The Big Bad Wolf Humans have viewed wolves as competitors, threats to personal safety, and symbols of evil throughout history. By the early part of the 20th century, gray wolves (Canis lupus) had been eradicated from 42% of their historic range in North America (Laliberte & Ripple, 2004). In Yellowstone National Park, gray wolves were hunted to local extinction by 1926, but were reintroduced in 1995 after a decades-long process involving biologists, politicians, ranchers, and the general public (Table 1). By the end of 2006, the wolf population in the park was at least 136 wolves in 13 packs (Smith et al., 2007). In this activity, high school students use mathematical models to explore how the presence of wolves buffers other carnivores and scavengers from the effects of climate change. By the end of the lesson, students should be able to: * define and give examples of keystone species. * demonstrate, using mathematical models, that ecosystems are more resilient to environmental change when they contain a full complement of species, including top carnivores. * recognize that math is a vital tool in scientific investigations. Background From an ecological perspective, it was important to restore the gray wolf to Yellowstone because it is a keystone species. Keystone species, which are usually top predators, affect their communities or ecosystems in a much larger way than expected based on abundance alone (Steneck, 2005). The presence, abundance, and productivity of a wide array of species in Yellowstone National Park are indirectly affected by interactions of wolves with elk and coyotes (Figure 1). For example, the reintroduction of wolves has facilitated the recovery of beavers in Yellowstone. In the 1800s, human trappers decimated beaver populations. After wolves were removed from the park, elk populations grew and competition for willow--the preferred food and construction material of beavers--became intense (Ripple & Beschta, 2003). Although beavers had been protected from trapping since the early 1920s, and beaver reintroduction efforts were underway, competitive pressure from elk suppressed recovery (Baker et al., 2005). Since wolves were restored to Yellowstone, predation, hunting, and drought have reduced elk populations. But elk also have changed their behavior (Creel et al., 2005): When elk detect wolves in a general area, they move toward conifer forests (where they have good protection from wolves) and away from open areas and streams (where they have less protection from wolves; Figure 2). Because of the combined effect of fewer elk and reduced use of willow habitat, the number of beaver colonies (five to six beavers per colony) on the northern range of Yellowstone increased from one in 1996 to nine in 2003 (D.W. Smith, 2006, personal communication). [FIGURE 1 OMITTED] [FIGURE 2 OMITTED] Winter on the Northern Range In Yellowstone National Park, daytime winter temperatures range from -40[degrees]C to -5[degrees]C and snow depths can exceed 7m at high elevations (NPS, 2006). Every autumn, the northern Yellowstone elk herd migrates from high-elevation summer habitat within the park to milder habitat in the northern range, a 1,530 [km.sup.2]-area that includes a portion of the park and some adjacent public and private land (Singer & Mack, 1999). Even so, winter isn't easy. Elk search for grasses and other herbaceous plants by digging in the snow with their hooves. When the snow is deep or covered by a hard crust, digging becomes more difficult, as does the simple act of moving through the snow (Gese et al., 1996). Furthermore, the plants under the snow are less nutritious than in the summer (Jenkins & Starkey, 2003). In severe winters, elk regularly starve to death (Wisdom & Cook, 2000). They also can suffer the same fate during mild winters if conditions during the previous summer were poor (Vucetich et al. …