The simple model of an "intelligence explosion" is the obscure equation
dx 2
-- = x
dt
which has the solution
1
x = -----
C-t
and is interesting in relation to the classic exponential growth equation
dx
-- = x
dt
because the rate of growth is proportional to x and represents the idea of an "intelligence explosion" AND a model of why small western towns became ghost towns, it is hard to start a new social network, etc. (growth is fast as x->C, but for xLike the exponential growth equation it is unphysical as well as unecological because it doesn't describe the limits of the Petri dish, and if you start adding realistic terms to slow the growth it qualitatively isn't that different from the logistic growth equation
dx
-- = (1-x) x
dt
thus it remains obscure. Hyperbolic growth hits the limits (electricity? intractable problems?) the same way exponential growth does.