Appendix C. Descriptions of bison road travel and bison off-road travel response variables.

*Road Travel Response Variable ( _{ij})*

To offer more insight into how we defined and calculated our road travel response variable (_{ij}) we examine each component of the response variable and provide plots of the data. Using data from our bison road use surveys we calculated the response as _{ij} = _{ij}_{ij} for each time interval (i) and year (j) to quantify the amount of bison road travel in units of bison groups observed per 100 km of road surveyed. We calculated _{ij} by summing the number of bison groups observed traveling on roads for the ij^{th} period and dividing by the total distance of road surveyed for that period. A plot of bison groups observed on roads over time for each period and for all eight years is presented in Fig. C1. Since varying amounts of survey effort between periods could lead to observing more (or fewer) groups, we divided the number of groups observed per period by the distance of road surveyed per period to obtain _{ij}. A plot of _{ij} vs. time is provided in Fig. C2.

Using _{ij} alone as the response variable was not sufficient because the size of traveling bison groups can vary throughout the season as bison migrate into the Madison-Firehole area (Fig. C3) and all groups would be treated equivalently whether the size of the group was one or 50 bison. Based upon our field observations we felt that _{ij} alone did not provide an accurate representation and quantification of the variability in the amount of bison road travel between periods. Therefore, we multiplied _{ij} by a unitless road use weighting factor (_{ij}) to account for the potential variability in group size over time. We calculated _{ij} as the total number of individual bison in road traveling groups for the ij^{th} period divided by the total number of individual bison documented in road traveling groups for the entire season. A plot of _{ij} over time is presented in Fig. C4. Our final road travel response variable (_{ij}) accounted for the number of traveling road groups observed, variability in group size, and survey effort as depicted in Fig. C5.

Given this background on how we calculated our response variable, we now examine the alternatives that we did not use. Two possibilities for the response were the number of individual bison traveling roads (Fig. C6) or the number of individual bison traveling roads per survey effort (Fig. C7). However, these would have given an overestimation of the amount of road travel because bison in groups do not travel independently. In large groups it is often the case that a few bison will begin traveling and the remainder of the group will follow until the lead animals stop. By this rationale we did not calculate
_{ij} as the average group size per period because this _{ij}, when multiplied by _{ij}, would have given a response variable of number of individual bison per survey effort.

*Off-Road Travel Response Variable ( _{ij})*

We defined and calculated our off-road travel response variable (_{ij}) using the same rationale as for _{ij}. Using travel data from our ground distribution surveys, we quantified the amount of bison off-road travel for each period by defining an off-road travel response variable as _{ij}, having units of bison groups observed traveling off-road per survey. We defined the off-road response as _{ij} = _{ij}_{ij}, where a_{ij} is the total number of bison groups observed traveling off-road per ground distribution survey for the ij^{th} period, calculated as the sum of off-road/off-trail and off-road/on-trail traveling groups. A plot of _{ij} over time for each period and for all eight years is presented in Fig. C8.

Again, using _{ij} alone as the response variable was not sufficient because the size of off-road traveling bison groups can vary throughout the season (Fig. C9). Therefore, we multiplied _{ij} by a unitless off-road travel weighting factor (_{ij}) to account for the potential variability in group size over time. We calculated _{ij} as the number of bison observed traveling off-road during ground distribution surveys during the ij^{th} period divided by the total number of bison observed traveling off-road for the entire season during ground distribution surveys. A plot of _{ij} over time is presented in Fig. C10. Our final off-road travel response variable (_{ij}) accounted for the number of off-road traveling groups observed, variability in group size, and survey effort as depicted in Fig. C11.

Appendix C Figures:

FIG. C1. The number of bison groups observed traveling on roads for each period and year during bison road use surveys from 1997–1998 through 2004–2005. |

FIG. C2. The number of bison groups observed traveling on roads per 100 km of road survey effort (_{ij}) for each period and year during bison road use surveys from 1997–1998 through 2004–2005. |

FIG. C4. The road travel weighting factor (_{ij}) for each period and year as calculated using data from bison road use surveys from 1997–1998 through 2004–2005. |

FIG. C5. The bison road travel response variable (_{ij}) for each period and year as calculated using data from bison road use surveys from 1997–1998 through 2004–2005. |

FIG. C6. The number of individual bison observed traveling on roads for each period and year during bison road use surveys from 1997–1998 through 2004–2005. |

FIG. C7. The number of individual bison observed traveling on roads per 100 km of road survey effort for each period and year during bison road use surveys from 1997–1998 through 2004–2005. |

FIG. C8. The number of bison groups observed traveling off-roads per survey effort (_{ij}) for each period and year during bison ground distribution surveys from 1997–1998 through 2004–2005. |

FIG. C10. The off-road travel weighting factor (_{ij}) for each period and year as calculated using data from bison ground distribution surveys from 1997–1998 through 2004–2005. |

FIG. C11. The bison off-road travel response variable (_{ij}) for each period and year as calculated using data from bison ground distribution surveys from 1997–1998 through 2004–2005. |