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Abstract Changes in environmental or system parameters often drive major biological transitions, including ecosystem collapse, disease outbreaks, and tumor development. Analyzing the stability of steady states in dynamical systems provides critical insight into these transitions. This paper introduces an algebraic framework for analyzing the stability landscapes of ecological models defined by systems of first-order autonomous ordinary differential equations with polynomial or rational rate functions. Using tools from real algebraic geometry, we characterize parameter regions associated with steady-state feasibility and stability via three key boundaries: singular, stability (Routh-Hurwitz), and coordinate boundaries. With these boundaries in mind, we employ routing functions to compute the connected components of parameter space in which the number and type of stable steady states remain constant, revealing the stability landscape of these ecological models. As case studies, we revisit the classical Levins-Culver competition-colonization model and a recent model of coral-bacteria symbioses. In the latter, our method uncovers complex stability regimes, including regions supporting limit cycles, that are inaccessible via traditional techniques. These results demonstrate the potential of our approach to inform ecological theory and intervention strategies in systems with nonlinear interactions and multiple stable states. 

Abstract A standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. This manuscript provides algorithms for computing the number of connected components, the Euler characteristic, and deciding the connectivity between two points for a smooth manifold arising as the complement of a real hypersurface of a real algebraic variety. When considering the complement of the set of singular points of a real algebraic variety, this yields an approach for determining smooth connectivity in a real algebraic variety. The method is based upon gradient ascent/descent paths on the real algebraic variety inspired by a method proposed by Hong, Rohal, Safey El Din, and Schost for complements of real hypersurfaces. Several examples are included to demonstrate the approach. 

Ramification points arise from singularities along solution paths of a homotopy. This paper considers ramification points of homotopies, elucidating the total number of ramification points and providing general theory regarding the properties of the set of ramification points over the same branch point. The general approach utilized in this paper is to view homotopies as lines in the parameter spaces of families of polynomial systems on a projective manifold. With this approach, the number of singularities of systems parameterized by pencils is computed under broad conditions. General conditions are given for when the singularities of the systems parameterized by a line in a space of polynomial systems have multiplicity two. General conditions are also given for there to be at most one singularity in the solution set of any system parameterized by such a line. Several examples are included to demonstrate the theoretical results. 

Newton's method is a classical iterative approach for computing solutions to nonlinear equations. To overcome some of its drawbacks, one often considers a continuous adjoint form of Newton's method. This paper investigates the geometric structure of the trajectories produced by the continuous adjoint Newton's method for bivariate quadratics, a system of two quadratic polynomials in two variables, via eigenanalysis at its equilibrium points. The main ideas are illustrated using plots generated by a Maple program.