Strata of Ecological Coexistence via Grassmannians

2025/08/29 by Türkü Özlüm Çelik, Pierre A. Haas, Çelik, Türkü Özlüm +7 · 2 voices
#math.AG

paper · pdf · doi:10.48550/arxiv.2509.00165

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 ℝ×ℝ^{n× n} with the feasible-stable semialgebraic sets. We encode them on the real Grassmannian \rm Gr_ℝ(n,2n) via a parameter matrix representation, and use oriented matroid theory to develop an algorithm, combining Grassmann--Plücker relations with branching under feasibility and stability constraints. This symbolic approach determines whether a given sign pattern in the parameter space ℝ×ℝ^{n× n} admits a consistent extension to Plücker 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 HypersurfaceRegions.jl to decompose the parameter space and detect rare feasible-stable sign patterns.

Citations

• Graph Rules for Recurrent Neural Network Dynamics
• The Riemann theta function solutions for the hierarchy of Bogoyavlensky lattices
• Conditions for a matrix to have only characteristic roots with negative real parts
• Routing functions for parameter space decomposition to describe stability landscapes of ecological models
• A vector bundle approach to Nash equilibria

Discussions

• New preprint: We study Lotka-Volterra ecological dynamics through computational algebraic geometry. By encoding feasibility & stability on the real Grassmannian, we prove certain ecological networks a [bsky, 16 points, 1 comments]
• New #preprint: "Strata of Ecological Coexistence via Grassmannians" with Türkü Özlüm Çelik, Georgy Scholten, Kexin Wang, Giulio Zucal - a great, first collaboration with #mathematics @csbdresden.bsky. [bsky, 12 points, 1 comments]

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