Joanna, I've had asthma on and off since I was a wee bun. Best advice I can give from experience is keep your inhaler hardy in case you need it and just keep running. Your lung capacity will increase with training and also your metabusism so your body will be more efficent with the oxygen but not as much as some other people's within the asthma. If it's any use I always carry a water bottle with me even if it's just a short run to help clear my throat a little.
If you want to give it a try and it helps there are some breathing exercises you can do from yoga and stuff. it's hardish to explain but i'll try. Breath in deeply pushing the air into your chest trying to keep your sholder's as still as possible count your breath for 10 seconds and try to hold your breath for also 10 seconds and then breath out again slowing aiming for 10 seconds and hold your breath after for a few seconds preferably 10 but don't aim for that at first. It should be easy to do a few times and try to do it in and out for a about 10 breaths or so.
If you are struggling to do that within that sort of time try 8 or 5 secords. It's best to try this laying down at first. Makes it easier and also means that if you get light headed or anything you won't fall over.
You shouldn't by the way if that happens ever I'd said that wrong or you might have health issue. It's a basic meditation tecniec so a goggle might be in order if I'm not explaining brilliantly. I'm dyslexia as funk.
Which tri are you doing chuck?
*edited due to rubbish expaination at first. definanlty check via goggle and not just take advice from a stranger off the internet.*
Edited: 28/11/2012 at 18:14