Estimating the Number of Distinct Darter Species in the Big Darby Watershed Using Percolation Theory

Erin Kingdom, Yvonne Vadeboncoeur, Allen Hunt
Wright State University
Dayton, OH 45435

The viability of fish species in inland waterways depends on the connectivity of the ranges of subpopulations. The ranges of such subpopulations extend up and downstream from their preferred spawning grounds due to their seasonal migrations. If these ranges do not overlap sufficiently, the species will either go extinct or develop into subspecies. Because stream networks can be modeled as fractals, it should be possible to address the question of the viability of species within the framework of percolation on a fractal, or to lowest order, percolation on a Bethe lattice. Treating a river as a Bethe lattice with Z=3 leads to a percolation probability of pc=1/(Z-1)=1/2. We then consider specifically darters in the Big Darby watershed in the state of Ohio, USA. We find that typical link lengths in this watershed are ca. L=8km; quoted migration distances of Etheostoma variatum for example, can be as high as R=5 km. The concentration, n, of available habitats is not yet established. In a very simple calculation we find that for this species p= the smaller of nR/L and 1. The typical habitat length or spawning territory for a darter varies for each species, but generally spawning occurs upstream and over an area around 40-50m depending on the length of the riffle or habitat unit utilized. Within this framework, we can find a rough estimate for the total number of darter species that could be supported in the watershed, provided that the few numbers we have can be applied for all species identically. The number of independent spawning areas in a link is thus 8000/50=160. If pc were 1, then it would be necessary for a minimum of approximately two such spawning areas to be found in each link, reducing the number by a factor of 2; pc=1/2 means that the maximum number of species that could be found is 160. Many factors that have not been included may affect the ultimate predicted limit of species: these include the assumption (which we will ultimately avoid) that the ranges are distributed evenly, and that spawning habitats are not shared, and we expect that this value is too high. Observation suggests that the present number of species is 18, almost a factor 10 lower, though numbers may well have been higher prior to the European immigration. Clearly the minimum number given by percolation theory, 1, if only one distinct habitat is available, is the result expected if the stream morphology is homogenized.