I second that the rats are after the chicken feed.
If getting a cat isn't a good option for you, here are some ideas:
Regular old Victor rat traps $2-$3 work alright with cheese or peanut butter, but you'll never catch all the rats (on a good night I've caught two in 1 trap at the same time - once I only found a tail...). Over time the trap springs also get less effective and can rust in the elements. Also if you are casual about checking traps, rebaiting and disposing of rat carcasses the remaining rats get wise and avoid the traps.
Rat poison also can work - provided you can keep it away from chickens/kids. The best brand is Just One Bite II Bars - they are a big yellowish bar that looks similar to a bird suet cake - you can see seeds etc in it. I get it at the local feed store. Thread wire through the hole and tie the wire to something - that way the rats can't carry it off and you can see if they've been eating it. Far more attractive than the green compressed rat baits at the grocery store. The poison ingredient is the same as some grocery brands, but the Bite II Bars are actually made of stuff rats like to eat - they're a little pricier in the NW, but well worth it.
Also know that some years there are just more rats - it depends on the spring weather.