He doesn't attack him after the first time. Because the first time Spirit Bear attacked him Cole was a threat, later in the book, Cole understands that for Spirit Bear to trust him he can't threaten Spirit Bear.
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The Spirit Bear trusts Cole after Cole shows compassion by saving a baby seagull from drowning. This act of kindness demonstrates to the bear that Cole has changed and is capable of caring for others. Additionally, Cole's efforts to heal and repent for his actions further earn the bear's trust.
cole stays longer
Cole kills a small black bear in the book "Touching Spirit Bear".
What changed Cole was the spirit bear after it mauled him.
Cole saw a massive white bear (the spirit bear) facing him.
Cole is sent to Alaska