatial pattern analysis. The proximity analyses we performed were as follows: <br />1. Comparison of stocking events vs. high backwater densities. <br />2. Comparison of high density pond locations vs. high backwater densiries. <br />Geostatistical analyses ' <br />We used three primary geostatistical approaches to evaluate further potential spatial <br />patterns between potential sources of nonnative fish in the ISA and nonnative fish densities in <br />critical habitat. Two of these approaches expand on analyses described above by focusing more <br />formally on explicit spatial relationships between sources and backwater populations. The third <br />method concentrates explicitly within the backwater populations to describe how these <br />populations are distributed in the ISA, and how their distribution may influence the identification <br />of nonnative fish sources and direct nonnative fish control efforts. <br />All three of the techniques we describe below and employ here are based on the fact that <br />the correlation between fish densities at any two locations may vary depending on the distance i <br />between those locations. For example, we may expect a riverine location that is close to a <br />floodplain pond to have similar densities of similar species, IF that pond is a potential source to <br />the riverine location. Similarly, fish densities at locations that are very far apart will not, on 1 <br />average, be correlated to one another. These geostatistical approaches can be more powerful <br />than visual and proximity approaches because they consider all densities of fish (not just high <br />ones) and many inter-site distances (not just the closest locations). , <br />1. The modified h-scattergram <br />The h-scattergram is a plot of all pairs of measurements of the same attribute (i.e. <br />fish density) at locations separated by a given distance (h) in a particular direction <br />(Goovaerts 1998). We have modified this approach here by plotting the fish density at a <br />backwater location vs. the sum of the density in ponds (or number of ponds contaimng <br />that species) within a specified buffer distance from the backwater location. This <br />analysis was performed by first buffering the backwater locations, then summing the <br />ponds and species-specific fish density within 250, 500, 750, and 1000 m buffers <br />extending out from the backwater locations. If fish densities at these backwater locations <br />are related to densities of fish or ponds at particular scales, there should be relationships <br /> <br />10 , <br />