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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. 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 ( W R 1 ) 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 W R 1 realizations of polynomial dynamical systems, whenever such realizations exist. [ABSTRACT FROM AUTHOR]

Complex-balanced mass-action systems are some of the most important types of mathematical models of reaction networks, due to their widespread use in applications, as well as their remarkable stability properties. We study the set of positive parameter values (i.e., reaction rate constants) of a reaction network G that, according to mass-action kinetics, generate dynamical systems that can be realized as complex-balanced systems, possibly by using a different graph G ′ . This set of parameter values is called the disguised toric locus of G. The R -disguised toric locus of G is defined analogously, except that the parameter values are allowed to take on any real values. We prove that the disguised toric locus of G is path-connected, and the R -disguised toric locus of G is also path-connected. We also show that the closure of the disguised toric locus of a reaction network contains the union of the disguised toric loci of all its subnetworks. [ABSTRACT FROM AUTHOR]