In this work we present a method for constructing the attainable region (AR) for reactor networks that exhibit dynamic behavior. In the past, our group has used the shrink-wrap algorithm to construct the AR for non-isothermal reactor networks, constant density reactor networks, variable density reactor networks, reactors with non-ideal compressibility factors, reactor networks with dispersion, etc. However, all of these models were considered at a steady state. In this work we will discuss how we can use limit-cycles to retain the pseudo-steady-state nature of a dynamically operated reactor network. Since we are interested in the infinite time behavior of the system, we do period averaging of the reactor outlet, which allows the reactor network equations to be written in a pseudo-steady-state form. We discuss a biological predator-prey reaction kinetic system example to show the difference between reactors that use limit cycles and those that do not.