In several documented cases, dogs have attracted bears back to their owners and the result has been a serious injury. Only specially trained dogs can protect you from bears. And besides, most national parks prohibit taking pets on backcountry trails. Many avid runners like to get off paved roads and running tracks and onto backcountry trails. But running on trails in bear country can be seriously hazardous to your health.
Most runners avoid running during the heat of the day.
This means they run early or late in the day when bears are most active. Fervent trail runners know that you tend to get closer to wildlife running than you do walking. Some people think that's because you cover distance faster than expected by wildlife. Other people think it's because you tend to be quieter when running. Whatever the reason, running on trails obviously increases your chance of surprising a bear.