# Stability and steady state of complex cooperative systems: a diakoptic approach

## TL;DR

This paper introduces a graph-theoretical diakoptic approach to analyze the stability of complex cooperative systems by decomposing their dependence graphs into strongly connected components and assessing eigenvalues.

## Contribution

It provides a novel divide-and-conquer method for stability analysis of large cooperative systems using graph decomposition and eigenvalue criteria.

## Key findings

- A linear cooperative system is Lyapunov stable if all SCCs have non-positive dominant eigenvalues.
- Stability requires no path connecting SCCs with zero eigenvalues.
- The approach simplifies stability analysis of complex ecological and population systems.

## Abstract

Cooperative dynamics are common in ecology and population dynamics. However, their commonly high degree of complexity with a large number of coupled degrees of freedom renders them difficult to analyse. Here we present a graph-theoretical criterion, via a diakoptic approach (`divide-and-conquer') to determine a cooperative system's stability by decomposing the system's dependence graph into its strongly connected components (SCCs). In particular, we show that a linear cooperative system is Lyapunov stable if the SCCs of the associated dependence graph all have non-positive dominant eigenvalues, and if no SCCs which have dominant eigenvalue zero are connected by a path.