Just a first thought but I would say persevere implies agency but persist doesn't. Both you and the rain may persist but only you can persevere. That is (probably) how I would use them. In the case of the mosquito I'd say, most likely, no agency is involved unless it's a Disney mosquito in which case, like cartoon ants at a picnic there definitely is.
"No man but a blockhead ever wrote except for money." Samuel Johnson.
Shu posted this on Extreme Pedantry, a FB group and they told him it wasn't a pedantic question. So glad I don't post on that group. From what I hear, most are Brits and they are a bit anti-American.
I may pop over to the group formed by my erstwhile English teacher devoted to pedantry - which I joined so that I could tell everyone there how wrong they are - and ask the question.
"No man but a blockhead ever wrote except for money." Samuel Johnson.