Why would the Perseids meteor rate fall off after maximum faster than the increase before maximum?
A very interesting question. Neither Bruce McClure at EarthSky nor I knew the answer, despite the fact that the observed gradual rise in numbers for the Perseids - and their quick drop-off - is something we've both observed for decades. So we asked a true expert on meteor showers, Robert Lunsford of the American Meteor Society. He also said: "Good question!" And, he said: "It seems that the particle density of the meteor stream [what you're calling here a 'fuzzy tube'] is greater prior to maximum than after. Perhaps the core of the stream, which produces maximum activity, is not quite centered in the stream. This is true for some other major showers such as the Geminids. The core of the Geminids must be really off-center as the display nearly disappears 48 hours after maximum. Gravitational effects from the sun may also play a role in this shift."

So here, as so often in astronomy, the answer suggests our knowledge of nature is - at present - inexact. Are any of you old enough to remember when the Voyager spacecraft first encountered Jupiter? Nowadays, we're used to seeing the very detailed images of Jupiter's cloud bands, but when we first saw that detail - via the Voyagers in the late 1970s - we were blown away! The situation with meteor streams in space is likely exactly the same. We picture them as symmetrical only because we don't have instruments refined enough to reveal their structure. All we have are the long-observed rates of meteors in annual showers - such as the Perseids and the Geminids - suggesting that meteor streams do indeed have a structure.

Hope this helps!