They will carry on producing new spears until well into the summer. When the bed is in full production you stop picking at Midsummer's Day; but they still produce. It's obviously dependent on temperature. Everyone's started early this year because April was so warm; now they've slowed down a lot (and there is quite a large proportion of thin spears), but as soon as the weather warms up again we should be back on course.
Yes, look out asparagus beetle: they like the tops. And we've had bird damage this year. I had to net mine at first. (And yes, it was birds: I caught them at it!)