DTM.4:5.1 - Rare growth and persistence give different answers

Consider a constructed reversible uptake model for the share f using a practice. The modeled adoption flow is s f²(1−f): encounters must provide the reinforcement represented by f². The modeled abandonment flow is s θ f(1−f): access to alternative users contributes to departure. Thus:

f' = s f(1−f)(f−θ),    0≤f≤1,    s>0,    0<θ<1.

The two flows are nonnegative on the stated region. Their forms are hypotheses for this example, not universal properties of social learning. With θ=0.4 and s=1 per chosen time unit, the fixed points are 0, 0.4 and 1.

For 0<f<0.4 the derivative is negative; for 0.4<f<1 it is positive. The endpoint regimes attract interior states on their respective sides, while 0.4 separates them. At f=0.2 the rate is −0.032; at f=0.6 it is +0.048.

The rare-variant test reports decline near zero. It does not establish that an existing majority will disappear. The model instead gives persistence from the higher initial range. The first useful result is this distinction, before any choice of intervention.