<p>The Lotka–Volterra system is the simplest model of the ecological interactions of <i>n</i> species. The sign pattern of its parameter space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^n\times \mathbb {R}^{n\times n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> defines the network structure of the competitive, mutualistic, and predator–prey interactions between these species. Here, we study the feasible and stable equilibria of the Lotka–Volterra system from the perspective of computational algebraic geometry. The feasibility and stability conditions stratify <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^n\times \mathbb {R}^{n\times n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> into feasible-stable semialgebraic sets. We encode them on the real Grassmannian <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\operatorname {Gr}_\mathbb {R}(n,2n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>Gr</mo> <mi mathvariant="double-struck">R</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}^n\times \mathbb {R}^{n\times n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> 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 <Emphasis FontCategory="NonProportional">HypersurfaceRegions.jl</Emphasis> to decompose the parameter space and detect rare feasible-stable sign patterns.</p>

Strata of Ecological Coexistence via Grassmannians

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2026年7月25日

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Türkü Özlüm Çelik,
Pierre A. Haas,
Georgy Scholten,
Kexin Wang,
Giulio Zucal

摘要

The Lotka–Volterra system is the simplest model of the ecological interactions of n species. The sign pattern of its parameter space $\mathbb {R}^n\times \mathbb {R}^{n\times n}$ $R^{n} \times R^{n \times n}$ defines the network structure of the competitive, mutualistic, and predator–prey interactions between these species. Here, we study the feasible and stable equilibria of the Lotka–Volterra system from the perspective of computational algebraic geometry. The feasibility and stability conditions stratify $\mathbb {R}^n\times \mathbb {R}^{n\times n}$ $R^{n} \times R^{n \times n}$ into feasible-stable semialgebraic sets. We encode them on the real Grassmannian $\operatorname {Gr}_\mathbb {R}(n,2n)$ ${Gr}_{R} (n, 2 n)$ 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 $\mathbb {R}^n\times \mathbb {R}^{n\times n}$ $R^{n} \times R^{n \times n}$ 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.