Strata of Ecological Coexistence via Grassmannians

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

Abstract The Lotka–Volterra system is the simplest model of the ecological interactions of n species. The sign pattern of its parameter space $$\mathbb {R}^n\times \mathbb {R}^{n\times n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> 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 $$\mathbb {R}^n\times \mathbb {R}^{n\times n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> into feasible-stable semialgebraic sets. We encode them on the real Grassmannian $$\operatorname {Gr}_\mathbb {R}(n,2n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>Gr</mml:mo> <mml:mi>R</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$\mathbb {R}^n\times \mathbb {R}^{n\times n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> 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 to decompose the parameter space and detect rare feasible-stable sign patterns.