  We propose a new overarching model for self-propelled particles that flexibly
generates a full family of "descendants". The general dynamics introduced in
this paper, which we denote as "parental" active model (PAM), unifies two
special cases commonly used to describe active matter, namely active Brownian
particles (ABPs) and active Ornstein-Uhlenbeck particles (AOUPs). We thereby
document the existence of a deep and close stochastic relationship between
them, resulting in the subtle balance between fluctuations in the magnitude and
direction of the self-propulsion velocity. Besides illustrating the relation
between these two common models, the PAM can generate additional offspring,
interpolating between ABP and AOUP dynamics, that could provide more suitable
models for a large class of living and inanimate active matter systems,
possessing characteristic distributions of their self-propulsion velocity. Our
general model is evaluated in the presence of a harmonic external confinement.
For this reference example, we present a two-state phase diagram which sheds
light on the transition in the shape of the positional density distribution,
from a unimodal Gaussian for AOUPs to a Mexican-hat-like profile for ABPs.
