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the evolution gives something like

P(t) = K/(1+ exp(-c*(t-t0)))

so that there is a biunivoque relationship between populations at t = 0

P(t=0) = K/(1+ exp(c*t0))

and populations at t, whatever t

therefore,

Prob(P(t) > K*q) = Prob(P(t=0) > K*q0)

with

q = 1/(1+ exp(-c*(t-t0)))

and

q0 = 1/(1+ exp(c*t0))

solve the above by eliminating t0 and get the explicit form of q0 as a function of q, say q0(q)

then, Prob(P(t) > K*q) = Prob(P(t=0) > K*q0(q))

if you know the distribution at t = 0, the above gives you the distribution at t

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