Improving Algebraic Tools to Study Bifurcation Sequences of Population Models

Since being introduced by Collins in the 1970s Cylindrical Algebraic Decomposition has found many applications. We focus on its use to decomposition the parameter space of a parametric system of polynomial equations, and possibly some polynomial inequality constraints, with respect to the number of real solutions that the system attains. Previous studies simplify the CAD computation by first computing the discriminant variety of the system, which usually involves Gr¨obner Basis computation. However, on some even very small applied examples this itself can become expensive. Thus we consider development of new algorithms to reduce the complexity of this approach, by adopting numerical ideas and some recent technical developments in CAD theory. In this extended abstract we outline our recent progress as evaluated on an example from population dynamics with the Allee effect.