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 x<<C it is glacial) It's an obscure equation because it never gets a good discussion in the literature (that I've seen, and I've looked) outside of an aside in one of Howard Odum's tomes on emergy.

Like 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.