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A Locally Stable Equilibria Criterion for the Generalized Lotka-Volterra

Michael Richard Livesay

TL;DR

This work addresses local stability of fixed points in the generalized Lotka--Volterra system under a non-degenerate regime. It develops a projected-subsystem framework and a Jacobian-based stability criterion, tying fixed-point stability to reduced-subsystem conditions and a saturation criterion, while revealing a deep link between transversal eigenvalues and Schur-complement structures of the full Jacobian. A central insight is that transversal dynamics are governed by the Schur complement, enabling a result that adding a single species to a stable fixed-point subset cannot preserve stability. The analysis also connects fixed points to linear complementarity problems and leverages symmetry and inertia concepts to derive practical stability tests across nested surviving-sets, clarifying how boundary dynamics influence the full-system behavior in LV ecosystems.

Abstract

The main result applies to non-degenerate cases of the generalized Lotka-Volterra model. A criterion is given that relates the stability of two fixed points with the associated Schur complement of there respective community matrices.

A Locally Stable Equilibria Criterion for the Generalized Lotka-Volterra

TL;DR

This work addresses local stability of fixed points in the generalized Lotka--Volterra system under a non-degenerate regime. It develops a projected-subsystem framework and a Jacobian-based stability criterion, tying fixed-point stability to reduced-subsystem conditions and a saturation criterion, while revealing a deep link between transversal eigenvalues and Schur-complement structures of the full Jacobian. A central insight is that transversal dynamics are governed by the Schur complement, enabling a result that adding a single species to a stable fixed-point subset cannot preserve stability. The analysis also connects fixed points to linear complementarity problems and leverages symmetry and inertia concepts to derive practical stability tests across nested surviving-sets, clarifying how boundary dynamics influence the full-system behavior in LV ecosystems.

Abstract

The main result applies to non-degenerate cases of the generalized Lotka-Volterra model. A criterion is given that relates the stability of two fixed points with the associated Schur complement of there respective community matrices.

Paper Structure

This paper contains 7 sections, 18 theorems, 53 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Projected Subsystems and Non-Degeneracy
  3. Jacobian Criterion for Locally Stable Equilibria
  4. Schur Complement and Local Stability
  5. Conclusion
  6. Appendix
  7. Appendix

Key Result

Theorem 2.6

Suppose $A$ is an $N \times N$ matrix and ${\mathbf {{r}}} \in \mathbb{R}^N$. Then $(A, {\mathbf {{r}}})$ is non-degenerate if and only if there are exactly $2^N$ distinct solutions to ${\mathbf {{x}}} \odot ({\mathbf {{r}}} - A {\mathbf {{x}}}) = \mathbf{0}$.

Figures (6)

  • Figure 1: Plotted are two instances with initial conditions near two different fixed points, showing each species' population converges to $1$. On the left $|S|=2$ and on the right $|S|=1$, in Ex. \ref{['xpl:not_stable_subs']}.
  • Figure 2: The fourth species is left out of the environment, and the other $3$ species prosper, in Ex. \ref{['xpl:bdry_fp']}.
  • Figure 3: To the left, the dynamics appears to converge to ${\mathbf {{p}}}_{\{1,2,3\}}$ at first, but over time the population of $x_4$ eventually prevails. To the right, the initial populations of each species is larger. The value of $x_4$ starts off diminishing even at a greater rate than the other species, but unavoidably, the population $x_4$ persists in defiance of the other three, in Ex. \ref{['xpl:bdry_fp']}.
  • Figure 4: LV$(A_1,{\mathbf {{r}}}_1)$ has stable points at $(2,0,0)$ and $(0.520833...,1.979166...,3.9166...)$, in Ex. \ref{['xpl:counter']}.
  • Figure 5: LV$(A_2,{\mathbf {{r}}}_2)$ has a stable point at $(1.805,0.263,0)$, in Ex. \ref{['xpl:counter']}.
  • ...and 1 more figures

Theorems & Definitions (48)

  • Remark 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Example 3.1
  • Example 3.2
  • Theorem 3.3
  • proof
  • ...and 38 more