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

## TL;DR

This paper introduces a new algebraic method to analyze how changes in parameters affect the stability of ecological models, helping to understand transitions like ecosystem collapse.

## Contribution

The novel algebraic framework uses routing functions and real algebraic geometry to decompose parameter spaces and reveal stability landscapes in ecological models.

## Key findings

- The method identifies parameter regions with constant numbers and types of stable steady states.
- Case studies reveal complex stability regimes, including regions with limit cycles, in ecological models.
- The approach uncovers stability landscapes inaccessible via traditional techniques in nonlinear ecological systems.

## 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.