Quick raspberry question

My 'self sown' raspberries (since I know not from whence they came), are for the 5th year doing brilliantly and I'm looking forward to a good harvest - assuming all the embryonic buds turn into fruit.

Although I cut out all the old wood and the majority of this years' canes are within the area I'd prefer they stayed in, they are inexorably 'walking' north to south and, if this continues, will encroach on areas where I don't want them.

I don't remember my Father's doing this so I'm asking, (before I start cutting down and/or grubbing out the unwanted ones) is this to be expected?

 

I planted a row of raspberry plants 3 years ago and the wood that the fruit has been on dies back at the end of the season to be replaced by the young shoots which are just starting to shoot up now. So far mine are being fairly well behaved and staying around the original plant; I have had a couple that have strayed though and popped up a couple of feet away between flags on the patio; I just remove them.

My mum had some raspberries in her garden a few years ago though and hers were popping up all over the place, some a good few feet away from the main plants....but that was a more mature clump