Strata of Ecological Coexistence via Grassmannians

We study the Lotka--Volterra system from the perspective of computational algebraic geometry, focusing on equilibria that are both feasible and stable. These conditions stratifies the parameter space in $\mathbb{R}\times\mathbb{R}^{n\times n}$ with the feasible-stable semialgebraic sets. We encode them on the real Grassmannian ${\rm Gr}_{\mathbb{R}}(n,2n)$ via a parameter matrix representation, and use oriented matroid theory to develop an algorithm, combining Grassmann--Pl{\"u}cker relations with branching under feasibility and stability constraints. This symbolic approach determines whether a given sign pattern in the parameter space $\mathbb{R}\times\mathbb{R}^{n\times n}$ admits a consistent extension to Pl{\"u}cker coordinates. As an application, we establish the impossibility of certain interaction networks, showing that the corresponding patterns admit no such extension satisfying feasibility and stability conditions, through an effective implementation. We complement these results using numerical nonlinear algebra with \texttt{HypersurfaceRegions.jl} to decompose the parameter space and detect rare feasible-stable sign patterns.