I'm assuming the Greg Bennett was your previous guitar. What brand is the one in the photos? Given the headstock inlay, I'd almost be willing to say it's a special Seagull model for the Australian market (a reference to the kangaroo inlay), LOL.
Anyway, you can go up or down a bit in string gauge (i.e., from .012's to .011's) without having to adjust anything. If you're especially picky, go ahead and install the new strings and let the guitar adjust to them, then go to The Annex, click on "Tech Stuff," and then on "Is My Guitar Sick?" for instructions on measuring neck relief. Heavier strings will tend to give more relief, lighter ones less. I went up a gauge without needing to adjust anything.