When the Moffat and Auto model functions are selected, DynamicPSF fits the beta parameter with the Levenberg-Marquadt algorithm. Note that this is not a search procedure; beta is fitted just as A, B, cx, cy, sx and sy (and r and theta when fitting an elliptic function).
In Auto mode, the process fits Gaussian and variable-beta Moffat functions, then selects the model with the least residual.