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

Changes in environmental or system parameters often drive major biological transitions, including ecosystem collapse, disease outbreaks, and tumor development. Analyzing the stability of steady states in dynamical systems provides critical insight into these transitions. This paper introduces an alg...

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Veröffentlicht in:Bulletin of mathematical biology Jg. 87; H. 12; S. 177
Hauptverfasser: Cummings, Joseph, Dahlin, Kyle J.-M., Gross, Elizabeth, Hauenstein, Jonathan D.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: United States Springer Nature B.V 01.12.2025
Schlagworte:
Algebra
Animals
Anthozoa - microbiology
Boundaries
Computer Simulation
Decomposition
Differential equations
Dynamical systems
Ecosystem
Eigenvalues
Epidemics
Geometry
Mathematical analysis
Mathematical Concepts
Models, Biological
Nonlinear Dynamics
Nonlinear systems
Ordinary differential equations
Parameters
Polynomials
Population Dynamics - statistics & numerical data
Stability
Steady state
Symbiosis
algebraic biology
competition-colonization
coral-bacteria symbiosis
stability analysis
Real numerical algebraic geometry
ISSN:0092-8240, 1522-9602, 1522-9602
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Zusammenfassung:Changes in environmental or system parameters often drive major biological transitions, including ecosystem collapse, disease outbreaks, and tumor development. Analyzing the stability of steady states in dynamical systems provides critical insight into these transitions. This paper introduces an algebraic framework for analyzing the stability landscapes of ecological models defined by systems of first-order autonomous ordinary differential equations with polynomial or rational rate functions. Using tools from real algebraic geometry, we characterize parameter regions associated with steady-state feasibility and stability via three key boundaries: singular, stability (Routh-Hurwitz), and coordinate boundaries. With these boundaries in mind, we employ routing functions to compute the connected components of parameter space in which the number and type of stable steady states remain constant, revealing the stability landscape of these ecological models. As case studies, we revisit the classical Levins-Culver competition-colonization model and a recent model of coral-bacteria symbioses. In the latter, our method uncovers complex stability regimes, including regions supporting limit cycles, that are inaccessible via traditional techniques. These results demonstrate the potential of our approach to inform ecological theory and intervention strategies in systems with nonlinear interactions and multiple stable states.
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ISSN:0092-8240
1522-9602
1522-9602
DOI:10.1007/s11538-025-01554-7