It is a myth that killing the pike you catch will solve any predatory problems.
Pike will regulate themselves to 10% of the biomass of the water. If you kill pike that you catch, you are removing the very predator than can regulate the myriad small pike and so you get an explosion of very hungry fish in the 1lb - 4lb bracket, eating anything and everything because they are eating for growth as well as maintenance. If you leave in the bigger pike 5lb up, they will be eating more of the smaller pike than you can catch. A 10lb pike is a female and will only eat once every couple of days. A 20lb pike doesn't even eat once a week, in still water, they figures are a little bit more often in running water because of energy expended, but they will never waste energy, even in rivers they will be sitting out of the current, in a slack, or behind some obstruction that affords them shelter from the current.
Fisheries that have left the pike alone have found the predation on the trout has LESSENED and catch rates have increased.