When you're jogging toward them on a path, and they get antsy, they fly away from you, in the same direction as you, fluttering away, stopping, fluttering some more, rather than simply taking into account the direction you're going and veering off to the side. Imagine my path below:
What the pigeon needs to do is take a little hop down the page to its right, and problem solved. It's like people in movies who are being chased by cars who try to run away from the car, as if outrunning the car is a viable solution, instead of waiting for the car to get close and veering or diving off to the side. Anyone who's ever been chased by a high-speed automobile knows that! Dur. You have to exploit your strengths against the cars weaknesses. Speed? Advantage: Car. Power: Car. Manueverability in a small space, say a box that's 10 yards per side? Advantage: Runner. Then, when you've dived out of the way, the car will come to a screeching halt and the driver, pissed now, will jump out and shoot you to death. (Unless you've found some providential door in the alley you're in that's unlocked that you've ducked into that will lead to either a foot chase or a cat-and-mouse stalking sequence.) But that's your fault for not carrying a gun.