Title: Strata of Ecological Coexistence via Grassmannians Show Chinese title

Title: 通过格拉斯曼的生态共存层

Abstract: 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. Show Chinese abstract

Abstract: 我们从计算代数几何的角度研究Lotka--Volterra系统，重点关注既可行又稳定的平衡点。 这些条件将参数空间分层为$\mathbb{R}\times\mathbb{R}^{n\times n}$中的可行稳定半代数集。 我们将它们通过参数矩阵表示编码到实格拉斯曼流形${\rm Gr}_{\mathbb{R}}(n,2n)$上，并利用定向拟阵理论开发一种算法，结合格拉斯曼--普吕克关系与可行性和稳定性约束下的分支。 这种符号方法可以确定参数空间中给定的符号模式$\mathbb{R}\times\mathbb{R}^{n\times n}$是否可以一致扩展为普吕克坐标。 作为应用，我们证明了某些相互作用网络是不可能的，表明相应的模式无法满足可行性和稳定性条件的扩展，通过有效的实现来展示这一点。 我们使用数值非线性代数与\texttt{超曲面区域.jl}相结合，对参数空间进行分解并检测罕见的可行稳定符号模式。