Under what conditions will an organism remain viable as numerous forces threaten its self-construction, and what does this abstract space of possibilities look like? A growing body of work has begun to confront this question by imposing viability limits on dynamical system models to separate sets of viable and nonviable states. Since the viability limits are not implicit in the equations that govern the dynamics, there is no guaranteed equivalence between the phase portrait and the basins of initial conditions that will remain viable. This means that the topology of a dynamical system model with imposed viability limits demands richer analyses, which we refer to as characterizing viability space. In this paper, we set the groundwork for such techniques using a protocell model governed by nonlinear ordinary differential equations, including the development of novel criteria for bifurcations so that entire classes of systems can be studied.