DTM.3:5.2 - Growth can reduce the support available to each learner

Suppose local readiness follows the conditional model:

x' = α(f)(1−x) − δx
α(f) = α₀/(1+kf).

Here f is the fraction adopting, α is the effective support rate per learner, δ a loss rate, and k specifies how adoption loads the support arrangement. The signs and functional form express the assumed sharing mechanism; they would have to change if adoption added support faster than demand.

For α₀=1, δ=0.25 and k=3 in the chosen time unit, the stationary readiness is α/(α+δ). It is about 0.714 at f=0.2 and 0.541 at f=0.8. If q=x² remains the receiving-event law, the corresponding contributions are about 0.510 and 0.292.

The loop is now visible: adoption changes support per participant, readiness changes the receiving-event rate, and that rate changes adoption. These stationary substitutions are useful only when readiness adjusts fast enough. Immediately after a large influx of beginners, replacing their states with these equilibria can be wrong.

A trial to increase adoption might therefore need additional teaching support. The equation does not prove that such support exists or how to teach the missing operation; those are provider questions.