Abstract
                                                                        Systems of differential equations with polynomial right-hand sides are very common in applications. In particular, when restricted to the positive orthant, they appear naturally (according  to the law of mass-action kinetics) in ecology, population dynamics, as models of biochemical  interaction networks, and models of the spread of infectious diseases. On the other hand, their  mathematical analysis is very challenging in general; in particular, it is very difficult to answer  questions about the long-term dynamics of the variables (species) in the model, such as questions about persistence and extinction. Even if we restrict our attention to mass-action systems,  these questions still remain challenging. On the other hand, if a polynomial dynamical system  has a weakly reversible single linkage class (WR1  ) realization, then its long-term dynamics is  known to be remarkably robust: all the variables are persistent (i.e., no species goes extinct),  irrespective of the values of the parameters in the model. Here we describe an algorithm for  finding WR1  realizations of polynomial dynamical systems, whenever such realizations exist.