Let us remember that a static character is a character that does not change or develop during the course of the story, and who stays the same. They are of course the opposite of dynamic characters, who definitely change and develop from how they are presented at the beginning of the story and the kind of character that they are shown to be at the end of the story.
Based on this understanding, we cannot say that Doodle or the narrator of this story are static characters, as they both develop and change during the course of the story. Doodle, for example, becomes much more physically active from the invalid that he started life as, and the narrator becomes more and more aware of the embarrassment and pride that drives his relationship with his brother until he has an epiphany when Doodle dies that shows him just how much he has been a victim of pride.
A static character would therefore be somebody like the father in this story, who plays a minor role, but certainly seems to remain the same. Consider how he is described when he faces the blight of 1918:
Doodle and I followed Daddy out into the cotton field, where he stood, shoulders sagging, surveying the ruin. When his chin sank down onto his chest, we were frightened, and Doodle slipped his hand into mine. Suddenly Daddy straightened his shoulders, raised a giant knuckly fist, and with a voice that seemed to rumble out of the earth itself began cursing heaven, hell, the weather, and the Republican party. Doodle and I, prodding each other and giggling, went back inot the house, knowing that everything would be alright.
The response of the children to this outburst strongly suggests that this was a normal defiant reaction of their father to adversity, and that he does not change through the course of the story, making him a static character.
how am i suppose to know