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Routing functions for parameter space decomposition to describe stability landscapes of ecological models

Joseph Cummings, Kyle J. -M. Dahlin, Elizabeth Gross, Jonathan D. Hauenstein

TL;DR

The paper develops an algebraic-geometric framework to map stability landscapes for ecological models described by $\dot{x}=f(x;a)$ with polynomial or rational rate functions, using routing functions to partition parameter space into connected components where the number and type of stable steady states are invariant. Boundaries are classified into singular, stability (Routh-Hurwitz), and coordinate types, determined via equilibrium ideals and elimination, enabling the computation of parameter-space regions corresponding to multistationarity and multistability. The authors illustrate the method on the Levins-Culver two-species competition–colonization model and a tripartite coral–bacteria symbiosis model, revealing regions with coexistence, bistability, and even regions that must exhibit limit cycles. The routing-function approach, supported by computational tools, offers a scalable way to analyze nonlinear ecological systems and informs theoretical and intervention strategies by revealing how stability landscapes shift with parameters like colonization or interaction strengths.

Abstract

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.

Routing functions for parameter space decomposition to describe stability landscapes of ecological models

TL;DR

The paper develops an algebraic-geometric framework to map stability landscapes for ecological models described by with polynomial or rational rate functions, using routing functions to partition parameter space into connected components where the number and type of stable steady states are invariant. Boundaries are classified into singular, stability (Routh-Hurwitz), and coordinate types, determined via equilibrium ideals and elimination, enabling the computation of parameter-space regions corresponding to multistationarity and multistability. The authors illustrate the method on the Levins-Culver two-species competition–colonization model and a tripartite coral–bacteria symbiosis model, revealing regions with coexistence, bistability, and even regions that must exhibit limit cycles. The routing-function approach, supported by computational tools, offers a scalable way to analyze nonlinear ecological systems and informs theoretical and intervention strategies by revealing how stability landscapes shift with parameters like colonization or interaction strengths.

Abstract

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.
Paper Structure (14 sections, 1 theorem, 28 equations, 20 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 1 theorem, 28 equations, 20 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Steady-state analysis
  3. Boundaries
  4. Singular boundary
  5. Stability (Routh-Hurwitz) boundary
  6. Coordinate boundary
  7. Routing function and connected components
  8. Levins and Culver model of competition-colonization trade-off
  9. Tripartite symbiosis
  10. Simulations
  11. Intrinsically viable coral population (b=2)
  12. Intrinsically non-viable coral population (b=0.5)
  13. Summary of results
  14. Conclusion

Key Result

Theorem 2.1

The connected components of the graph $G$ computed by Algorithm alg:Connectivity are in one-to-one correspondence with the connected components of $\mathbb{R}^k_{>0}\setminus\mathcal{B}$.

Figures (20)

  • Figure 1: Computing the 6 regions in $\mathbb{R}^2_{>0}\setminus\mathcal{B}$ where the boundaries are the two ellipses (thick blue) and coordinate axes. Routing points (index 0 in blue, index 1 in red) and gradient paths (thin blue) are also plotted demonstrating $6$ connected components.
  • Figure 2: The coral-bacteria symbioses compartmental model represented by system \ref{['eq:xyz-ODEs']}.
  • Figure : (a) $\beta_z = 2.5$
  • Figure : (a) $\beta_z = 2.5$
  • Figure : (a) $\beta_z = 2.5$
  • ...and 15 more figures

Theorems & Definitions (2)

  • Theorem 2.1
  • Example 1