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Routing Functions for Parameter Space Decomposition to Describe Stability Landscapes of Ecological Models.

Joseph Cummings1, Kyle J-M Dahlin2, Elizabeth Gross3

  • 1Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, United States.

Bulletin of Mathematical Biology
|November 14, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a new algebraic framework to analyze the stability of ecological models. It reveals complex stability landscapes and provides insights into biological transitions, aiding ecological theory and intervention strategies.

Keywords:
Real numerical algebraic geometryalgebraic biologycompetition-colonizationcoral-bacteria symbiosisstability analysis

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Area of Science:

  • Ecology
  • Mathematical Biology
  • Dynamical Systems Theory

Background:

  • Biological transitions, such as ecosystem collapse and disease outbreaks, are often triggered by changes in environmental parameters.
  • Analyzing the stability of steady states in dynamical systems is crucial for understanding these critical transitions.
  • Existing methods may not fully capture the complexity of stability in ecological models with nonlinear interactions.

Purpose of the Study:

  • To introduce a novel algebraic framework for analyzing the stability landscapes of ecological models.
  • To characterize parameter regions related to steady-state feasibility and stability.
  • To reveal the connected components of parameter space with constant numbers and types of stable steady states.

Main Methods:

  • Utilizing tools from real algebraic geometry to analyze systems of first-order autonomous ordinary differential equations.
  • Defining stability landscapes through singular, stability (Routh-Hurwitz), and coordinate boundaries.
  • Employing routing functions to compute connected components of parameter space.

Main Results:

  • Characterization of parameter regions for steady-state feasibility and stability.
  • Identification of the stability landscape by computing connected components.
  • Uncovering complex stability regimes, including limit cycles, in a coral-bacteria symbiosis model.

Conclusions:

  • The developed algebraic framework offers a powerful approach to understanding ecological model stability.
  • The method reveals complex stability regimes previously inaccessible to traditional techniques.
  • This approach has the potential to inform ecological theory and intervention strategies for systems with nonlinear interactions and multiple stable states.