In this paper, we present a formal model, named PobSAM (Policy-based Self-Adaptive Model), for developing and modeling self-adaptive evolving systems. In this model, policies are used as a mechanism to direct and adapt the behavior of self-adaptive systems. A PobSAM model is a collection of autonomous managers and managed actors. The managed actors are dedicated to the functional behavior while the autonomous managers govern the behavior of managed actors by enforcing suitable policies. A manager has a set of configurations including two types of policies: governing policies and adaptation policies. To adapt the system behavior in response to the changes, the managers switch among different configurations. We employ the combination of an algebraic formalism and an actor-based model to specify this model formally. Managed actors are expressed by an actor model. Managers are modeled as meta-actors whose configurations are described using a multi-sorted algebra called CA. We provide an operational semantics for PobSAM using labeled transition systems. Furthermore, we provide behavioral equivalence of different sorts of CA in terms of splitting bisimulation and prioritized splitting bisimulation. Equivalent managers send the same set of messages to the actors. Using our behavioral equivalence theory, we can prove that the overall behavior of the system is preserved by substituting a manager by an equivalent one.