I believe regional population counts are stored for vermin (e.g. cave spiders) as well as other creatures. Like other creatures, they don't reproduce and when a region runs out of creatures, it's out forever.
I was about to ask a silly question about trapped (and even tamed) vermin being viable population sustainers, before I realised you included "don't reproduce". And then I was going to ask an equally silly question about whetehr trapping some in advance could be a precautionary measure. Only, I was thinking that you could do so before allowing other vermin to perish completely, then get rid of your cats (seeing as you couldn't stop vermin from entering/leaving any cat-holding/cat-proof areas, while they got back onto the job of spinning webs for you) and be all happy and jolly about things, at least until the next Urist McDickWittington arrives witht he next cowd of immigrants, knotted handkerchief on stick and pet cat following on behind, expectantly awaiting elevation to mayor while your precious last few spiders succumb to the prowling puss.
So... given this isn't possible, maybe the lesson is to identify the spider-biome areas straight away (underground only?) and cat-proof those areas. Put your stores in the cat-accessible areas, and do something (not sure what) to ensure that any Capture Small Animal tasks don't encage the spiders (if you're running low on them) even while you attempt to do your bit against the rats, roaches, etc...
But I don't have enough experience with efficiently handling vermin, so am far from an expert and have surely abstracted the mechanisms beyond 'reality'.